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Journal of Inequalities and Applications

Erschienen in:

Open Access 01.12.2011 | Research

Sharp Cusa and Becker-Stark inequalities

verfasst von: Chao-Ping Chen, Wing-Sum Cheung

Erschienen in: Journal of Inequalities and Applications | Ausgabe 1/2011

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Abstract

We determine the best possible constants θ,ϑ,α and β such that the inequalities

{(\frac{2 + cos x}{3})}^{θ} < \frac{sin x}{x} < {(\frac{2 + cos x}{3})}^{ϑ}

and

{(\frac{π^{2}}{π^{2} - 4 x^{2}})}^{α} < \frac{tan x}{x} < {(\frac{π^{2}}{π^{2} - 4 x^{2}})}^{β}

are valid for 0 < × < π/ 2. Our results sharpen inequalities presented by Cusa, Becker and Stark.

Mathematics Subject Classification (2000): 26D05.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors read and approved the final manuscript

1. Introduction

For 0 < × < π/ 2, it is known in the literature that

\frac{sin x}{x} < \frac{2 + cos x}{3} .

(1)

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 [1], 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 [2].

It is the first aim of present paper to establish sharp Cusa's inequality.

Theorem 1. For 0 < × < π/ 2,

{(\frac{2 + cos x}{3})}^{θ} < \frac{sin x}{x} < {(\frac{2 + cos x}{3})}^{ϑ}

(2)

with the best possible constants

θ = \frac{ln (π ∕ 2)}{ln (3 ∕ 2)} = 1.11373998 \dots and ϑ = 1 .

Becker and Stark [6] obtained the inequalities

\frac{8}{π^{2} - 4 x^{2}} < \frac{tan x}{x} < \frac{π^{2}}{π^{2} - 4 x^{2}} (0 < x < \frac{π}{2}) .

(3)

The constant 8 and π² are the best possible.

Zhu and Hua [7] 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 [8] extended the tangent function to Bessel functions.

It is the second aim of present paper to establish sharp Becker-Stark inequality.

Theorem 2. For 0 < × < π/ 2,

{(\frac{π^{2}}{π^{2} - 4 x^{2}})}^{α} < \frac{tan x}{x} < {(\frac{π^{2}}{π^{2} - 4 x^{2}})}^{β}

(4)

with the best possible constants

α = \frac{π^{2}}{12} = 0.822467033 \dots and β = 1 .

Remark 1. There is no strict comparison between the two lower bounds

\frac{8}{π^{2} - 4 x^{2}}

and

{(\frac{π^{2}}{π^{2} - 4 x^{2}})}^{π^{2} ∕ 12}

in (3) and (4).

The following lemma is needed in our present investigation.

Lemma 1 ([9‐11]). Let - ∞ < a < b < ∞, and f, g : [a, b] → ℝ be continuous on [a, b] and differentiable in (a, b). Suppose g' ≠ 0 on (a; b). If f'(x)/g' (x) is increasing (decreasing) on (a, b), then so are

[f (x) - f (a)] ∕ [g (x) - g (a)] and [f (x) - f (b)] ∕ [g (x) - g (b)] .

If f'(x) = g'(x) is strictly monotone, then the monotonicity in the conclusion is also strict.

2. Proofs of Theorems 1 and 2

Proof of Theorem [1]. Consider the function f(x) defined by

\begin{gathered} F (x) = \frac{ln (\frac{sin x}{x})}{ln (\frac{2 + cos x}{3})}, 0 < x < \frac{π}{2}, \\ F (0) = 1 and F (\frac{π}{2}) = \frac{ln(π /2)}{ln(3/2)} . \end{gathered}

For 0 < x < π/2, let

F_{1} (x) = ln (\frac{sin x}{x}) and F_{2} (x) = ln (\frac{2 + cos x}{3}) .

Then,

\frac{{F^{'}}_{1} (x)}{{F^{'}}_{2} (x)} = \frac{- 2 x cos x - x {cos}^{2} x + 2 sin x + sin x cos x}{x {sin}^{2} x} = \frac{F_{3} (x)}{F_{4} (x)},

where

F_{3} (x) = - 2 x cos x - x {cos}^{2} x + 2 sin x + sin x cos x and F_{4} (x) = x {sin}^{2} x .

Differentiating with respect to x yields

\frac{{F^{'}}_{3} (x)}{{F^{'}}_{4} (x)} = \frac{2 x + 2 x cos x - sin x}{sin x + 2 x cos x} ≜ F_{5} (x) .

Elementary calculations reveal that

F_{5}^{'} (x) = \frac{2 F_{6} (x)}{2 x sin (2 x) + 4 x^{2} {cos}^{2} x + {sin}^{2} x},

where

F_{6} (x) = sin (2 x) + (2 x^{2} + 1) sin x - 2 x - x cos x .

By using the power series expansions of sine and cosine functions, we find that

F_{6} (x) = x^{3} - \frac{1}{10} x^{5} - \frac{19}{2520} x^{7} + 2 \sum_{n = 4}^{\infty} {(- 1)}^{n} u_{n} (x),

where

u_{n} (x) = \frac{4^{n} - 4 n^{2} - 3 n}{(2 n + 1)!} x^{2 n + 1} .

Elementary calculations reveal that, for 0 < × < π/ 2 and n ≥ 4,

\begin{aligned} \frac{u_{n + 1} (x)}{u_{n} (x)} & = \frac{x^{2}}{2} \frac{2^{2 n + 2} - 4 n^{2} - 11 n - 7}{(n + 1) (2 n + 3) (4^{n} - 4 n^{2} - 3 n)} \\ < \frac{1}{2} {(\frac{π}{2})}^{2} \frac{2^{2 n + 2} - 4 n^{2} - 11 n - 7}{(n + 1) (2 n + 3) (4^{n} - 4 n^{2} - 3 n)} \\ = \frac{π^{2}}{8 (n + 1)} \frac{4^{n + 1} - 4 n^{2} - 11 n - 7}{(2 n + 3) (4^{n} - 4 n^{2} - 3 n)} \\ < \frac{π^{2}}{8 (n + 1)} < 1 . \end{aligned}

Hence, for fixed x ∈ (0, π/ 2), the sequence n↦ u_n(x) is strictly decreasing with regard to n ≥ 4. Hence, for 0 < × < π/2,

F_{6} (x) = x^{3} - \frac{1}{10} x^{5} - \frac{19}{2520} x^{7} > 0 (0 < x < \frac{π}{2}),

and therefore, the functions F₅(x) and

\frac{{F^{'}}_{3} (x)}{{F^{'}}_{4} (x)}

are both strictly increasing on (0, π/2).

By Lemma 1, the function

\frac{{F^{'}}_{1} (x)}{{F^{'}}_{2} (x)} = \frac{F_{3} (x)}{F_{4} (x)} = \frac{F_{3} (x) - F_{3} (0)}{F_{4} (x) - F_{4} (0)}

is strictly increasing on (0, π/2). By Lemma 1, the function

F (x) = \frac{F_{1} (x)}{F_{2} (x)} = \frac{F_{1} (x) - F_{1} (0)}{F_{2} (x) - F (0)}

is strictly increasing on (0, π/2), and we have

1 = F (0) < F (x) = \frac{ln (\frac{sin x}{x})}{ln (\frac{2 + cos x}{3})} < F (\frac{π}{2}) = \frac{ln (π ∕ 2)}{ln (3 ∕ 2)} \forall x \in (0, \frac{π}{2}) .

By rearranging terms in the last expression, Theorem 1 follows.

Proof of Theorem 2. Consider the function f(x) defined by

\begin{align} f (x) & = \frac{ln (\frac{tan x}{x})}{ln (\frac{π^{2}}{π^{2} - 4 x^{2}})}, 0 < x < \frac{π}{2}, \\ f (0) & = \frac{π^{2}}{12} and f (\frac{π}{2}) = 1 . \end{align}

For 0 < x < π/2, let

f_{1} (x) = ln (\frac{tan x}{x}) and f_{2} (x) = ln (\frac{π^{2}}{π^{2} - 4 x^{2}}) .

Then,

\frac{{f^{'}}_{1} (x)}{{f^{'}}_{2} (x)} = \frac{(π^{2} - 4 x^{2}) (2 x - sin (2 x))}{8 x^{2} sin (2 x)} ≜ g (x) .

Elementary calculations reveal that

4 x^{3} {sin}^{2} (2 x) g^{'} (x) = - (π^{2} + 4 x^{2}) x sin (2 x) - 2 (π^{2} - 4 x^{2}) x^{2} cos (2 x) + π^{2} {sin}^{2} (2 x) ≜ h (x) .

Motivated by the investigations in [12], we are in a position to prove h(x) > 0 for x ∈ (0, π/2).Let

H (x) = \{\begin{matrix} λ, & x = 0, \\ \frac{h (x)}{x^{6} {(\frac{π}{2} - x)}^{2}} & 0 < x < \frac{π}{2}, \\ μ, & x = \frac{π}{2}, \end{matrix}

Where λ and μ are constants determined with limits:

\begin{aligned} λ & = lim_{x \to 0^{+}} \frac{h (x)}{x^{6} {(\frac{π}{2} - x)}^{2}} = \frac{224 π^{2} - 1920}{45 π^{2}} = 0.654740609 . . ., \\ μ & = lim_{t \to {(π ∕ 2)}^{-}} \frac{h (x)}{x^{6} {(\frac{π}{2} - x)}^{2}} = \frac{128}{π^{4}} = 1.31404572 . . . . \end{aligned}

Using Maple, we determine Taylor approximation for the function H(x) by the polynomial of the first order:

P_{1} (x) = \frac{32 (7 π^{2} - 60)}{45 π^{2}} + \frac{128 (7 π^{2} - 60)}{45 π^{3}} x,

which has a bound of absolute error

ε_{1} = \frac{- 1920 - 1920 π^{2} + 224 π^{4}}{15 π^{4}} = 0.650176097 . . .

for values x ∈ [0,π/2]. It is true that

H (x) - (P_{1} (x) - ℰ_{1}) \geq 0, P_{1} (x) - ℰ_{1} = \frac{64 (60 π^{2} + 90 - 7 π^{4})}{45 π^{4}} + \frac{128 (7 π^{2} - 60)}{45 π^{3}} x > 0

for x ∈ [0, π/ 2]. Hence, for x ∈ [0, π/ 2], it is true that H (x) > 0 and, therefore, h (x) > 0 and g'(x) > 0 for x ∈ [0, π/ 2]. Therefore, the function

\frac{{f^{'}}_{1} (x)}{{f^{'}}_{2} (x)}

is strictly increasing on. (0, π/ 2).By Lemma 1, the function

f (x) = \frac{f_{1} (x)}{f_{2} (x)}

is strictly increasing on (0, π/ 2), and we have

\frac{π^{2}}{12} = f (0) < f (x) = \frac{ln (\frac{tan x}{x})}{ln (\frac{π^{2}}{π^{2} - 4 x^{2}})} < f (\frac{π}{2}) = 1 .

By rearranging terms in the last expression, Theorem 2 follows.

Acknowledgements

Research is supported in part by the Research Grants Council of the Hong Kong SAR, Project No. HKU7016/07P.

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors read and approved the final manuscript

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Titel: Sharp Cusa and Becker-Stark inequalities
verfasst von: Chao-Ping Chen
Wing-Sum Cheung
Publikationsdatum: 01.12.2011
Verlag: Springer International Publishing
Erschienen in: Journal of Inequalities and Applications / Ausgabe 1/2011
Elektronische ISSN: 1029-242X
DOI: https://doi.org/10.1186/1029-242X-2011-136

Springer Professional

Abstract

Competing interests

Authors' contributions

1. Introduction

2. Proofs of Theorems 1 and 2

Acknowledgements

Competing interests

Authors' contributions

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