Sharp bounds for the lemniscatic mean by the weighted Hölder mean

Zhao, Tie-Hong; Wang, Miao-Kun

doi:10.1007/s13398-023-01429-3

Cited by 2 publications

(2 citation statements)

References 23 publications

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“…(iii) In the case of p ∈ (1/3, 1), since ν(x) = p − 1 + Φ tan (x) is increasing on (0, 1) with ν(0 + ) = p − 1 < 0 and ν(1 − ) = p − 1/3 > 0, there is an x 0 ∈ (0, 1) such that ν(x) < 0 for x ∈ (0, x 0 ) and ν(x) > 0 for x ∈ (x 0 , 1). This, by (52), implies that U tan /V > 0 for…”

Section: Proofs Of Theorems 1 Andmentioning

confidence: 91%

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Sharp Power Mean Bounds for Two Seiffert-like Means

Yang,

Zhang

2023

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The mean is a subject of extensive study among scholars, and the pursuit of optimal power mean bounds is a highly active field. This article begins with a concise overview of recent advancements in this area, focusing specifically on Seiffert-like means. We establish sharp power mean bounds for two Seiffert-like means, including the introduction and establishment of the best asymmetric mean bounds for symmetric means. Additionally, we explore the practical applications of these findings by extending several intriguing chains of inequalities that involve more than ten means. This comprehensive analysis provides a deeper understanding of the relationships and properties of these means.

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Section: Proofs Of Theorems 1 Andmentioning

confidence: 91%

“…Propositions 3 and 4 are very efficient to study for certain special functions, see for example [43][44][45][46][47][48][49][50][51][52].…”

Section: Preliminaries 21 Toolsmentioning

confidence: 99%

Sharp Power Mean Bounds for Two Seiffert-like Means

Yang,

Zhang

2023

Axioms

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New refinements of Becker-Stark inequality

Wang,

Zhao

2024

MATH

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<abstract>This paper deals with the well-known Becker-Stark inequality. By using variable replacement from the viewpoint of hypergeometric functions, we provide a new and general refinement of Becker-Stark inequality. As a particular case, the double inequality <disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \frac{\pi^2-(\pi^2-8)\sin^2x}{\pi^2-4x^2}<\frac{\tan x}{x}<\frac{\pi^2-(4-\pi^2/3)\sin^2x}{\pi^2-4x^2} \end{equation*} $\end{document} </tex-math></disp-formula> for $ x\in(0, \pi/2) $ will be established. The importance of our result is not only to provide some refinements preserving the structure of Becker-Stark inequality but also that the method can be extended to the case of generalized trigonometric functions.</abstract>

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Sharp bounds for the lemniscatic mean by the weighted Hölder mean

Cited by 2 publications

References 23 publications

Sharp Power Mean Bounds for Two Seiffert-like Means

Sharp Power Mean Bounds for Two Seiffert-like Means

New refinements of Becker-Stark inequality

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