Results 31 to 40 of about 5,576,128 (87)

Sharp One‐Parameter Mean Bounds for Yang Mean

Mathematical Problems in Engineering, Volume 2016, Issue 1, 2016., 2016
We prove that the double inequality Jα(a, b) < U(a, b) < Jβ(a, b) holds for all a, b > 0 with a ≠ b if and only if α≤2/(π-2)=0.8187⋯ and β ≥ 3/2, where U(a,b)=(a-b)/[2arctan⁡((a-b)/2ab)], and Jp(a, b) = p(ap+1 − bp+1)/[(p + 1)(ap − bp)] (p ≠ 0, −1), J0(a, b) = (a − b)/(log⁡a − log⁡b), and J−1(a, b) = ab(log⁡a − log⁡b)/(a − b) are the Yang and pth one ...
Wei-Mao Qian, Yu-Ming Chu, Xiao-Hui Zhang, Yann Favennec   +3 more
wiley   +1 more source

Two inequalities for means

International Journal of Mathematics and Mathematical Sciences, 1995
We prove two new inequalities for the identric mean and a mean related to the arithmetic and geometric mean of two numbers.
J. Sandor
doaj   +1 more source

A Sharp Lower Bound for Toader‐Qi Mean with Applications

Journal of Function Spaces, Volume 2016, Issue 1, 2016., 2016
We prove that the inequality TQ(a, b) > Lp(a, b) holds for all a, b > 0 with a ≠ b if and only if p ≤ 3/2, where TQ(a,b)=2/π∫0π/2acos2θbsin2θdθ, Lp(a, b) = [(bp − ap)/(p(b − a))] 1/p (p ≠ 0), and L0(a,b)=ab are, respectively, the Toader‐Qi and p‐order logarithmic means of a and b.
Zhen-Hang Yang, Yu-Ming Chu, Kehe Zhu
wiley   +1 more source

Best Possible Bounds for Yang Mean Using Generalized Logarithmic Mean

Mathematical Problems in Engineering, Volume 2016, Issue 1, 2016., 2016
We prove that the double inequality Lp(a, b) < U(a, b) < Lq(a, b) holds for all a, b > 0 with a ≠ b if and only if p ≤ p0 and q ≥ 2 and find several sharp inequalities involving the trigonometric, hyperbolic, and inverse trigonometric functions, where p0 = 0.5451⋯ is the unique solution of the equation (p+12) 1/p=2π/ on the interval (0, ∞), U(a,b)=(a-b)
Wei-Mao Qian, Yu-Ming Chu, Kishin Sadarangani   +2 more
wiley   +1 more source

The geometric mean is a Bernstein function [PDF]

, 2013
In the paper, the authors establish, by using Cauchy integral formula in the theory of complex functions, an integral representation for the geometric mean of $n$ positive numbers.
Li, Wen-Hui, Qi, Feng, Zhang, Xiao-Jing
core   +2 more sources

A simple proof of inequalities related to means [PDF]

, 2009
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Gou-Sheng Yang, Shuoh-Jung Liu
core   +2 more sources

Sharp Power Mean Bounds for the One‐Parameter Harmonic Mean

Journal of Function Spaces, Volume 2015, Issue 1, 2015., 2015
We present the best possible parameters α = α(r) and β = β(r) such that the double inequality Mα(a, b) < Hr(a, b) < Mβ(a, b) holds for all r ∈ (0, 1/2) and a, b > 0 with a ≠ b, where Mp(a, b) = [(ap + bp)/2] 1/p (p ≠ 0) and M0(a, b)=ab and Hr(a, b) = 2[ra + (1 − r)b][rb + (1 − r)a]/(a + b) are the power and one‐parameter harmonic means of a and b ...
Yu-Ming Chu, Li-Min Wu, Ying-Qing Song, David R. Larson   +3 more
wiley   +1 more source

A Survey on Operator Monotonicity, Operator Convexity, and Operator Means

International Journal of Analysis, Volume 2015, Issue 1, 2015., 2015
This paper is an expository devoted to an important class of real‐valued functions introduced by Löwner, namely, operator monotone functions. This concept is closely related to operator convex/concave functions. Various characterizations for such functions are given from the viewpoint of differential analysis in terms of matrix of divided differences ...
Pattrawut Chansangiam, Julien Salomon
wiley   +1 more source

On a Hierarchy of Means [PDF]

, 2013
For a class of partially ordered means we introduce a notion of the (nontrivial) cancelling mean. A simple method is given which helps to determine cancelling means for well known classes of Holder and Stolarsky ...
Simic, Slavko
core   +2 more sources