A Sharp Lower Bound for Toader-Qi Mean with Applications

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Open Access

Publisher website

1 January 2016

journal article
research article
Published by Wiley in Journal of Function Spaces

Vol. 2016, 1-5
DOI:10.1155/2016/4165601

Abstract

We prove that the inequality

T Q (a, b) > L_{p} (a, b)

holds for all

a, b > 0

with

a \neq b

if and only if

p \leq 3 / 2

, where

T Q (a, b) = (2 / π) \int_{0}^{π / 2} a^{{c o s}^{2} θ} b^{{s i n}^{2} θ} d θ

,

L_{p} (a, b) = [(b^{p} - a^{p}) / (p (b - a))]^{1 / p} (p \neq 0)

, and

L_{0} (a, b) = \sqrt{a b}

are, respectively, the Toader-Qi and

p

-order logarithmic means of

a

and

b

. As applications, we find two fine inequalities chains for certain bivariate means.

Keywords

Funding Information

National Natural Science Foundation of China (61374086, LY13A010004)
Natural Science Foundation of Zhejiang Province (61374086, LY13A010004)

This publication has 12 references indexed in Scilit:

A double inequality for an integral mean in terms of the exponential and logarithmic means
Periodica Mathematica Hungarica, 2016
Sharp bounds for the arithmetic-geometric mean
Journal of Inequalities and Applications, 2014
Sharp bounds for Seiffert and Neuman-Sándor means in terms of generalized logarithmic means
Journal of Inequalities and Applications, 2013
Bounds for the Ratio of Two Gamma Functions
Journal of Inequalities and Applications, 2010
Studying Baskakov–Durrmeyer operators and quasi-interpolants via special functions
Journal of Approximation Theory, 2007
Some Mean Values Related to the Arithmetic–Geometric Mean
Journal of Mathematical Analysis and Applications, 1998
Inequalities for Compound Mean Iterations with Logarithmic Asymptotes
Journal of Mathematical Analysis and Applications, 1993
Inequality of the AGM and the Logarithmic Mean
SIAM Review, 1991
On the identric and logarithmic means
Aequationes mathematicae, 1990
Ungleichungen f r Mittelwerte
Archiv der Mathematik, 1986