Optimal inequalities for bounding Toader mean by arithmetic and quadratic means

Tie Hong Zhao; Yu Ming Chu; Wen Zhang

doi:10.1186/s13660-017-1300-8

Optimal inequalities for bounding Toader mean by arithmetic and quadratic means

Tie Hong Zhao, Yu Ming Chu, Wen Zhang

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

In this paper, we present the best possible parameters α(r) and β(r) such that the double inequality (Formula presented.) holds for all r ≤ 1 and a, b > 0 with a ≠ b, and we provide new bounds for the complete elliptic integral (Formula presented.) of the second kind, where (Formula presented.) and (Formula presented.) are the Toader, arithmetic, and quadratic means of a and b, respectively.

Original language	English
Article number	26
Journal	Journal of Inequalities and Applications
Volume	2017
Issue number	1
DOIs	https://doi.org/10.1186/s13660-017-1300-8
State	Published - 1 Dec 2017
Externally published	Yes

Keywords

Toader mean
arithmetic mean
complete elliptic integral
quadratic mean

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10.1186/s13660-017-1300-8

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N2 - In this paper, we present the best possible parameters α(r) and β(r) such that the double inequality (Formula presented.) holds for all r ≤ 1 and a, b > 0 with a ≠ b, and we provide new bounds for the complete elliptic integral (Formula presented.) of the second kind, where (Formula presented.) and (Formula presented.) are the Toader, arithmetic, and quadratic means of a and b, respectively.

AB - In this paper, we present the best possible parameters α(r) and β(r) such that the double inequality (Formula presented.) holds for all r ≤ 1 and a, b > 0 with a ≠ b, and we provide new bounds for the complete elliptic integral (Formula presented.) of the second kind, where (Formula presented.) and (Formula presented.) are the Toader, arithmetic, and quadratic means of a and b, respectively.

KW - Toader mean

KW - arithmetic mean

KW - complete elliptic integral

KW - quadratic mean

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Optimal inequalities for bounding Toader mean by arithmetic and quadratic means

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Keywords

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