New inequalities of Wilker's type for hyperbolic functions

1.   Introduction

Wilker ^[1] proposed two open problems as the following statements:

(a) If $0 < x < \pi /2$, then

(b) There exists a largest constant $c$ such that

for $0 < x < \pi /2$.

Sumner et al. ^[2] affirmed the truth of two problems above and obtained a further results as follows

where $16/{\pi }^{4}$ and $8/45$ are the best constants in ${(1.3)}$.

Some refreshing proofs of inequalities ${(1.1)}$ and ${(1.3)}$ can be found in Pinelis ^[3]. In 2007, the author of this paper ^[4] established a new Wilker-type inequality involving hyperbolic functions as follows:

where $8/45$ can not be replaced by any larger number.

In 2011, Sun and Zhu ^[5] showed another new Wilker-type inequality involving hyperbolic functions and obtained the following result.

where $2/45$ can not be replaced by any smaller number.

The purpose of this paper is to give three new inequalities of Wilker-type for hyperbolic functions.

Theorem 1. Let $x\neq 0$. Then

holds if and only if $\ \alpha \leq 8/45$.

Theorem 2. The function

has only one maximum point $x_{0} = 1.54471\ldots$ on $\left( 0, \infty \right)$, so the function $G(x)$ has the maximum value

Specifically, for $x\neq 0,$

holds if and only if $\theta \geq \theta _{0}$.

Theorem 3. Let $x\neq 0$. Then

holds if and only if $\ \beta \geq 2/45.$

2.   A lemma

In order to prove Theorem $2$, we need the following lemma. We introduce a useful auxiliary function $H_{f, g}$. For $-\infty \leq a < b\leq \infty$, let $f$ and $g$ be differentiable on $(a, b)$ and $g^{\prime }\neq 0$ on $(a, b)$. Then the function $H_{f, g}$ is defined by

The function $H_{f, g}$ has some well properties and play an important role in the proof of a monotonicity criterion for the quotient of power series.

Lemma 1. (^[6]) Let $f\left(t\right) = \sum_{k = 0}^{\infty }a_{k}t^{k}$ and $g\left(t\right) = \sum_{k = 0}^{\infty }b_{k}t^{k}$ be two real power series converging on $\left(-r, r\right)$ and $b_{k} > 0$ for all $k$. Suppose that for certain $m\in \mathbb{N}$, the non-constant sequence $\left\{ a_{k}/b_{k}\right\}$ is increasing (resp. decreasing) for $0\leq k\leq m$ and decreasing (resp. increasing) for $k\geq m$. Then the function $f/g$ is strictly increasing (resp. decreasing) on $\left(0, r\right)$ if and only if $H_{f, g}\left(r^{-}\right) \geq 0$ (resp. $\leq 0$). Moreover, if $H_{f, g}\left(r^{-}\right) < 0$ (resp. $> 0$), then there exists $t_{0}\in \left(0, r\right)$ such that the function $f/g$ is strictly increasing (resp. decreasing) on $\left( 0, t_{0}\right)$ and strictly decreasing (resp. increasing) on $\left(t_{0}, r\right)$.

3.   Proofs of Theorems 1–3

3.1.   Proof of Theorem 1

Since

we can let

Then

where

By substituting the power series expansions of all hyperbolic functions involved in the above formula into $h(x)$, we obtain that

where

Since $l\left(5\right) = 77\, 856\, 768 > 0$ and for $n\geq 6,$

we have that $l\left(n\right) > 0$ for $n\geq 5$. So $h(x) > 0$ and $F^{\prime }(x) > 0$ for $x > 0$. Then the function $F(x)$ is strictly increasing on $\left(0, \infty \right)$. Therefore, $F(x) > {F(0^{+})} = 0$. At the same time, we find that

Then the proof of Theorem 1 is complete.

3.2.   Proof of Theorem 2

Let

Rewrite $G(x)$ as

By substituting the power series expansions of the hyperbolic functions involved in the above formula into $f(x)$ and $g(x)$ we have

and

Let

We can calculate that

and show that $\{c_{n}\}_{n\geq 4}$ is decreasing:

In fact,

where

Since for $n\geq 4$, we have $108n^{2}-189n-324\geq 648 > 0$. By mathematical induction it is easy to prove that

holds for $n\geq 4$. So $p(n) > 0$ for $n\geq 4$. This leads to $\{c_{n}\}_{n\geq 4}$ is decreasing. Therefore

We compute to get

By Lemma 1 we obtain that there exists $x_{0}\in \left(0, \infty \right)$ such that the function $G(x) = f/g$ is strictly increasing on $\left(0, x_{0}\right)$ and strictly decreasing on $\left(x_{0}, \infty \right)$. That is, $x_{0}$ is the only maximum point of the function $G(x)$ on $\left(0, \infty \right)$. Let us determine the maximum point $x_{0}$ and the maximum value $G(x_{0})$. We compute

where

We find out that

So $x_{0} = 1.54471\ldots$ and

Considering the reasons

we have

The proof of Theorem 2 is completed.

3.3.   Proof of Theorem 3

Let

where $0 < x < \infty.$ Then

where

By substituting the power series expansions of the hyperbolic functions involved in the above formula into $q(x)$, we have

where

We can find $d_{5} = \ 13\, 516\, 800 > 0$ and $d_{n} > 0$ holds for $n\geq 6$ due to

So $H^{\prime }(x) >$ $0$. Then $H(x)$ is increasing on $(0, {\infty })$. Therefore, $H(x) > {H(0^{+})} = 0$. At the same time, we find that

Then the proof of Theorem 3 is complete.

Acknowledgments

This paper is supported by the Natural Science Foundation of China grants No.61772025.

Conflict of interest

The author declares no conflict of interest in this paper.