New refinements of Becker-Stark inequality

Suxia Wang; Tiehong Zhao; Suxia Wang; Tiehong Zhao

doi:10.3934/math.2024960

AIMS Mathematics

2024, Volume 9, Issue 7: 19677-19691. doi: 10.3934/math.2024960

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New refinements of Becker-Stark inequality

Suxia Wang ¹,
Tiehong Zhao ^{2
,
,}

1.
School of Finance and Mathematics, Huainan Normal University, Huainan 232038, China
2.
School of Mathematics, Hangzhou Normal University, Hangzhou 311121, China

Received: 28 April 2024 Revised: 05 June 2024 Accepted: 11 June 2024 Published: 14 June 2024
MSC : 26D05, 26D15, 33B10

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

$ \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*} $

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.
- Becker-Stark inequality,
- power series,
- Gaussian hypergeometric function,
- monotonicity
Citation: Suxia Wang, Tiehong Zhao. New refinements of Becker-Stark inequality[J]. AIMS Mathematics, 2024, 9(7): 19677-19691. doi: 10.3934/math.2024960

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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

$ \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*} $

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.

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