Sharp Cusa and Becker-Stark inequalities
© Chen and Cheung; licensee Springer. 2011
Received: 8 June 2011
Accepted: 7 December 2011
Published: 7 December 2011
We determine the best possible constants θ,ϑ,α and β such that the inequalities
are valid for 0 < × < π/ 2. Our results sharpen inequalities presented by Cusa, Becker and Stark.
Mathematics Subject Classification (2000): 26D05.
KeywordsInequalities trigonometric functions
Inequality (1) was first mentioned by the German philosopher and theologian Nicolaus de Cusa (1401-1464), by a geometrical method. A rigorous proof of inequality (1) was given by Huygens , who used (1) to estimate the number π. The inequality is now known as Cusa's inequality [2–5]. Further interesting historical facts about the inequality (1) can be found in .
It is the first aim of present paper to establish sharp Cusa's inequality.
The constant 8 and π2 are the best possible.
Zhu and Hua  established a general refinement of the Becker-Stark inequalities by using the power series expansion of the tangent function via Bernoulli numbers and the property of a function involving Riemann's zeta one. Zhu  extended the tangent function to Bessel functions.
It is the second aim of present paper to establish sharp Becker-Stark inequality.
Remark 1. There is no strict comparison between the two lower bounds and
in (3) and (4).
The following lemma is needed in our present investigation.
If f'(x) = g'(x) is strictly monotone, then the monotonicity in the conclusion is also strict.
2. Proofs of Theorems 1 and 2
and therefore, the functions F5(x) and are both strictly increasing on (0, π/2).
By rearranging terms in the last expression, Theorem 1 follows.
By rearranging terms in the last expression, Theorem 2 follows.
Research is supported in part by the Research Grants Council of the Hong Kong SAR, Project No. HKU7016/07P.
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