Sharp Cusa and Becker-Stark inequalities

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

and

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

Mathematics Subject Classification (2000): 26D05.

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

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,

with the best possible constants

Becker and Stark [6] obtained the inequalities

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,

with the best possible constants

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.

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

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

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

For 0 < x < π/2, let

Then,

where

Differentiating with respect to x yields

Elementary calculations reveal that

where

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

where

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

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,

and therefore, the functions F₅(x) and are both strictly increasing on (0, π/2).

By Lemma 1, the function

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

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

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

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

For 0 < x < π/2, let

Then,

Elementary calculations reveal that

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

Where λ and μ are constants determined with limits:

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

which has a bound of absolute error

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

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

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

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

The authors declare that they have no competing interests.

All authors read and approved the final manuscript

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

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