New Cusa-Huygens type inequalities

1.   Introduction

It is well known inequality plays an irreplaceable role in the development of mathematics. Very recently, many inequalities such as Hermite-Hadamard type inequality ^{[1,2,3,4,5,6]}, Petrović type inequality ^[7], Pólya-Szegö and Ćebyýev type inequalities ^[8], Ostrowski type inequality ^[9], reverse Minkowski inequality ^[10], Jensen type inequality ^[11,12,13], Cauchy-Schwarz type inequality ^[14], Bessel function inequality ^[15], trigonometric and hyperbolic functions inequalities ^{[16,17,18,19]}, Grötzsch ring function inequality ^[20], Ramanujan transformation inequality ^[21], fractional integral inequality ^{[22,23,24,25,26,27]}, complete and generalized elliptic integrals inequalities ^{[28,29,30,31,32,33]}, generalized convex function inequality ^[34,35,36] and mean values inequality ^[37,38,39] have attracted the attention of many researchers.

The classical and well-known Cusa-Huygens inequality states that

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

Chen and Cheung ^[40] gave the bounds for $\sin x/x$ in term of $\left(\left(2+\cos x\right) /3\right) ^{\delta }$ as follows

for $0 < x < \pi /2$, where $\vartheta _{0} = 1$ and $\theta _{0} = (\ln \pi -\ln 2)/(\ln 3-\ln 2)$ are the best possible constants such that the double inequality (1.2) holds for all $0 < x < \pi /2$. Inequality (1.2) was proved by Sun and Zhu in ^[41]. Recently, the generalizations, improvements and variants for the Cusa-Huygens inequality (1.1) have been the subject of much research.

Inspired by inequalities (1.1) and (1.2), the first aim of this paper is to improve the Cusa-Huygens inequality by considering the monotonicity of the functions

and

on a wider range $(0, \pi)$ instead of $\left(0, \pi /2\right)$. Our first aim of the article is to prove the following Theorems 1.1 and 1.2.

Theorem 1.1. Let $U\left(x\right)$ be defined by (1.3). Then the following statemsnts are true:

$(i)$ There exists $x_{0}\in\left(\pi/2, \pi\right)$ such that $U\left(x\right)$ is increasing on $\left(0, x_{0}\right)$ and decreasing on $\left(x_{0}, \pi\right)$, and the double inequality

holds for $x\in\left(0, \pi /2\right)$ with the best possible constants $\alpha_{1} = 16\left(\pi-3\right)/\pi^{5} = 0.007403\cdots$ and $\beta _{1} = 1/180 = 0.005555\cdots$. Moreover, the right hand side inequality of (1.5) also holds for $x\in\left(0, \pi\right)$.

$(ii)$ The function

is increasing on $\left(0, \pi \right)$, and the inequality

holds for $x\in\left(0, \pi \right)$.

From Theorem 1.1, we get Corollary 1.1 immediately.

Corollary 1.1. The double inequality

holds for all $x\in\left(0, \pi\right)$ with the best possible constants $\pi^{3}$ and $180$.

Theorem 1.2. The function $G\left(x\right)$ defined by (1.4) is strictly increasing on $\left(0, \pi\right)$.

Let

and

Then Theorem 1.2 leads to Corollary 1.2 immediately.

Corollary 1.2. $(i)$ The double inequality

holds for all $x\in(0, \pi /2)$ with the best possible constants $\vartheta_{1}$ and $\theta_{1}$ given in (1.9) and (1.10).

$(ii)$ The inequality

holds for all $x\in (\pi /2, \pi)$ with the best constant $\theta _{1}$ given by (1.9).

A real-valued function $f$ is said to be absolutely monotonic on the interval $I$ if $f$ has derivatives of all orders on $I$ such that

for all $x\in I$ and $n\geq 0$.

The second aim of the article is to provide an absolute monotonicity result for a special function and derive a new Cusa-Huygens type inequality.

Theorem 1.3. The function

is absolutely monotonic on $\left(0, 2\pi\right)$, and

From Theorem 1.3, we can easily obtain the following Corollary 1.3.

Corollary 1.3. Let $J(x)$ be defined by (1.13) Then the function

is absolutely monotonic on $\left(0, 2\pi\right)$, and the double inequality

holds for all $x\in (0, \pi /2)$ with the best possible constants

and

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

Remark 1.1. Let $n = 1$ and $n = 2$. Then Corollary 1.3 leads to the conclusion that

and

for $0 < x < \pi/2$ with $\lambda _{2} = 1/840 = 0.001190\cdots$ and $\mu_{2} = \left(16/\pi^{4}\right)\left(2-6/\pi-\pi ^{2}/120\right) = 0.001296\cdots$.

2.   Lemmas

In order to prove our main results, we need the monotone form of the L'Hô spital rule ^[42,43,44].

Lemma 2.1. (See ^[42,43,44]) Let $f, g:[a, b]\rightarrow \mathbb{R}$ be continuous on $[a, b]$ and differentiable on $(a, b)$ such that $g^{\prime }\neq 0$ on $(a, b)$ and $f^{\prime }/g^{\prime }$ is (strictly) increasing (decreasing) on $(a, b)$. Then both the functions $(f(x)-f(b))/(g(x)-g(b))$ and $(f(x)-f(a))/(g(x)-g(a))$ are (strictly) increasing (decreasing) on $[a, b]$.

Lemma 2.2. Let $B_{n}$ be the Bernoulli number. Then we have the following power series formulas

and

for all $x\in (0, \pi)$.

Proof. The power series formulas (2.1) and (2.3) can be found in the literature ^[45], and the power series formulas (2.2) and (2.4) can be obtained from (2.1) and (2.3) together with the facts that

and

3.   Proofs of Theorems 1.1–1.3

3.1.   Proof of Theorem 1.1

$(i)$ We clearly see that the function $U\left(x\right)$ can be rewritten as

Differentiation yields

Expanding in power series leads to

which gives

It follows from the identities

and

given in ^[44] that $H_{p, q}^{\prime } > 0$ due to $\left(p^{\prime }/q^{\prime }\right) ^{\prime } > 0$ and $q > 0$.

From the formula

we get

which implies that $H_{p, q}\left(x\right) < 0$ for $x\in\left(0, \pi/2\right)$, and there exists $x_{0}\in\left(\pi /2, \pi\right)$ such that $H_{p, q}\left(x\right) < 0$ for $x\in\left(0, x_{0}\right)$ and $H_{p, q}\left(x\right) > 0$ for $x\in\left(x_{0}, \pi\right)$. It follows from $q^{\prime} = -\sin x < 0$ and (3.1) that $\left(p/q\right)^{\prime} > 0$ on $\left(0, \pi /2\right)$, and $\left(p/q\right)^{\prime} > 0$ on $\left(0, x_{0}\right)$ and $\left(p/q\right)^{\prime } < 0$ on $\left(x_{0}, \pi\right)$.

Therefore, the double inequality (1.3) follows from the monotonicity of $U\left(x\right)$ on $\left(0, \pi/2\right)$.

Using the piecewise monotonicity of $U\left(x\right)$ on $\left(0, \pi\right)$, we arrive at

which prove that the right hand side inequality of (1.3) also holds for $x\in\left(0, \pi\right)$.

$(ii)$ Differentiation yields

where

It follows from

for $x\in\left(0, \pi\right)$ and $V\left(0\right) = 0$ that $V\left(x\right) > 0$ for $x\in\left(0, \pi\right)$, and so is $\left(xU\right)^{\prime }$. Therefore, the inequality

holds for $x\in \left(0, \pi \right)$.

3.2.   Proof of Theorem 1.2

Let

Then from Lemma 2.1 we clearly see that it suffices to prove that $b^{\prime }(x)/a^{\prime }(x)$ is strictly decreasing on $(0, \pi)$ due to $a(0^{+}) = b(0^{+}) = 0$.

Elaborated computations lead to

where

and

Next, we prove that $h_{i}(x)$ is decreasing on $(0, \pi)$ for $i = 1, 2, 3$ and $h_{i}(x)$ is positive for $i = 1, 2$.

$(i)$ Let

Then

and

is clearly increasing on $\left(0, \pi\right)$. It follows from Lemma 2.1 that $h_{1}(x)$ is decreasing on $(0, \pi)$.

$(ii)$ To prove that $h_{2}(x)$ is positive and decreasing on $\left(0, \pi\right)$, it suffices to prove that $1/h_{2}\left(x\right)$ is positive and increasing on $\left(0, \pi\right)$. Note that

where

and $xU(x)$ is defined as (1.6), which is strictly increasing on $\left(0, \pi \right)$ by Theorem 1.1. We clearly see that it suffices to prove that $K(x)$ is strictly increasing on $(0, \pi)$. Indeed, by Lemma 2.2 we have

which is obviously increasing on $(0, \pi)$.

$(iii)$ To prove that $h_{3}(x)$ is decreasing on $\left(0, \pi\right)$, it suffices to prove that $1/h_{3}\left(x\right)$ is positive and increasing on $\left(0, \pi \right)$. Note that

It follows from Lemma 2.2 that

and

which is evidently positive and increasing on $\left(0, \pi\right)$. The proof of Theorem 1.2 is completed.

3.3.   Proof of Theorem 1.3

It is obviously that $J(x)$ can be rewritten as

which is clearly absolutely monotonic on $(0, 2\pi)$.

4.   Remarks

Remark 4.1. One of the referees asserted that the Cusa-Huygens inequality (1.1) holds for all $x\neq 0$. In fact, inequality (1.1) is equivalent to

Differentiation yields

for all $x\in\mathbb{R}$. If $x > 0$, then $D\left(x\right) < D\left(0\right) = 0$ and inequality (1.1) holds for $x > 0$. If $x < 0$, then $D\left(x\right) > D\left(0\right) = 0$ and inequality (1.1) also holds for $x < 0$.

Remark 4.2. We clearly see that the right hand side inequality of (1.5) is stronger than the Cusa-Huygens inequality (1.1).

Remark 4.3. Our double inequality (1.11) is clearly better than the inequality (1.2). Moreover, by Theorem 1.2 we deduce that the function

is also strictly increasing on $\left(0, \pi \right)$. This conclusion immediately leads to the inequality (1.2) and the following new result: the inequality

holds for all $x\in (\pi /2, \pi)$ with the best constant $\theta _{0} = (\ln \pi -\ln 2)/(\ln 3-\ln 2)$.

Remark 4.4. The right-hand side inequality of (1.14) is stronger than the Cusa-Huygens inequality (1.1) due to

for all $x\in\left(0, \pi /2\right)$.

Remark 4.5. Numerical calculations and computer simulation experiments show that the double inequality (1.15) is stronger than the inequalities (1.5) and (1.11) on $(0, \pi /2)$.

Final, the following power series formula

inspires us to propose the Conjecture 4.1.

Conjecture 4.1. The function $G^{\ast}\left(x\right)$ above mentioned is absolutely monotonic on $\left(0, \pi/2\right)$.

5.   Conclusions

In the article, we have discussed the monotonicity of the functions $U\left(x\right)$, $xU\left(x\right)$ and $G(x)$ defined by (1.3) and (1.4) on the interval $\left(0, \pi\right)$, and the absolute monotonicity of the function $J\left(x\right)$ given in (1.13) on the interval $\left(0, 2\pi\right)$. Consequences, we have discovered several new Cusa-Huygens type inequalities, which are the improvements and refinements of some earlier known results.

Acknowledgments

The author 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 author declares that he has no competing interest.