Abstract
Keywords:
Inequalities; trigonometric functions1. Introduction
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 (14011464), 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 [25]. 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 π^{2 }are the best possible.
Zhu and Hua [7] established a general refinement of the BeckerStark 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 BeckerStark 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
The following lemma is needed in our present investigation.
Lemma 1 ([911]). 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.
2. Proofs of Theorems 1 and 2
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 F_{5}(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.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors read and approved the final manuscript
Acknowledgements
Research is supported in part by the Research Grants Council of the Hong Kong SAR, Project No. HKU7016/07P.
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