Means in the Trapezoid Following is a list of several means of a pair of positive numbers, a and b: 1. arithmetic mean: A(a,b) = (a + b)/2 2. geometric mean: G(a,b) = sqrt(ab) 3. harmonic mean: H(a,b) = 2ab/(a + b) 4. heronian mean: N(a,b) = (a + sqrt(ab) + b)/3 5. contraharmonic mean: C(a,b) = (a^2 + b^2)/(a + b) 6. root-mean-square: R(a,b) = sqrt(a^2 + b^2)/2) 7. centroidal mean: T(a,b) = 2(a^2 + ab + b^2)/(3(a + b)) 8. square-mean-root: S(a,b) = ((sqrt(a) + sqrt(b))/2)^2 9. logarithmic mean: L(a,b) = (a - b)/(log(a) - log(b)) Power means: M_p = ((a^p + b^p)/2)^(1/p) for p != 0 and M_0 = sqrt(ab). Gini means: S_p = ((a^(p-1) + b^(p-1))/(a + b))^(1/(p-2)) for p != 2 and S_2 = (a^a b^b)^(1/(a+b)). p-Logarithmic means: L_p = ((a^(p+1) - b^(p+1))/((p+1)(a-b)))^(1/p) for p != 0 and p != -1 L_0 = exp(-1) (a^a/b^b)^(1/(a-b)) the identric mean, L_-1= (a - b)/(log(a) - log(b)). Symmetric means: Q_p = (a^s b^t + a^t b^s)/2 with s = (1+sqrt(p))/2 and t = (1-sqrt(p))/2. Heron means: H_p = ((a^p + (ab)^(p/2) + b^p)/3)^(1/p) for p != 0 and H_0 = sqrt(ab). Contraharmonic means: N_p = (a^p + b^p)/(a^(p-1) + b^(p-1)) We notice A(a,b) = M_1(a,b) = S_1(a,b) = L_1(a,b) = Q_1(a,b) = N_1(a,b) G(a,b) = M_0(a,b) = S_0(a,b) = L_-2(a,b) = Q_0(a,b) = N_1/2(a,b) H(a,b) = M_-1(a,b) = N_0(a,b) C(a,b) = S_3(a,b) = N_2(a,b) R(a,b) = M_2(a,b) S(a,b) = M_1/2(a,b) = L_-1/2(a,b) In the following we assume a >= b > 0. :: b <= A(a,b) <= a :: G(a,b) <= A(a,b) with equality iff a=b sqrt(ab) <= (a + b)/2 <=> 4ab <= (a + b)^2 <=> 0 <= (a - b)^2 :: H(a,b) <= G(a,b) with equality iff a=b or ab=0 2ab/(a + b) <= sqrt(ab) // divide by sqrt(ab) if ab is not zero. <=> G(a,b) / A(a,b) <= 1 :: H(a,b) <= C(a,b) with equality iff a=b 2ab/(a + b) <= (a^2 + b^2)/(a + b) // multiply by (a + b) <=> 2ab <= a^2 + b^2 <=> 0 <= (a - b)^2 :: A(a,b) = A(H(a,b), C(a,b)) :: H(a,b) <= A(a,b) <= C(a,b) <=> H(a,b) <= A(H(a,b), C(a,b)) <= C(a,b) <=> A(H(a,b), H(a,b)) <= A(H(a,b), C(a,b)) <= A(C(a,b), C(a,b)) <=> H(a,b) <= C(a,b) :: N(a,b) = (G(a,b) + 2A(a,b))/3 :: S(a,b) = A(G(a,b), A(a,b)) :: G(a,b) <= S(a,b) <= N(a,b) <= A(a,b) Applying means to means is usefull in showing T(a,b) <= R(a,b). :: T(a,b) = (2A(a,b) + C(a,b))/3 = (H(a,b) + 2C(a,b))/3 :: T(a,b) <= H(A(a,b), C(a,b)) Let A = A(a,b) and C = C(a,b) then (2A + C)/3 <= 2AC/(A + C) <=> (2A - C)(A - C) <= 0 <=> A <= C <= 2A <= a/2 <= A <= C <= a <= 2A :: H(A(a,b), C(a,b)) <= G(A(a,b), C(a,b)) :: R(a,b) = G(A(a,b), C(a,b)) :: R(a,b) = sqrt(((a+b)/2)^2 + ((a-b)/2)^2) (a^2 + b^2)/2 = ((a+b)/2)^2 + ((a-b)/2)^2 Now we place a mean of means between G(a,b) <= S(a,b). :: A(H(a,b), A(a,b)) <= S(a,b) Let A = A(a,b), G = G(a,b) and H = H(a,b) then A(H,A) <= A(G,A) <=> H <= G :: G(a,b) = G(H(a,b), A(a,b)) Logarithmic mean inequalities: ------------------------------ :: x <= 1/2 log((1+x)/(1-x)) <= x/(1-x^2) for 0<=x<1 The Taylor series of 1/2 log((1+x)/(1-x)) is x + x^3/3 + x^5/5 + x^7/7 + O(x^9). As the coefficients are not negative we get the lower bound. As the coefficients are bounded by 1 we get the upper bound. :: L(a,b) <= A(a,b) log((1+x)/(1-x)) >= 2x with 0<=x<1 <=> log(z) >= 2(z-1)/(z+1) with z = (1+x)/(1-x) <=> log(a/b) >= 2(a-b)/(a+b) with z = a/b <=> (a+b)/2 >= (a-b)/(log(a)-log(b)) :: H(a,b) <= L(a,b) log((1+x)/(1-x)) <= 2x/(1-x^2) with 0<=x<1 <=> log(z) <= (z - 1/z)/2 with z = (1+x)/(1-x) <=> log(a/b) <= (a/b - b/a)/2 with z = a/b <=> 2ab/(a+b) <= (a-b)/(log(a)-log(b)) A second proof of these inequalities use the area of the hyperbola. See [Arthur Engel (1988), Exc 7.63] [Roger B. Nelson (October 1995), Proof Without Words]. Look at the area below function 1/x in the interval {b,a}. As the 1/x is convex the trapezoid rule is an overestimate and the midpoint rule is an underestimate. (a-b) / A(a,b) <= Integral_{b,a} 1/x dx <= (a-b) A(1/a, 1/b). Therefore we have again the inequalities (a-b) / A(a,b) <= log(a) - log(b) <= (a-b) / H(a,b). These inequalities can be improved by a simple scaling. :: L(a,b) <= S(a,b) <=> (a+b)/2 >= (a-b)/(log(a)-log(b)) <=> ((a+b)/2)^2 >= (a^2-b^2)/(2log(a)-2log(b)) <=> ((a+b)/2)^2 >= (a^2-b^2)/(log(a^2)-log(b^2)) <=> ((sqrt(a) + sqrt(b))/2)^2 <= (a-b)/(log(a)-log(b)) :: G(a,b) <= L(a,b) H(a,b) <= L(a,b) <=> 2ab/(a+b) <= (a-b)/(log(a)-log(b)) <=> ab <= (a^2-b^2)/(2log(a)-2log(b)) <=> ab <= (a^2-b^2)/(log(a^2)-log(b^2)) <=> sqrt(ab) <= (a-b)/(log(a)-log(b)) There is a third way to get this result. [Frank Burk (1985)] :: tanh(x) <= x <= sinh(x) for x>=0 with equality iff x=0. We are using the criterium: if f(0) >= g(0) and f'(x) >= g'(x) for x>=0 then f(x) >= g(x) for x>=0. Let f(x) = sinh(x) then f'(x) = cosh(x). Let g(x) = x then g'(x) = 1. Now cosh(x) = A(exp(x),exp(-x)) >= G(exp(x),exp(-x)) = 1. Let f(x) = x then f'(x) = 1. Let g(x) = tanh(x) then g'(x) = 1/cosh^2(x). Therefore the same argument shows 1 >= 1/cosh^2(x). :: L(a,b) <= C(a,b) x >= tanh(x) for x>0 <=> 1/x <= (exp(x)+exp(-x))/(exp(x)-exp(-x)) let x = log(a/b) <=> 1/log(a/b) <= (a^2 + b^2)/(a^2 - b^2) with a>b>0 <=> (a-b)/(log(a)-log(b)) <= (a^2 + b^2)/(a + b) :: H(a,b) <= L(a,b) sinh(x) >= x for x>0 <=> 2/(exp(x)-exp(-x)) <= 1/x let x = log(a/b) <=> 2ab/(a^2 - b^2) <= 1/log(a/b) with a>b>0 <=> 2ab/(a+b) <= (a-b)/(log(a)-log(b)) Using the substitution x = log(a/b) we relate the equations tanh(x) <= 2tanh(x/2) <= 4tanh(x/4) <= x <= 2sinh(x/2) <= sinh(x) with the mean inequalities H(a,b) <= G(a,b) <= L(a,b) <= S(a,b) <= A(a,b) <= C(a,b) :: x <= sinh(x)/cosh(x/sqrt(3)) for x>=0 with equality iff x=0. cosh(x/sqrt(3)) <= sinh(x)/x <=> 1 + x^2/6 + ... + 1/((2n)! 3^n) x^(2n) <= 1 + x^2/6 + ... + 1/(2n+1)! x^(2n) <=> 3^n >= 2n+1 for n>=0 with equality iff n=0 or n=1. :: Q_1/3(a,b) <= L(a,b) cosh(x/sqrt(3)) <= sinh(x)/x :: L(a,b) <= M_1/3(a,b) log((1+x)/(1-x))/2 >= x + x^3/3 with 0<=x<1 <=> log(z)/2 >= (z-1)/(z+1) + ((z-1)/(z+1))^3/3 with z = (1+x)/(1-x) <=> 3log(a/b) >= (a^3-b^3)/((a+b)/2)^3 with z = a/b <=> M_1/3(a^3,b^3) = ((a+b)/2)^3 >= (a^3-b^3)/(log(a^3)-log(b^3)) = L(a^3,b^3) :: L(a,b) <= M_1/3(a,b) Proof by Harley Flanders. [T. P. Lin (1974)] Consider f(z) = 3/8 log(z) - (z^3 - 1)/(z + 1)^3 for z >= 1. Differentiation yields f'(z) = 3/8 (z - 1)^4/(z(z + 1)^4). Hence f'(z) > 0 for z > 1. Since f(1) = 0, we conclude that f(z) > 0 for z > 1. Apply f(z) > 0 to z = (a/b)^(1/3). :: L(a,b) <= M_1/3(a,b) Exp Proof. Let a = exp(x) and b = exp(-x). L(a,b) = L(exp(x),exp(-x)) = sinh(x)/x. M_1/3(a,b) = M_1/3(exp(x),exp(-x)) = (cosh(x/3))^2 = 1/4cosh(x) + 3/4cosh(x/3) >= 1 + 1/6x^2 + 1/4(cosh(x) - 1 - 1/2x^2) >= sinh(x)/x. To show the last inequality we use the Taylor series. The coefficient of x^(2n) for n>=2 of the rhs is 1/((2n+1)!) and of the lhs is 1/(4(2n)!). So the rhs is termwise lower than the lhs. :: log((1+x)/(1-x))/2 >= x/(1 - x^2/3) with 0<=x<1 <=> x + x^3/3 + ... + 1/(2n+1) x^(2n+1) >= x + x^3/3 + ... + 1/3^n x^(2n+1) <=> 3^n >= 2n+1 for n>=0 with equality iff n=0 or n=1. :: L(a,b) <= A(G(a,b),G(a,b),A(a,b)) log((1+x)/(1-x))/2 >= x/(1 - x^2/3) with 0<=x<1 <=> log(z) >= 3(z^2 - 1)/(z^2 + 4z + 1) with z = (1+x)/(1-x) <=> 2log(a/b) >= 6(a^2 - b^2)/(a^2 + 4ab + b^2) with z = a/b <=> 2/3 G(a^2,b^2) + 1/3 A(a^2,b^2) = (a^2 + 4ab + b^2)/6 >= (a^2 - b^2)/(log(a^2)-log(b^2)) = L(a^2,b^2) :: L(a,b) <= A(G(a,b),G(a,b),A(a,b)) Exp Proof. Let a = exp(x) and b = exp(-x). L(a,b) = L(exp(x),exp(-x)) = sinh(x)/x. A(G(a,b),G(a,b),A(a,b)) = 2/3 + 1/3cosh(x) = 1 + 1/3(cosh(x) - 1) >= sinh(x)/x. To show the last inequality we use the Taylor series. The coefficient of x^(2n) for n>=1 of the rhs is 1/((2n+1)!) and of the lhs is 1/(3(2n)!). So the rhs is termwise lower than the lhs. :: sinh(x)/x <= cosh(x) <= (sinh(x)/x)^3 for x>=0 with equality iff x=0. cosh(x) <= (sinh(x)/x)^3 = (sinh(3x) - 3sinh(x))/(4x^3) <=> 1 + x^2/2! + ... + 1/(2n)! x^(2n) <= 1 + ... + (3^(2n+3) - 3)/(4(2n+3)!) x^(2n) <=> 3^(2n+3) >= 4(2n+3)(2n+2)(2n+1) + 3 for n>=0 with equality iff n=0 or n=1. As (2n+3)(2n+1) = (2n+2)(2n+2) - 1 we get with m = n+1 the condition 9^m >= 32/3 m^3 - 8 m + 1 for integers m >= 1. For m=1 and m=2 this is ok. So we are left with m >= 3. Now we fill in an inequalitiy which can be checked more easily. 9^m >= 2^(3m) >= (8/3 m)^3 >= 32/3 m^3 - 8 m + 1 for integers m >= 3. Now 2^m >= 8/3 m for m >= 3 which is easily checked. :: G(G(a,b),G(a,b),A(a,b)) <= L(a,b) cosh(x) <= (sinh(x)/x)^3 let x = log(a/b) :: log((1+x)/(1-x))/2 <= x/(1 - x^2)^(1/3) with 0<=x<1 <=> x + x^3/3 + ... + 1/(2n+1) x^(2n+1) <= x + x^3/3 + 2x^5/9 + 14x^7/81 + ... coefficients are equal iff n=0 or n=1. The ratio of coefficients on the lhs is (2n-1)/(2n+1). The ratio of coefficients on the rhs is (3n-2)/(3n). Now (2n-1)/(2n+1) < (3n-2)/(3n) for n>1. :: log((1+x)/(1-x))/2 <= x(1 + x^2/3)/(1 - x^4) with 0<=x<1 <=> x + x^3/3 + ... + 1/(2n+1) x^(2n+1) <= x + x^3/3 + x^5 + x^7/3 + ... coefficients are equal iff n=0 or n=1. :: N_2/3(a,b) = (a b^(1/3) + b a^(1/3))/(a^(1/3) + b^(1/3)) <= L(a,b) log((1+x)/(1-x))/2 <= x(1 + x^2/3)/(1 - x^4) with 0<=x<1 set (a/b)^(1/3) = (1+x)/(1-x). :: L(a,b) <= H_1/2(a,b) Exp Proof. Let a = exp(x) and b = exp(-x). L(a,b) = L(exp(x),exp(-x)) = sinh(x)/x. H_1/2(a,b) = H_1/2(exp(x),exp(-x)) = (2/3cosh(x/2) + 1/3)^2 = 2/9cosh(x) + 4/9cosh(x/2) + 1/3 >= 1 + 1/6x^2 + 2/9(cosh(x) - 1 - 1/2x^2) >= sinh(x)/x. To show the last inequality we use the Taylor series. The coefficient of x^(2n) for n>=2 of the rhs is 1/((2n+1)!) and of the lhs is 1/(4.5(2n)!). So the rhs is termwise lower than the lhs. :: L(a,b) <= H_1/2(a,b) Proof by [Jia and Cao (2003)] Consider f(z) = 4/9 log(z) - (z^4 - 1)/(z^2 + z + 1)^2 for z >= 1. Differentiation yields f'(z) = 2(z - 1)^4(2z^2 + 5z + 2)/(9z(z^2 + z + 1)^3). Hence f'(z) > 0 for z > 1. Since f(1) = 0, we conclude that f(z) > 0 for z > 1. Apply f(z) > 0 to z = (a/b)^(1/4). :: H_1/2(a,b) <= M_1/3(a,b) Young Inequality: ----------------- Let a, b > 0 and p, q > 1 with 1/p + 1/q = 1. Then ab <= 1/p a^p + 1/q b^q. Proof by Jenssen's inequality. log(ab) = 1/p log(a^p) + 1/q log(b^q) <= log(1/p a^p + 1/q b^q). Arithmetic-Geometric Mean Inequality: ------------------------------------- George Polya constructed a proof using only: For all real values of x we have exp(x) >= 1 + x. Let a_1, a_2, ..., a_n denote positive real numbers. A = ( a_1 + a_2 + ... + a_n )/n, G = ( a_1 a_2 ... a_n ) ^ (1/n). Given x in turn the values (a_i/A)-1, i=1,2,...,n, we obtain the n relations exp(a_1/A - 1) >= a_1/A, exp(a_2/A - 1) >= a_2/A, . . . exp(a_n/A - 1) >= a_n/A. Multiplying all these together we get exp((a_1 + a_2 + ... + a_n)/A - n) >= ( a_1 a_2 ... a_n )/A^n, which is simply exp(n-n) >= G^n/A^n, or 1 >= G^n/A^n, from which it follows that A >= G. Observe that A=G only if equality holds in all n relations. This requires a_i/A - 1 = 0 in all cases, showing that A=G only when all the a_i are equal (to A). [Honsberger; Mathematical Morsels, 1978, problem 26] Horst Alzer [1996] gave this proof. Let f(x) be a continous function with f(x) > f(x0) for x > x0 and f(x) < f(x0) for x < x0. Then we have the integral inequality Integral_{x0,x} f(t) dt >= Integral_{x0,x} f(x0) dt with equality if and only if x=x0. Apply this to the function f(t) = -1/t. Integral_{x0,x} 1/x0 dt >= Integral_{x0,x} 1/t dt Now plug in a_1, a_2, ..., a_n and sum it up. 1/n Sum_{i=1..n} Integral_{x0,a_i} 1/x0 dt >= 1/n Sum_{i=1..n} Integral_{x0,a_i} 1/t dt The left hand side evaluates to ( a_1 + a_2 + ... + a_n )/n / x0 - 1. The right hand side evaluates to log( ( a_1 a_2 ... a_n ) ^ (1/n) / x0 ). We get A/x0 - 1 >= log( G/x0 ). Now let x0 be A or G. Then on side will vanish and you get A >= G. It is even possible to select x0 in the interval (G,A). Assume that G > A then select x0 such that G > x0 > A. We get A/x0 - 1 < 0 < log( G/x0 ) a contradiction. O. Schlömilch [1858] gave an elementary proof. Start with the polynomial identity (1-z)^2 (1 + 2z + 3z^2 + ... + n z^(n-1)) = 1 - (n+1) z^n + n z^(n+1) which you get by evaluation of d/dx (1 - z^(n+1))/(1 - z) in two ways. The left hand side is non negative for z >= 0. Therefore 1 + n z^(n+1) >= (n+1) z^n for z>=0 with equality for z=1. The substitution z^(n+1) = x/y gives n x + y >= (n+1) (x^n y)^(1/(n+1)) with equality iff x = y. This is a weighted AM-GM inequality. Now use induction in n. For n=1 we have trivial inequality a_1 >= a_1. The n -> n+1 case: a_1 + a_2 + ... + a_(n+1) >= n ( a_1 a_2 ... a_n ) ^ (1/n) + a_(n+1) >= (n+1) ( a_1 a_2 ... a_n a_(n+1) ) ^ (1/(n+1)) And the proof is complete. Fergus Gaines constructed a proof using Schur's inequality. Let A be a square matrix with Frobinus norm ||A||_F then Schur says ||A||_F >= |lambda_1|^2 + |lambda_2|^2 + ... + |lambda_n|^2. Apply it on the special Matrix 0 a_1 0 . . . 0 0 0 a_2 . . . 0 . . . . . . . 0 0 0 . . . a_n-1 a_n 0 0 . . . 0 As A^n = (a_1 a_2 ... a_n)*I we have lambda_1 = ( a_1 a_2 ... a_n ) ^ (1/n). We get the inequality a_1^2 + a_2^2 + ... + a_n^2 >= n (a_1 a_2 ... a_n) ^ (2/n). Power Mean Inequalities: ------------------------ Let a_1, a_2, ..., a_n be positive numbers. Define the function P(t) = ( (a_1^t + a_2^t + ... + a_n^t)/n )^(1/t). Theorem: lim_{t->0} P(t) = (a_1 a_2 ... a_n) ^ (1/n). Proof: Define Q(t) = (a_1^t + a_2^t + ... + a_n^t)/n. P(t) = Q(t)^(1/t) = exp( log(Q(t))/t ). lim_{t->0} P(t) = exp( lim_{t->0} log(Q(t))/t ) = exp( lim_{t->0} d/dt log(Q(t)) ) = exp( ((a_1^0 log(a_1) + a_2^0 log(a_2) + ... + a_n^0 log(a_n))/n) / Q(0) ) = exp( (log(a_1) + log(a_2) + ... + log(a_n))/n ) = (a_1 a_2 ... a_n) ^ (1/n). Theorem: P(t) is a monotonic. [Pólya, Szegö 1924, Problem 2.82] [Wilf 1985, Section 6] Proof: Let t not equal 0 and let A_k = a_k^t then t^2 P'(t)/P(t) = t^2 d/dt log(P(t)) = t^2 d/dt ( 1/t log( (A_1 + A_2 + ... + A_n)/n ) ) = t d/dt log( (A_1 + A_2 + ... + A_n)/n ) - log( (A_1 + A_2 + ... + A_n)/n ) = (A_1 log(A_1) + A_2 log(A_2) + ... + A_n log(A_n))/ (A_1 + A_2 + ... + A_n) - log( (A_1 + A_2 + ... + A_n)/n ) >= 0 <=> (A_1 log(A_1) + A_2 log(A_2) + ... + A_n log(A_n))/n >= (A_1 + A_2 + ... + A_n)/n log( (A_1 + A_2 + ... + A_n)/n ) This is an application of Jenssen's inequality to the function f(x) = x log x. This is convex for x>0 as f''(x) = 1/x > 0. Define the function P(t,x) = ( ((a_1 + x)^t + (a_2 + x)^t + ... + (a_n + x)^t)/n )^(1/t). Q(t,x) = ((a_1 + x)^t + (a_2 + x)^t + ... + (a_n + x)^t)/n. P(t,x) = Q(t,x)^(1/t) = exp( log(Q(t,x))/t ). d/dx P(t,x) = d/dx exp( log(Q(t,x))/t ) = P(t,x) d/dx log(Q(t,x))/t = P(t,x) Q(t-1,x) / Q(t,x) Series expansions of means: --------------------------- M_1(1+x,1-x) = S_1(1+x,1-x) = Q_1(1+x,1-x) = L_1(1+x,1-x) = N_1(1+x,1-x) = 1 M_-1(1+x,1-x) = 1 - x^2 S_3(1+x,1-x) = 1 + x^2 M_p(1+x,1-x) = 1 + (p-1)/2 x^2 - (p-1)(p+1)(2p-3)/24 x^4 + O(x^6) S_p(1+x,1-x) = 1 + (p-1)/2 x^2 - (p-1)(p-3)(2p+1)/24 x^4 + O(x^6) Q_p(1+x,1-x) = 1 + (p-1)/2 x^2 + (p-1)(p+3)/24 x^4 + O(x^6) L_p(1+x,1-x) = 1 + (p-1)/6 x^2 - (p-1)(2p^2+5p-13)/360 x^4 + O(x^6) H_p(1+x,1-x) = 1 + (2p-3)/6 x^2 - (2p^3-4p^2-4p+9)/72 x^4 + O(x^6) N_p(1+x,1-x) = 1 + (p-1) x^2 - p(p-1)(p-2)/3 x^4 + O(x^6) Arithmetic-Geometric Mean AGM: M_1/2(1+x,1-x) = 1 - 1/4 x^2 - 1/16 x^4 + O(x^6) Q_1/2(1+x,1-x) = 1 - 1/4 x^2 - 7/96 x^4 + O(x^6) AGM(1+x,1-x) = 1 - 1/4 x^2 - 5/64 x^4 - 11/256 x^6 + O(x^8) N_3/4(1+x,1-x) = 1 - 1/4 x^2 - 5/64 x^4 - 23/512 x^6 + O(x^8) S_1/2(1+x,1-x) = 1 - 1/4 x^2 - 5/48 x^4 + O(x^6) Radius of a circle having the same perimeter as an ellipse PR: M_3/2(1+x,1-x) = 1 + 1/4 x^2 + 1/192 x^6 + O(x^8) PR(1+x,1-x) = 1 + 1/4 x^2 + 1/64 x^4 + 1/256 x^6 + O(x^8) N_5/4(1+x,1-x) = 1 + 1/4 x^2 + 5/64 x^4 + O(x^6) A(A,G,G) = 1 - 1/3 x^2 - 1/12 x^4 + O(x^6) // 12 M_1/3(1+x,1-x) = 1 - 1/3 x^2 - 7/81 x^4 + O(x^6) // 11.75 H_1/2(1+x,1-x) = 1 - 1/3 x^2 - 25/288x^4 + O(x^6) // 11.52 L(1+x,1-x) = 1 - 1/3 x^2 - 4/45 x^4 + O(x^6) // 11.25 Q_1/3(1+x,1-x) = 1 - 1/3 x^2 - 5/54 x^4 + O(x^6) // 10.8 N_2/3(1+x,1-x) = 1 - 1/3 x^2 - 8/81 x^4 + O(x^6) // 10.12 G(A,G,G) = 1 - 1/3 x^2 - 1/9 x^4 + O(x^6) // 9 S_1/3(1+x,1-x) = 1 - 1/3 x^2 - 10/81 x^4 + O(x^6) // 8.1 A(A,G,G) = 1 + 1/6 x^2 + 1/72 x^4 + O(x^6) // 72 M_1/3(exp(x),exp(-x)) = 1 + 1/6 x^2 + 7/648x^4 + O(x^6) // 92.57 H_1/2(exp(x),exp(-x)) = 1 + 1/6 x^2 + 1/96 x^4 + O(x^6) // 96 L(exp(x),exp(-x)) = 1 + 1/6 x^2 + 1/120x^4 + O(x^6) // 120 Q_1/3(exp(x),exp(-x)) = 1 + 1/6 x^2 + 1/216x^4 + O(x^6) // 216 N_2/3(exp(x),exp(-x)) = 1 + 1/6 x^2 - 1/648x^4 + O(x^6) //-648 G(A,G,G) = 1 + 1/6 x^2 - 1/72 x^4 + O(x^6) // -72 References: Dumitru Acu; Some Inequalities for Certain Means in Two Arguments, General Mathematics 9:1-2 (2001) 11-14 http://arhimede.ulbsibiu.ro/gm/vol9nr1_2/dacu.htm http://arhimede.ulbsibiu.ro/gm/vol9nr1_2/DAcu.pdf - the relation of Power and Gini Means. M_p = S_p for p in {0,1}, M_p >= S_p for 0 < p < 1, M_p <= S_p for p < 0 or 1 < p. János Aczél, Zs. Páles; The behaviour of means under equal increments of their variables, American Mathematical Monthly 95, No.9 (1988) 856-860, Zbl 0671.26008 [L. Hoehn, I. Niven; Averages on the Move, 1985] Claudio Alsina; Cauchy-Schwarz Inequality, Mathematics Magazine (Feb. 2004) 30 Proof Without Words: We present a sequence of pictures showing Cauchy-Schwarz inequality in the plane by means of some appropriate area-preserving transformations. Horst Alzer; Ungleichungen für Mittelwerte. Arch. Math. 47 (1986) 422-426 Zbl 0585.26014 - inequalities for the logarithmic and identric mean in two arguments G(G(a,b),A(a,b)) <= G(L(a,b),I(a,b)) <= A(L(a,b),I(a,b)) <= A(G(a,b),A(a,b)) Horst Alzer; Inequalities for arithmetic, geometric and harmonic means, Bull. London Math. Soc. 22:4 (1990) 362-366 Zbl 0707.26014 - pairs 0 < x <= x' < 1 with x + x' = 1. 1/H' - 1/H <= 1/G' - 1/G <= 1/A' - 1/A Horst Alzer; A Proof of the Arithmetic Mean-Geometric Mean Inequality, American Mathematical Monthly 103 (1996) 585 Zbl 0867.26014 - A very short proof of the arithmetic mean - geometric mean inequality is given. Horst Alzer; A new refinement of the arithmetic mean -- geometric mean inequality, Rocky Mountain Journal of Mathematics 27:3, (1997) 663-667 Zbl 0897.26004 - refinment of Variance <= 2Max*(A - G). Alzer, Horst; Ruscheweyh, Stephan On the intersection of two-parameter mean value families, Proc. Am. Math. Soc. 129, No.9, 2655-2662 (2001). Zbl 0979.26015 Alzer, Horst; Ruscheweyh, Stephan; Salinas, Luis; Inequalities for cyclic functions, J. Approximation Theory 112, No.2, 216-225 (2001), Zbl 0992.26012 - generalizing sinh(x)/x <= cosh(x) <= (sinh(x)/x)^3 with x>=0 Horst Alzer, Song-liang Qiu; Inequalities for means in two variables, Archiv der Mathematik, 80:2 (2003) 201-215 Zbl 1020.26011 David H. Anderson, The Generalized Arithmetic-Geometric Mean Inequality, College Mathematics Journal 10:2 (1979) 113-114 G. Aumann; Eine geometrische Bemerkung zum arithmetischen und harmonischen Mittel, Unterrichtsblätter, 41 (1935) 88. JFM 61.0196.01 - means and the hyperbola E. F. Beckenbach; A class of mean value functions, American Mathematical Monthly 57 (1950) 1-6 Zbl 0035.15704 Die Bemerkung, dass bei einem Zahlen-n-tupel (a)=(a_1,a_2,...,a_n) die Relativstreuung S(x) = Sum_{j=1..n} ((a_j - x)/x)^2 ihr Minimum fuer das gewoehnliche antiharmonische Mittel x = Sum_{j=1..n} a_j^2 / Sum_{j=1..n} a_j = N_2(a) annimmt, gibt dem Verfasser die Gelegenheit, eine zusammenhaengende Theorie der allgemeinen antiharmonischen Mittel N_t(a) = Sum_{j=1..n} a_j^t / Sum_{j=1..n} a_j^(t-1), die nicht der meist betrachteten Klasse der Mittelwerte, den quasiarithmetischen Mitteln f^(-1)((f(x_1) + f(x_2) + ... + f(x_n))/n) angehoeren (ausser fuer t=0 und t=1, wo sie mit dem harmonischen bzw. mit dem arithmetischen Mittel zusammenfallen) zu entwickeln, aehnlich der allgemeinen Theorie der Potenzmittel M_t(a) = (1/n Sum_{j=1..n} a_j^t)^(1/t), wie sie z. B. in Hardy-Littlewood-Polya: Inequalities, 1934, Ch. II, Ch. III zu finden ist; naturgemäss sind die Betrachtungen hier bei den antiharmonischen Mitteln wesentlich schwieriger. - Es werden folgende wichtige Saetze bewiesen. Seien 1/a = (1/a_1, 1/a_2, ..., 1/a_n) und k*a = (k*a_1, k*a_2, ..., k*a_n) mit k>0. lim_{t->-oo} N_t(a) = min(a), lim_{t->+oo} N_t(a) = max(a) N_{-t}(a) = 1/N_{t+1}(1/a), N_t(k a) = k N_t(a), N_t(a) <= M_t(a) fuer t <= 1 [ N_t(a) <= M_{t-1}(a) <= M_t(a) fuer t <= 0 ] und N_t(a) >= M_t(a) fuer t >= 1 N_t(a_1,...,a_j,...,a_n) wächst mit t, und waechst mit jedem a_j, falls 0<=t<=1 a_j <= N_t(a), falls t <= 0, und a_j >= N_t(a), falls t >= 1. Das Analogon der Minkowskischen Ungleichung fuer M_t(a), hier N_t(a_1,...,a_n) + N_t(b_1,...,b_n) <= N_t(a_1+b_1,...,a_n+b_n) bzw. N_t(a) + N_t(b) >= N_t(a+b) gilt nur für 0<=t<=1 bzw. 1<=t<=2; die schwaecheren Ungleichungen N_t(a_1,...,a_n) + c <= N_t(a_1+c,...,a_n+c) bzw. N_t(a_1,...,a_n) + c >= N_t(a_1+c,...,a_n+c) gelten aber fuer alle t<=1 bwz. t>=1. Ueberall wird auch der Gleichheitsfall untersucht. Die Resultate werden meist durch Derivation hergeleitet. Es werden Verallgemeinerungen fuer Gewichtsmittel und Integralmittel derselben Art angedeutet. E. F. Beckenbach, R. Bellman; An introduction to inequalities, New York-Toronto: Random Hous, IX, 133 p. (1961). ISBN: 0394015592 Zbl 0186.09606 Edwin F. Beckenbach, Richard Bellman; Inequalities, (Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge: Heft 30.) Berlin- Göttingen-Heidelberg: Springer-Verlag. XII, 198 p. (1961). ISBN: 0387032835 Zbl 0097.26502 Alexander Bogomolny; The Means, http://www.cut-the-knot.org/arithmetic/Means.shtml - 4 means in the trapezoid J. M. Borwein, P. B. Borwein; The way of all means, American Mathematical Monthly 94 (1987) 519-522 J. L. Brenner; Limits of means for large values of the variables, Pi Mu Epsilon J. 8, 160-163 (1985). Zbl 0601.26012 [L. Hoehn, I. Niven; Averages on the Move, 1985] - This article concerns the limit of the difference between two power means when all the variables become large. For each real number t>=0, define f(t) = M(t+a_1,t+a_2,...,t+a_n) - t for the power mean M. f(t) is monotonic and lim_{t->oo} f(t) = A(a_1,a_2,...,a_n). P. S. Bullen; Averages still on the move, Mathematics Magazine, 63, No.4, (1990) 250-255 Zbl 0736.26009 [L. Hoehn, I. Niven; Averages on the Move, 1985] P. S. Bullen, D. S. Mitrinovic, and P. M. Vasic; Means and Their Inequalities, Reidel, Dordrecht, 1988. MR 89d:26003 P. S. Bullen; Handbook of Means and Their Inequalities, Second Edition Kluwer Academic Publishers, Dordrecht, 2003 MATHEMATICS AND ITS APPLICATIONS : Volume 560 ISBN 1-4020-1522-4 I: Introduction. II: The Arithmetic, Geometric and Harmonic Means. III: The Power Means. IV: Quasi-Arithmetic Means. V: Symmetric Polynomial Means. VI: Other Topics. - over seventy for the inequality between the arithmetic and geometric means Frank Burk; Mean Inequalities, College Mathematics Journal 14:5 (1983) 431-434 Frank Burk; By all means, American Mathematical Monthly 92 (1985) 50 - shows H(a,b) <= G(a,b) <= L(a,b) <= A(a,b) <= R(a,b) Frank Burk; The geometric, logarithmic, and arithmetic mean inequality. American Mathematical Monthly 94 (1987) 527-528 Zbl 0632.26008 - elegant proof of G(a,b) <= L(a,b) <= M_1/3(a,b). B. C. Carlson; A hypergeometric mean value, Proc. Am. Math. Soc. 16, 759-766 (1965). Zbl 0137.26802 B. C. Carlson; M. D. Tobey; A property of the hypergeometric mean value, Proc. Am. Math. Soc. 19, 255-262 (1968). Zbl 0173.06902 B. C. Carlson; The logarithmic mean, American Mathematical Monthly 79 (1972) 615-618 MR 46:1985 - shows G(a,b) <= G(G(a,b),M_{1/2}(a,b)) <= L(a,b) <= M_{1/2}(a,b) <= A(a,b) - cites L(a,b) <= (2G(a,b) + A(a,b))/3. B. C. Carlson; Algorithms involving arithmetic and geometric means, American Mathematical Monthly, 78 (1971) 496-505 Augustin Louis Cauchy; Analyse algebrique, (Note 2; Oeuvres completes, Serie 2, Vol 3, p375-377; Paris, Gauthier-Villars 1897) - gives an elegant elemanetary proof of the AM-GM-Inequality B. C. Diestel, R. A. Gordon; Using tangent lines to define means, Mathematics Magazine, 76 (2003) 52-61 D. E. Dobbs; A proof of the arithmetic-geometric mean inequality using non-Euclidean geometry, International Journal of Mathematical Education in Science and Technology, 32:5, (1 September 2001) 778-782 Edward L. Dodd; Definitions and properties of the median, quartiles, and other positional means, American Mathematical Monthly, 45 (1938) 302-306 JFM 64.0545.01 Zbl 0018.41502 As shown by D. Jackson, the indefiniteness in the definition of the median of a set of real numbers can be avoided by the use of a limit process. The author gives a corresponding definition of quatiles and other "k-iles". The properties of these unique k-iles are discussed under reference to earlier papers (see also Zbl 10.108). Edward L. Dodd; Interior and exterior means obtained by the method of moments, Ann. Math. Stat. 9, 153-157 (1938). Zbl 0020.04104 Edward L. Dodd; The substitutive mean and certain subclasses of this general mean. Ann. Math. Stat. 11, (1940) 163-176. Zbl 0023.34004 Ziel der vorliegenden Abhandlung ist eine Definition des Begriffs der Mittelwerte. Von den bisherigen Autoren sind je nach den Bedingungen, die sie als notwendig fuer einen Mittelwert erachten, verschiedene Definitionen gegeben worden; bekanntlich sind die hauptsaechlichsten dieser Bedingungen die beiden folgenden: 1. Der Mittelwert soll ein innerer Wert der Reihe sein; 2. der Mittelwert einer Reihe, deren saemtliche Glieder gleich einer Konstanten c sind, soll auch c sein. Verf. legt seiner Definition des "substitutive mean" nur die zweite als notwendige Bedingung zugrunde und knuepft damit an eine bekannte und vielbesprochene Definition von Chisini an. Auf Grund seiner Definition klassifiziert Verf. die verschiedenen Mittelbildungen, die bisher untersucht wurden, unter besonderer Beachtung einer neuerdings von Gini gefundenen zusammenfassenden Formel fuer die Mittelbildungen; er zeigt weiterhin, wie die Ginische Formel sich in die von ihm zugrunde gelegte Definition des Mittels einfuegt; hierzu ist zu bemerken, dass, waehrend die Definition des Verf. rein logischen und theoretischen Charakter traegt, die Ginische Formel vom Standpunkt des Praktikers aus gesehen, den grossen Vorteil bietet, in algebraischer Form alle Formeln der hauptsaechlichsten analytischen Mittelbildungen zu umfassen. Die Abhandlung schliesst mit einer ins einzelne gehenden Schrifttumsuebersicht. Edward L. Dodd; Some generalization of the logarithmic mean and of similar means of two variates which become indeterminate when the two variates are equal, Ann. Math. Statist., 12 (1941) 422-428. MR 3:170e Zbl 0063.01124 Heinrich Dörrie; Ein neues elementares Verfahren zur Lösung von Extremwertaufgaben, MU 18 (1972) Heft 5, 23-51 (Reprint of: Zeitschrift für den mathematischen und naturwissenschaftlichen Unterricht aller Schulgattungen, 50 (1919) 153-177) Heinrich Dörrie; Über einige Anwendungen des Satzes vom arithmetischen und geometrischen Mittel, Zs. f. math. u. naturw. Unterr. 52 (1921) 103-108. JFM 48.0073.05 - the exp function and the arithmetic geometric mean inequality M. Drescher; Moment spaces and inequalities, Duke Math Journal 20 (1953) 261-271 Arthur Engel; Mathematische Olympiadeaufgaben aus der UDSSR, Klett Verlag, Stuttgart, 1965, ISBN 3-12-710300-X - problem 151 Slices of the trapezoid taken parallel to parallel sides of lengths a and b. geometric_mean(a, b) divides the trapezoid into two similar trapezoids. harmonic_mean(a, b) slice passes through the intersection of the diagonals. root_mean_square(a, b) bisects the area of the trapezoid. arithmetic mean(a, b) bisects the sides of the trapezoid. Arthur Engel; Problem-solving strategies, Problem Books in Mathematics. New York, NY: Springer. x, 403 p. (1998) ISBN 0-387-98219-1 Zbl 887.00002 - chapter 7: Inequalities Howard W. Eves; Introduction to the History of Mathematics. Sixth edition. Saunders College Publishing, NY, 1990 p201 Howard W. Eves; Means Appearing in Geometrical Figures, Mathematics Magazine, 76:4 (October 2003) 292-294 Practically every student of mathematics is acquainted with the arithmetic and geometric means of two given positive numbers, a and b. Not so many students realize that there are many other means of two given positive numbers. A short list includes the arithmetic mean, the geometric mean, the harmonic mean, the heronian mean, the contraharmonic mean, the root-mean-square, and the centroidal mean. We give many examples where these means arise in geometric figures. In particular, each of the means in our list has a special interpretation as the lengths of slices of the trapezoid taken parallel to parallel sides of lengths a and b. Supplement to Means Appearing in Geometric Figures, by Howard Eves in the October 2003 issue of the Magazine: An animated demonstration (in Geometer's Sketchpad) of the means in a trapezoid, by Shannon Umberger Patton. http://www.maa.org/pubs/Umberger.gsp Mabrouck K. Faradj; WHICH MEAN DO YOU MEAN? AN EXPOSITION ON MEANS, These, August 2004 etd.lsu.edu/docs/available/etd-07082004-091436/unrestricted/thesis5.pdf Paul Fjelstad and Ivan Ginchev; Volume, Surface Area, and Harmonic Mean, Mathematics Magazine, 76:2 (April 2003) Fergus Gaines; On the arithmetic mean-geometric mean inequality, American Mathematical Monthly 74 (1967), 305-306. Zbl 0187.01202 Schur's inequality is used to show the arithmetic mean-geometric mean inequality. Corrado Gini; Di una formula comprensiva delle medie. (Italian) Metron 13, No.2, 3-22 (1938). Zbl 0018.41404 H. W. Gould, M. E. Mays; Series expansions of means, J. Math. Anal. Appl. 101 (1984), 611-621. MR 86a:26025 Zbl 0582.41026 Jutta Gut; Mittelwerte, http://members.chello.at/gut.jutta.gerhard/mittel.htm G.H. Hardy, J.E. Littlewood, and G. Pólya; Inequalities, Cambridge Univ. Press, Cambridge, 1934. Zbl 0010.107 Wilfried Herget; Bemerkungen zum Mittelwertsatz der Differenzialrechnung, PM 21 (1979) 39-41 Wilfried Herget; Der Zoo der Mittelwerte, Mittelwerte-Familien, Mathematik Lehren, (Themenheft Mittelwerte) (Feb 1985) (no. 8) 50-51 Horst Hischer; "Fundamentale Ideen" und "Historische Verankerung" dargestellt am Beispiel der Mittelwertbildung, Mathematica Didactica. Zeitschrift fuer Didaktik der Mathematik. 21:1 (1998) 3-20 MathDi 1999b.00866 http://hischer.de/uds/forsch/publikat/hischer/index.htm Horst Hischer; Mittelwertbildung. Eine der aeltesten mathematischen Ideen. Primarstufe, Sekundarstufe I/II, 3.-13. Schuljahr, Mathematik Lehren, (Aug 2003) (no.119) 40-46 MathDi 2004a.00225 Horst Hischer; Viertausend Jahre Mittelwertbildung? Eine fundamentale Idee der Mathematik und didaktische Implikationen, Mathematica Didactica. Zeitschrift fuer Didaktik der Mathematik. 25 (2002) 3-51 http://hischer.de/uds/forsch/publikat/hischer/index.htm Larry Hoehn; A Geometrical Interpretation of the Weighted Mean, College Math Journal, 15:2 (March 1984) 135-?? L. Hoehn, I. Niven; Averages on the Move, Mathematics Magazine, 58 (1985) 151-156 Zbl 0601.26011 Ross Honsberger; Mathematical Morsels, The Dolciani Mathematical Expositions No. 3 The Mathematical Association of America, 1978 (german: Gitter-Reste-Wuerfel, Vieweg, 1984) problem 26: a^b and b^a - the AM-GM inequality, p60 Alan Horwitz; Means, generalized divided differences, and intersections of osculating hyperplanes, J. Math. Anal. Appl. 200 (1996) 126–148 - multi varialbe logarithmic mean Joachim Jäger; Verknüpfungsmittelwerte, Mathematische Semesterberichte 52 (2005) 63-79 http://www.htw-saarland.de/fb-wi/personal/dozenten/publikationen-jaeger Walther Janous; Crux Mathematicorum 16 (1990) ?, Problem 1584 Crux Mathematicorum 17 (Dec. 1991) ?, Problem 1584 (solution) Prove that for x > 1 (log(x)/(x-1))^3 < 2/(x(x+1)) (this shows (ab(a+b)/2)^(1/3) <= L(a,b)) Jun Ji, Charles Kicey; The Slope Mean and Its Invariance Properties, Mathematics Magazine, 78:2 (April 2005) 139-144 Gao Jia, Jinde Cao; A new upper Bound of the Logarithmic Mean, Journal of Inequalities in Pure and Applied Mathematics, 4:4 (2003) Article 80 L(a,b) <= H_{1/2}(a,b) <= M_{1/3}(a,b) Erwin Just, Norman Schaumberger; Two More Proofs of a Familiar Inequality, The TWO-YEAR COLLEGE MATH JOURNAL 6:2 (1975) 45 - what is bigger e^pi or pi^e? Hidefumi Kaatsura; Generalization of the Arithmetic-Geometric Mean Inequality and a Three Dimensional Puzzle, College Mathematics Journal (Sep 2003) It is possible to pack twenty-seven a by b by c boxes in a cube of side a + b + c. In the course of trying to pack a five-dimensional cube (it's known that it can be done in four dimensions) the author found an inequality that allowed him to solve a different three-dimensional problem. P. Kahlig and J. Matkowski; Functional equations involving the logarithmic mean, Z. Angew. Math. Mech. 76 (1996) 385-390. MR 97e:39017 J. Karamata; Sur quelques problèmes posés par Ramanujan, J. Indian Math. Soc., n. Ser. 24 (1961) 343-365. Zbl 0217.32101 - shows log(x)/(x - 1) <= (1 + x^(1/3))/(x + x^(1/3)) for x > 0 with equality iff x = 1. P. P. Korowkin; Ungleichungen, Berlin, Deutscher Verlag der Wissenschaften, 19?? Kleine Ergänzungsreihe zu den Hochschulbüchern der Mathematik, Band 4. (russion edition: Moskow 1952) - gives Dörrie's proof of the AM-GM-Inequality. Sidney Kung; Harmonic, Geometric, Arithmetic, Root Mean Inequality, College Math Journal, 21:3 (1990) 227 Marcin E. Kuczma; Crux Mathematicorum 17 (Dec. 1991) ?, Problem 1622 Let n be a positive integer. (a) Prove the inequality (a^(2n) + b^(2n))/2 <= ( ((a+b)/2)^2 + (2n-1)((a-b)/2)^2 )^n for real a, b, and find conditions for equality. (b) Show that the constant 2n-1 is the right-hand expression is the best possible, in the sense that on replacing it by a smaller one we get an inequality which fails to hold for some a, b. E. Neumann, ZS. Páles; On comparison of Stolarsky and Gini means, J. Math. Anal. Appl. 278 (2003) 274-284 Leach, E.B.; Sholander, M.C. Extended mean values, American Mathematical Monthly, 85 (1978) 84-90 Zbl 0379.26012 Leach, E.B.; Sholander, M.C. Corrections to "Extended mean values", American Mathematical Monthly, 85 (1978) 656 Zbl 0389.26008 Leach, E.B.; Sholander, M.C.; Extended mean values. II. J. Math. Anal. Appl. 92, 207-223 (1983). Zbl 0517.26007 logarithmic; identric; arithmetic; geometric; harmonic; reverse means D. H. Lehmer; On the compounding of certain means, J. Math. Anal. Appl. 36 (1971) 183-200 Zbl 0222.26018 MR 43:7411 - Lehmer's means are contraharmonic means [Beckenbach, 1950] H. Leinfelder; Vorgegebene Zwischenstellen im Mittelwertsatz der Differentialrechnung, Mathematische Semesterberichte 30 (1983) 7-17 T. P. Lin; The power mean and the logarithmic mean, American Mathematical Monthly, 81 (1974) 879-883. MR 50:7449 - shows G(a,b) <= L(a,b) <= M_1/3(a,b). Viktors Linis; Crux Mathematicorum ? (197?) ?, Problem 98 Prove that, if a > b > 0, then log a^2/b^2 < a/b - b/a. (this shows G(a,b) <= L(a,b)) Viktors Linis; Crux Mathematicorum 4:1 (1978) 11, Problem 304 Crux Mathematicorum 4 (1978) 178-179, Problem 304 (solution) Prove the inequality: (Karamata's inequality) log(x)/(x - 1) <= (1 + x^(1/3))/(x + x^(1/3)) for x > 0 with equality iff x = 1. (this shows N_2/3(a,b) <= L(a,b)) Gunter Lorenzen; Why means in two arguments are special, Elemente der Mathematik, 49:1 (1994) 32-37 Zbl 0860.26014 László Losonczi; Subadditive Mittelwerte. Archiv der Mathematik, 22, 168-174 (1971). Zbl 0226.26023 László Losonczi; Restricted subadditivity of homogeneous means. J. Math. Anal. Appl. 222, No.1, 167-176 (1998). Zbl 0955.26014 E. Maor; A Mathematician's Repertoire of Means, Mathematics Teacher, 70 (January 1977) 20-25 D. S. Mitrinovics; Elementary Inequalities, P. Noordhoff Ltd, Groningen, The Netherlands, 1964 - p 272 Karamata's inequality log(x)/(x - 1) <= (1 + x^(1/3))/(x + x^(1/3)) for x > 0 with equality iff x = 1. Mitrinovic, D.S.; Pecaric, J.E.; Fink, A.M.; Classical and new inequalities in analysis, Mathematics and Its Applications. East European Series. 61. Dordrecht: Kluwer Academic Publishers. xvii, 740 p. (1993). ISBN 0-7923-2064-6 MR 94c:00004, Zbl 0771.26009 David Moskovitz; An alignment chart for various means, American Mathematical Monthly 40 (1933) 592-596 JFM 59.0537.01 - Class of means (f(a)*b + f(b)*a)/(f(a) + f(b)) Vedula N. Murty; Crux Mathematicorum 24:4 (May 1998) 236, Problem 2345 Let a and b denote distict positive real numbers. (a) Show that if x>1 then log(x) > 3(x^2 - 1)/(x^2 + 4x + 1) (b) Use (a) to deduce Pólya's inequality: (a-b)/(log(a)-log(b)) < (2sqrt(ab) + (a+b)/2)/3. T. Nagell; Über das arithmetische und das geometrische Mittel. Norsk Mat. Tidsskrift 14 (1932) 54-55. JFM 58.0210.05 Ein neuer elementarer Beweis, der zugleich eine positive untere Schranke für die Differenz zwischen dem arithmetischen und dem geometrischen Mittel angibt. I. P. Natanson; Einfachste Maxima- und Minimaaufgaben, Berlin, Deutscher Verlag der Wissenschaften, 1960 Kleine Ergänzungsreihe zu den Hochschulbüchern der Mathematik, Band 9. (russion edition: Moskow 1951) - gives Cauchy's proof of the AM-GM-Inequality. Roger B. Nelson; Proof Without Words: The Harmonic Mean-Geometric Mean-Arithmetic Mean-Root Mean Square Inequality, Mathematics Magazine, 60:3 (1987) 158 Roger B. Nelson; The Root Mean Square-Arithmetic Mean-Geometric Mean-Harmonic Mean Inequality, College Math Journal, 20:3 (1989) 231 Roger B. Nelson; The Mean of the Squares Exceeds the Square of the Means (Proof Without Words), College Math Journal, 26:5 (1995) 368 Roger B. Nelson; Proof Without Words: The Arithmetic-Logarithmic-Geometric Mean Inequality, Mathematics Magazine, 68:4 (October 1995) 305 Edward Neuman, József Sándor; On Certain Means of two Arguments and their Extensions, International Journal of Mathematics and Mathematical Sciences, 16 (2003) 981-993 Inequalities for three means introduced by H.-J. Seiffert are obtained. Generalizations of these means, their basic properties, and inequalities satisfied by the new class of means are also included. Edward Neuman, József Sándor; Inequalities involving Stolarsky and Gini means, Math. Pannonica 14, No.1 (2003) 29-44 Zbl 1026.26012 Constantin P. Niculescu; A note on the Hermite-Hadamard Inequality, The Mathematical Gazette, July 2001, 48-50 Constantin P. Niculescu, Lars-Erik Persson; Old and new on the Hermite-Hadamard inequality, Real Analysis Exchange, 29 (2003/04), no. 2, 663-686 MR 2083805 Zsolt Páles; Comparison of two variable homogeneous means, in: General Inequalities 6, Internat. Ser. Numer. Math., Vol. 103, Birkhäuser, Basel, 1992, 59-70. MR 94b:26016 M. Perisastry, V. N. Murty; Bounds for the Ratio of the Arithmetic Mean to the Geometric Mean, The College Mathematical Journal 13:2 (1982) 160-161 A. O. Pittenger; The logarithmic mean in n variables, American Mathematical Monthly, 92 (1985), 99-104. MR 86h:26012 George Pólya; On the harmonic mean of two numbers, American Mathematical Monthly, 57 (1950) 26-28 Zbl 0035.03503 Von einer unbekannten Groesse x weiss man, dass sie zwischen zwei positiven Schranken liegt: 0 < a <= x <= b. Bei Wahl eines Naeherungswertes p fuer x entsteht ein Fehler p-x und ein relativer Fehler (p-x)/x. Die Aufgabe, das Risiko des Fehlers oder des relativen Fehlers zu einem Minimum zu machen, gibt zu folgenden Fragestellungen Anlass: (I) Gesucht Min_{p} Max_{x} |p - x| (II) Gesucht Min_{p} Max_{x} |p - x|/x Die Loesung von (I) ist bekannt: p = (a+b)/2, Min_{p} Max_{x} |p - x| = (b-a)/2. Als Loesung von (II) findet Verf. p = 2ab/(a+b), Min_{p} Max_{x} |p - x|/x = (b-a)/(b+a). George Pólya and Gabor Szegö; Problems and Theorems in Analysis, vol. I, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen. Band 193. Springer-Verlag, 1972, (german: Aufgaben und Lehrsätze aus der Analysis I, Springer-Verlag, 1924) Zbl 236.00003 Section 2.2: Inequalities - Problem 2.82: George Pólya, Gabor Szegö; Isoperimetric inequalities in mathematical physics. Princeton University Press, 1951 MR 13:270d - introduces the logarithmic mean Claus Michael Ringel; Eine Ungleichung und viele Gleichungen, Bielefeld 2000, www.mathematik.uni-bielefeld.de/~ringel/schueler.ps - Cauchy Schwarz inequality J. Rooin; Another proof of the arithmetic-geometric mean inequality, The Mathematical Gazette 285 (July 2001) Note 85.32 Dieter Ruthing; Eine allgemeine logarithmische Ungleichung, Elemente der Mathematik, 41 (1996) 14-16 - shows G(a,b) <= L(a,b) <= M_{1/3}(a,b) József Sándor, I. Rasa; Inequalities for certain means in two arguments, Nieuw Archief voor Wiskunde, 15:1-2 (1997) 51-55 Zbl 0938.26011 József Sándor, Tiberiu Trif; Some new inequalities for means of two arguments, International Journal of Mathematics and Mathematical Sciences, 25:8 (2001) 525-532 Zbl 1002.26018 The authors consider special means of two arguments, among them the logarithmic, identric, arithmetic and geometric means. Several interesting inequalities connecting them are known; the authors give new such inequalities, some of them improving older ones. Doris Schattschneider; Proof Without Words: The arithmetic mean-geometric mean inequality, Mathematics Magazine, 59:1 (1986) 11 Norman Schaumberger; Still Another Proof of the Arithmetic-Geometric Mean Inequality, College Mathematics Journal 13:2 (1982) 159-160 Norman Schaumberger; Another Proof of the Inequality Between Power Means, College Mathematics Journal 19:1 (1988) 56-58 Norman Schaumberger; A General Form of the Arithmetic-Geometric Mean Inequality via the Mean Value Theorem, College Mathematics Journal 19:2 (1988) 172-173 Norman Schaumberger; Another Proof of Chebysheff's Inequality, College Mathematics Journal 20:2 (1989) 141-142 Norman Schaumberger; The AM-GM Inequality via x^(1/x), College Mathematics Journal 20:4 (1989) 320 Norman Schaumberger; The AM-GM inequality via one of its consequences, Pi Mu Epsilon Journal 10, No.2, (1995) 90-91 Norman Schaumberger and Bert Kabak; Another Proof of Jensen's Inequality, College Mathematics Journal 20:1 (1989) 57-58 Norman Schaumberger; A Monte Carlo AM e GM (Mathematics Without Words), College Mathematics Journal 31:1 (2000) 68 Norman Schaumberger and Bert Kabak, Another Proof of Jensen's Inequality, College Mathematics Journal 20:1 (1989) 57-58 N. Schaumberger, M. Steiner; The Power Means Theorem via the Weighted AM-GM Inequity, Pi Mu Epsilon Journal 10:5 (1996) 405 Harald Scheid; Wo ist bloss die Mitte? In: T. Devendran (editor); Neues aus dem Mathematischen Kabinett, Hugendubel Verlag, München 1985, ISBN 3-88034-259-8 Section IV.4: Wo ist bloss die Mitte?, p144-152 O. Schlömilch; Z. Math. Physik 3 (1858) 301-308. - die Potenzmittel wachsen mit dem Exponenten. O. Schlömilch; Ueber das arithmetische, das geometrische und das harmonische Mittel aus beliebig vielen positiven Zahlen. Title in English: On the arithmetic, the geometric, and the harmonic mean of arbitrary many positive numbers. Hoffmann Z. XI. (1880) 361-362. JFM 12.0340.01 Karl Schuler; Maxima-Minima-Aufgaben, Archimedes, Sonderheft 1966 - gives Schlömilch's (1858) proof of the AM-GM-Inequality. Hans Schupp; Optimieren, Extremwertbestimmung im Mathematikunterricht, BI-Wiss.-Verlag, Mannheim, 1992, ISBN 3-411-15771-2 Series: Lehrbücher und Monographien zur Didaktik der Mathematik Band 20 - gives Dörrie's proof of the AM-GM-Inequality. Allen Schwenk; Distortion of average class size: The Lake Wobegon effect. College Mathematics Journal 37 (September 2006) 293-296 H.-J. Seiffert; P 918. Ungleichungsverallgemeinerung, PM = Praxis der Mathematik (1/1989) 55 H.-J. Seiffert; Ungleichungen für einen bestimmten Mittelwert, Nieuw Archief voor Wiskunde, 13 (1995) 195-198. MR 96h:26025 H.-J. Seiffert; Ungleichungen für elementare Mittelwerte, Arch. Math. (Basel), 64 (1995) 129-131. MR 95j:26026 Heinz-Jürgen Seiffert; Crux Mathematicorum 23:1 (Feb. 1997) 46, Problem 2206 Crux Mathematicorum 24:1 (Feb. 1998) 60-62, Problem 2206 (solution I by J. Bradley) Crux Mathematicorum 24:5 (Sep. 1998) 311-312, Problem 2206 (solution III by Seiffert) Let a and b denote distict positive real numbers. (a) Show that if 0 < p < 1, p != 1/2, and q = 1-p then (a^p b^q + a^q b^p)/2 < 4pq sqrt(ab) + (1-4pq) (a+b)/2 (b) Use (a) to deduce Pólya's inequality: (a-b)/(log(a)-log(b)) < (2sqrt(ab) + (a+b)/2)/3. O. Shisha, B. Mond; Bounds on difference of means, Inequalities, (O.Shisha, Ed.). Academic Press, New York, 1967, 293-308. Eckard Specht; geometria - scientiae atlantis. 300+ Aufgaben zur Geometrie und zu Ungleichungen insbesondere zur Vorbereitung auf Mathematik-Olympiaden, Otto-von-Guericke-Universität Magdeburg, 2001, ISBN 3-929757-39-7 Kapitel U: Ungleichungen Kapitel G: Geometrische Ungleichungen Eckard Specht; 470+ Mathematik-Aufgaben zum Training in Vorbereitung auf Olympiaden und Wettbewerbe http://www.math4u.de/ (online version) Kapitel U: Ungleichungen Kapitel G: Geometrische Ungleichungen Wilhelm Specht; Zur Theorie der elementaren Mittel, Mathematische Zeitschrift, 74 (1960), 91-98. Zbl 0095.03801 - a sharp upper bound for M_t/M_s. J. Michael Steele; The Cauchy-Schwarz Master Class, Mathematical Association of America, 2004, ISBN 0-521-54677-X Chapter 2: Cauchy's Second Inequality: The AM-GM Bound Chapter 8: The ladder of Power Means Kenneth B. Stolarsky; Generalizations of the logarithmic mean, Mathematics Magazine, 48 (1975) 87-92. MR 50:10186 Kenneth B. Stolarsky; The power and generalized logarithmic means, American Mathematical Monthly, 87 (1980) 545-548. MR 82g:26029 Kenneth B. Stolarsky; Hölder means, Lehmer means, and x-1 log cosh x, J. Math. Anal. Appl., 202 (1996) 810-818 Charles W. Trigg; Mathematical Quickies, New York 1967, (reprint: Dover Publ., 1985) Q 49. Three Means W. Türke; Eine geometrische Deutung der Mittelwert-Ungleichungen, alpha, 13:3 (1979) 52-53 Slices of the trapezoid taken parallel to parallel sides of lengths a and b. geometric_mean(a, b) divides the trapezoid into two similar trapezoids. harmonic_mean(a, b) slice passes through the intersection of the diagonals. root_mean_square(a, b) bisects the area of the trapezoid. arithmetic mean(a, b) bisects the sides of the trapezoid. Shannon Umberger; Some "Mean" Trapezoids, EMAT 6690 Essays (Summer 2001) Essay # 3 http://jwilson.coe.uga.edu/EMT668/EMAT6680.2000/Umberger/Shannon.html Stephen R. Wassell; Rediscovering a Family of Means, The Mathematical Intelligenzer 24:2 (2002) 58-65 Dieter Wickmann; Mittelwert: Ein Beispiel aus der Sprachforschung, Mathematik Lehren (Themenheft Mittelwerte) (Feb. 1985) (no. 8) 58-59 wikipedia.de Ungleichung vom arithmetischen und geometrischen Mittel, http://de.wikipedia.org/wiki/Ungleichung_vom_arithmetischen_und_geometrischen_Mittel Umordnungs-Ungleichung, http://de.wikipedia.org/wiki/Umordnungs-Ungleichung Herbert S. Wilf; Some examples of combinatorial averaging, American Mathematical Monthly 92:4 (1985) 250-261 MR 87b:05014 Section 6: The hierachy of average powers. - The Power-Mean Theorem. Heinrich Winter; Mittelwerte - eine grundlegende mathematische Idee, Mathematik Lehren (Themenheft Mittelwerte) (Feb. 1985) (no. 8) 4-14 Heinrich Winter; Dreieck und Dreiklang - woher das harmonische Mittel seinen Namen hat, Mathematik Lehren (Themenheft Mittelwerte) (Feb. 1985) (no. 8) 48 Alfred Witkowski; A New Proof of the Monotonicity of Power Means, Journal of Inequalities in Pure and Applied Mathematics, 5:1 (2004) article 6 http://jipam.vu.edu.au/article.php?sid=358 Peter Y. Woo; Crux Mathematicorum, Problem 2730 Let AM(x_1,x_2,...,x_n) and GM(x_1,x_2,...,x_n) denote the arithmetic mean and the geometric mean of the real numbers x_1,x_2,...,x_n respectively. Given positive real numbers a_1,a_2,...,a_n, b_1,b_2,...,b_n, prove that (a) GM(a_1+b_1, a_2+b_2, ..., a_n+b_n) >= GM(a_1,a_2,...,a_n) + GM(b_1,b_2,...,b_n). For each real number t>=0, define f(t) = GM(t+b_1,t+b_2,...,t+b_n) - t. (b) Prove that f(t) is a monotonic increasing function of t, and that lim_{t->oo} f(t) = AM(b_1,b_2,...,b_n). Zhen-Gang Xiao, Zhi-Hua Zhang; The Inequalities G <= L <= I <= A in n Variables, Journal of Inequalities in Pure and Applied Mathematics, 4:2 (2003) article 39 http://jipam.vu.edu.au/article.php?sid=277 Elmar Zemgalis, Another Proof of the Arithmetic-Geometric Mean Inequality, College Mathematics Journal 10:2 (1979) 112-113 Klamkin, M.S; Elementary approximations to the area of n-dimensional ellipsoids, American Mathematical Monthly 78 (1971) 280-283, Zbl 0208.32503 The purpose of this note is to present an elementary self-contained derivation of sharp bounds for the perimeter of an ellipse and the surface area of an ellipsoid. The method can be easily extended for ellipsoids of higher dimension. Pi(a+b) <= P <= Pi Sqrt[2(a^2+b^2)]; Klamkin, M.S; Corrections to 'Elementary approximations to the area of n-dimensional ellipsoids', American Mathematical Monthly 83 (1976) 478, Zbl 0325.26009 Richard E. Pfiefer; Bounds on the Perimeter of an Ellipse via Minkowski Sums, College Mathematics Journal 19:4 (1988) 348-350 Geographische Mitte: -------------------- www.mittelpunkt-deutschlands.de http://www.mittelpunkt-deutschlands.de/ David Austin; The Center of Population of the United States, Feature Column from the AMS, http://www.ams.org/featurecolumn/archive/population-center.html Andreas Engel; Emsige Suche nach der deutschen Mitte, Mathematik Lehren (Themenheft Mittelwerte) (Feb. 1985) (no. 8) 60 - Herbstein ist Schwerpunkt der Karte der Bundesrepublik. J. F. Hayford; What is the Center of an Area, or the Center of a Population? Publications of the American Statistical Association, 8 (58): 47 - 58, 1902. Die Mitte (The Center); Ein Film von Stanislaw Mucha, http://strandfilm.com/ Wo befindet sich die geographische Mitte Europas? Regisseur Stanislaw Mucha begibt sich auf eine kurzweilige, manchmal burleske, manchmal tragikomische Odyssee auf der Suche nach der einzigen, der "wahren Mitte" Europas... - - - RGMIA; Journal of Inequalities in Pure and Applied Mathematics, Inequalities for Means, http://rgmia.vu.edu.au/InequalitiesMeans.htm -- mailto:Torsten.Sillke@uni-bielefeld.de http://www.mathematik.uni-bielefeld.de/~sillke/