In this paper, we present and prove a new Wilker-type inequality for hyperbolic functions. We also give a simple proof of the counterpart of the above inequality for the circular functions.
1. Introduction
Wilker [1] proposed two open questions as the following statements.
Problem 1. If , then
Problem 2. There exists a largest constant such that
for .
Sumner et al. [2] affirmed the truth of the problems above and obtained a further result as follows.
Theorem 1.1. If , then
Furthermore, and are the best constants in (1.3).
Guo et al. [3] gave new proofs of inequalities (1.1) and (1.2). In recent years, Zhu [4] showed a new simple proof of inequality (1.1); Pinelis [5] got other proof of inequalities (1.3) by using L'Hospital rules for monotonicity; Zhang and Zhu [6] gave a new elementary proof of double inequalities (1.3).
Besides, in article [7], Wang offered another type of the inequality and proved it by the power series expansions of circular functions as follows
Theorem 1.2. Let . Then,
Furthermore, and are the best constants in (1.4).
Zhu [8] established a new Wilker-type inequality involving hyperbolic functions as follows:
where the constant is the optimum constant in (1.5).
In fact, we can show a new Wilker-type inequality involving hyperbolic functions referring to the result above.
Theorem 1.3. Let . Then
Furthermore, is the best constant in (1.6).
The purpose of this paper is to present a concise proof of inequality (1.6) by using the power series expansion of hyperbolic functions and to give an elementary proof of inequality (1.4) in another way.
Let , where ,ββand . We get that the existence of Theorem 1.3 can be ensured when proving the following two statements: the first is or , and the second is .
Since
where , , , and , according to the analysis above, the proof of Theorem 1.3 can be completed when proving that
that is,
for .
Let for . We compute
where
Then let ; we compute , where
As we can see, all coefficients of the polynomial are positive integers. When , we have and . In view of that , we have that and for . We can also conclude that is increasing on , respectively. Since , we have for . So the proof of inequality (2.3) is completed. Furthermore, ; the proof of Theorem 1.3 is completed.
We simplify the double inequality (1.4) into another form:
Here we will discuss the monotonicity of the function . If the function is increasing for and both and are right, the proof of equality (1.4) is completed.
From [9] or [10, page 75], we know that for the equality
holds. So we can also know that for
hold. Then we compute by using the results above, where
When , we can compute by using and . Furthermore, there is an obvious fact that for . So far, we can demonstrate that for all . So the function is increasing on . Evidently, and are true. So the proof of the double inequality of (1.4) is completed.
Remark 3.1. Wilker's inequalities (1.1) and (1.2) have been further refined by many scholars in the past few years; the readers can refer to [11β16].
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