New inequalities of Wilker’s type for circular functions

1.   Introduction

Inequalities are ubiquitous in all branches of pure and applied mathematics, while the trigonometric and hyperbolic functions inequalities are the indispensable parts in the whole theory of inequality, there are numerous researchers devoted to the study of mathematical inequalities. Recently, there are many novel inequalities in different areas have been discovered, for example, the Hermite-Hadamard type inequalities ^{[1,2,3,4,5,6]}, bivariate means inequalities ^{[7,8,9,10,11]}, gamma function inequalities ^[12], complete elliptic integrals inequalities ^[13,14,15], Bessel functions inequalities ^[16], Jensen type inequalities ^[17,18,19], Ostrowski type inequalities ^[20], Cauchy-Schwarz inequality ^[21], reverse Minkowski inequality ^[22], Petrović-type inequalities ^[23], Pólya-Szegö and Ćebyšev type inequalities ^[24], delay dynamic double integral inequalities ^[25], integral majorization type inequalities ^[26], generalized convex functions inequalities ^[27], generalized proportional fractional integral operators inequalities ^[28], generalized trigonometric and hyperbolic functions inequalities ^[29], exponentially convex inequalities ^[30] and so on.

In 1989, Montgomery et al. ^[31] proposed two open problems as follows:

$(1)$ Does the inequality

holds for all $0 < x < \pi/2$?

$(2)$ Is there a largest constant $c$ such that the inequality

holds for all $0 < x < \pi/2$?

Wilker et al. ^[32] gave positive answers to the inequalities (1.1) and (1.2), and proved that the double inequality

holds for all $x\in(0, \pi/2)$ with the best constants $16/\pi^{4}$ and $8/45$.

For different proofs of inequalities (1.1) and (1.3), we recommend the literature ^[33,34] to the interested readers.

In ^[35], Zhu proved that the Wilker type inequality

for hyperbolic functions holds whenever $x\neq 0$, and the constant $8/45$ in the right of inequality (1.4) can not be replaced by any larger number.

Sun and Zhu ^[36] provided a new version of the Wilker type inequality for hyperbolic functions as follows:

for all $x\neq 0$, and the constant $2/45$ in the right of inequality (1.5) can not be replaced by any smaller number.

Very recently, Zhu ^[37] established the following novel results (Propositions 1.1–1.3) involving the Wilker type inequality for hyperbolic functions by use of the power series formulas and monotonicity criteria of the quotient of two power series ^[38].

Proposition 1.1. The inequality

takes place for all $x\neq 0$ with the best possible constant $8/45$ in the right of inequality (1.6).

Proposition 1.2. $x_{0} = 1.54471\cdots$ is the unique point on the interval $(0, \infty)$ such that the function

attain its maximum

and $\theta_{0}$ is the best possible constant such that the following inequality (1.7)

holds for all $x\in (0, \infty)$.

Proposition 1.3. $2/45$ is the best possible constant such that the inequality (1.8)

holds for all $x\neq 0$.

This main purpose of the article is to find new Wilker type inequalities for trigonometric functions. Our main results are the following Theorems 1.1–1.3.

Theorem 1.1. The inequality

is valid for all $x\in (0, \pi/2)$ with the best possible constant $8/45$ in the right of inequality (1.9).

Theorem 1.2. $\alpha = 0$ and $\beta = 2/45$ are the best possible constants such that the two-sided inequality (1.10)

takes place for all $x\in (0, \pi/2)$.

Theorem 1.3. Let $0 < x < \pi /2$. Then the double inequality

holds with the best constants $\alpha = 0$ and $\beta = 2/45$ in the left and right sides of inequality (1.11), respectively.

2.   Lemmas

In order to prove our main results, we need four lemmas which we present in this section.

Lemma 2.1. Let $\left\vert x\right\vert < \pi /2$. Then we have the following power series formulas

and

where $B_{n}$ is the Bernoulli number.

Proof. The power series formula (2.1) can be found in the literature [39, 1.3.1.4(2)]. Next, we prove the remain power series formulas.

It follows from (2.1) that

Making use of (2.1) and (2.2) we get

and

Note that

The desired power series formulas for the functions $\sec^{2}x\tan x$, $\sec^{2}x\tan^{2}x$ and $\sec^{2}x\tan^{3}x$ can be derived easily.

Lemma 2.2. (See ^[40]) The power series formula

holds for all $x\in(-\pi, 0)\cup(0, \pi)$, where $B_{n}$ is the Bernoulli number.

Lemma 2.3. (See ^[41]) The double inequality

holds for $n = 1, 2, 3, \cdots$, where $B_{n}$ is the Bernoulli number.

Lemma 2.4. (See ^[38]) Let $a_{n}$ and $b_{n}$ $(n = 0, 1, 2, 3, \cdots)$ be real numbers, and the power series $A(x) = \sum_{n = 0}^{\infty }a_{n}x^{n}$ and $B(x) = \sum_{n = 0}^{\infty }b_{n}x^{n}$ be convergent for $\left\vert x\right\vert < R$ $(R\leq\infty)$. If $b_{n} > 0$ for all $n\geq 0$ and the sequence $\{a_{n}/b_{n}\}_{n = 0}^{\infty}$ is increasing (decreasing), then the function $A(x)/B(x)$ is strictly increasing (decreasing) on $(0, R)$.

3.   Proof of Theorems 1.1–1.3

Proof of Theorem 1.1. Let $0 < \left\vert x\right\vert < \pi /2$ and

Then elaborated computations lead to

where

By substituting the power series formulas in Lemma 2.1 into $f(x)$, we obtain

where

It follows from (2.4) that

which is equivalent to

where

It is easy to prove that $2^{2n} > v_{1}(n)/u_{1}(n)$ for all $n\geq 6$ by using mathematical induction. Therefore, we deduce that

$F(x)$ is strictly increasing on $(0, \pi /2)$ and $F(x) > {F(0^{+})} = 0$. Note that

Which completes the proofs of Theorem 1.1.

Proof of Theorem 1.2. Let $0 < x < \pi /2$, and

Then we clearly see that

It follows from Lemma 2.1 and inequality (2.3) that

and

Note that

Next, we prove that the sequence $\left\{a_{n}/b_{n}\right\}$ is decreasing for $n\geq 2$. Indeed, from Lemma 2.3 we clearly see that

and

Therefore, the sequence $\left\{a_{n}/b_{n}\right\}$ is decreasing for $n\geq 2$ follows from the fact that

for $n\geq 2$.

Therefore, it follows from Lemma 2.4 that the function $G(x) = A(x)/B(x)$ is strictly increasing on $(0, \pi /2)$, and Theorem 1.2 follows from the limit values

Proof of Theorem 1.3. Let $0 < x < \pi /2$ and

Then elaborated computations lead to

where

It follows from Lemmas 2.1 and 2.2 that

where

From Lemma 2.3 we have

that is

and

where

It is not difficulty to prove that $q(n) > 0$ for all $n\geq 5$ by use of the mathematical induction. Therefore, $H^{\prime }(x) > 0$, $F(x)$ is strictly increasing on $(0, \pi /2)$ and $H(x) > {H(0^{+})} = 0$. Note that

Which completes the proofs of Theorem 1.3.

4.   Remarks

Remark 4.1. The first inequality of (1.3) and our inequality (1.9) are not comparable. Indeed, experiments and numerical simulation show that our inequality (1.9) is better than the first inequality of (1.3) for $x\in (1.0828, \pi /2)$, but the first inequality of (1.3) is stronger than the inequality (1.9) for $x\in (0, 1.0828)$.

Remark 4.2. Let $\alpha = 0$. Then inequality (1.10) or (1.11) leads to

for $x\in (0, \pi/2)$, which also was proved by Zhu ^[42]. In addition, the second inequality of (1.11) is stronger than that of (1.10) due to the fact $\tan x > x$ for all $x\in (0, \pi/2)$.

Remark 4.3. It is worth pointing out that it is very meaningful and important to improve the existing inequalities. We recommend the interested readers to read the literature ^[43,44,45] where one can found many new inequalities for the circular, hyperbolic, inverse circular, inverse hyperbolic and exponential functions as well as their related special functions.

5.   Conclusions

We have established three sharp inequalities of Wilker type for trigonometric functions:

where $0 < x < \pi /2,$ $\alpha = 0,$ and $\beta = 8/45$. The above three inequalities improve and develop the known famous results.

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions, which led to considerable improvement of the article.

The research is supported by the Natural Science Foundation of China (Grant No. 61772025).

Conflict of interest

The authors declare that they have no competing interests.