High Precision Wilker-Type Inequality of Fractional Powers

Abstract

This paper established a new high precision Wilker-type inequality with fractional powers for the function $2-[{\left(x/\left(sinx\right)\right)}^{6/5}+{\left(x/\left(tanx\right)\right)}^{3/5}]$ bounded by the function $x^6{\left(\left(tanx\right)/x\right)}^{5/4}.$

1. Introduction

We know the following fact: If $0<x<\pi /2$, then the sine function and tangent function satisfy the following simultaneous inequalities

$$sinx<x<tanx,$$

which is

$$\frac{sinx}{x}<1<\frac{tanx}{x}.$$

(1)

One can consider the asymptotic expansion of the sum of the square of the first function and the last one in (1) as follows:

$${\left(\frac{sinx}{x}\right)}^2+\frac{tanx}{x}=2+\frac{8}{45}{x}^4+O\left(x^6\right).$$

(2)

In 1989, inspired by the above formula (2), Wilker [1] proposed two open problems as follows:

(1)

If $0<x<\pi /2$, then

$${\left(\frac{sinx}{x}\right)}^2+\frac{tanx}{x}>2.$$

(3)

(2)

There exists a largest constant c such that

$${\left(\frac{sinx}{x}\right)}^2+\frac{tanx}{x}>2+c{x}^{3}tanx$$

(4)

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

Sumner et al. [2] affirmed the truth of two problems above and obtained a further result as follows: For $x\in \left(0,\pi /2\right)$, the double inequality

$$\frac{16}{{\pi }^4}{x}^{3}tanx<{\left(\frac{sinx}{x}\right)}^2+\frac{tanx}{x}-2<\frac{8}{45}{x}^{3}tanx$$

(5)

holds with the best constants $16/{\pi }^4$ and $8/45$.

For the promotion and development of inequalities (3) and (5), interested readers can see [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43]. Recently, Zhu [44] established two new Wilker-type inequality for circular functions as follows.

Proposition 1.

Let $0<x<\pi /2$. Then

$${\left(\frac{sinx}{x}\right)}^2+\frac{tanx}{x}-2>\frac{8}{45}{x}^4{\left(\frac{tanx}{x}\right)}^{6/7}$$

(6)

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

Proposition 2.

Let $0<x<\pi /2$. Then

$${\left(\frac{x}{sinx}\right)}^2+\frac{x}{tanx}-2<\frac{2}{45}{x}^4{\left(\frac{tanx}{x}\right)}^{4/7}$$

(7)

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

Resently, the author of this paper [45] shown two Wilker’s inequalities of exponential type for the functions ${\left(\left(sinx\right)/x\right)}^2+\left(tanx\right)/x-2$ and ${\left(x/\left(sinx\right)\right)}^2+x/\left(tanx\right)-2$ bounded by the function $x^4{\left(tan\left(\lambda x\right)/\left(\lambda x\right)\right)}^{\alpha }$, and obtained the following results.

Proposition 3.

Let $0<\left|x\right|<\left(7\pi \right)/$$\left(2\sqrt{61}\right)\approx$$1.4078.$ Then

$${\left(\frac{sinx}{x}\right)}^2+\frac{tanx}{x}-2<\frac{8}{45}{x}^4{\left(\frac{tan(\sqrt{61}x/7)}{\sqrt{61}x/7}\right)}^{42/61}$$

(8)

holds with the best constant $8/45$.

Proposition 4.

Let $0<\left|x\right|<\pi /$2. Then

$${\left(\frac{x}{sinx}\right)}^2+\frac{x}{tanx}-2>\frac{2}{45}{x}^4{\left(\frac{tan\left(\sqrt{46}x/14\right)}{\sqrt{46}x/14}\right)}^{56/23}$$

(9)

holds with the best constant $2/45$.

In this paper we consider the asymptotic expansion of the following function

$${\left(\frac{x}{sinx}\right)}^{2a}+{\left(\frac{x}{tanx}\right)}^a-2=\frac{1}{45}a\left(5a-3\right)x^4+O\left(x^6\right).$$

(10)

In order to obtain some more precise Wilker-type inequalities, letting $a=3/5$ in (10) we obtain the following result:

$$2-\left[{\left(\frac{x}{sinx}\right)}^{\frac{6}{5}}+{\left(\frac{x}{tanx}\right)}^{\frac{3}{5}}\right]=\frac{16}{7875}{x}^6+\frac{4}{4725}{x}^8+\frac{5392}{19490625}{x}^{10}+\frac{34562816}{399070546875}{x}^{12}+O\left(x^{14}\right).$$

Then we obtain the following precise Wilker-type inequalities with fractional powers.

Theorem 1.

Let $0<x<\pi /2$. Then

$$2-\left[{\left(\frac{x}{sinx}\right)}^{\frac{6}{5}}+{\left(\frac{x}{tanx}\right)}^{\frac{3}{5}}\right]<\frac{16}{7875}{x}^6{\left(\frac{tanx}{x}\right)}^{\frac{5}{4}}$$

(11)

holds with the best constant $16/7875$.

Due to the high accuracy of the inequality established in this paper, we demonstrate it by the following technical route: using the power series expansions of the correlation functions whose Taylor coefficients are connected with Bernoulli numbers we realize our whole proof by estimating the ratio of two adjacent even-indexed Bernoulli numbers. Here, I would like to mention two conclusions of Euler, a famous mathematician in the history of mathematics, on the expansion of power series of cotangent function and Riemann zeta function:

$$\begin{array}{ccc}\hfill cotx& =& \frac{1}{x}-\sum _{n=1}^{\infty }\frac{2^{2n}}{\left(2n\right)!}|B_{2n}|x^{2n-1},0<\left|x\right|<\pi ,\hfill \\ \hfill \zeta \left(2n\right)& =& \frac{{\left(2\pi \right)}^{2n}}{2\left(2n\right)!}\left|B_{2n}\right|,n\in \mathbb{N},\hfill \end{array}$$

which are the bridges between cotangent and Bernoulli number, Bernoulli number and Riemann zeta function. Some very interesting recent developments (see [46,47,48,49,50,51,52]) involving cotangent function have shown the connection of such functions to really essential open problems in Mathematics such as the Riemann Hypothesis, as well as to the study of important Number Theoretic functions, such as the Euler totient function. It is beneficial for readers to understand the internal relationship between the functions mentioned above in the number theory and other mathematical fields.

2. Lemmas

In order to prove Theorem 1, we need the following lemmas.

Lemma 1.

Let $B_{2n}$ be the even-indexed Bernoulli numbers, we have the following power series expansions

$$\begin{array}{ccc}\hfill \frac{1}{{cos}^{2}x}& =& {sec}^{2}x=\sum _{n=1}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-2},\hfill \\ \hfill \frac{1}{{cos}^{4}x}& =& \frac{2}{3}\sum _{n=1}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-2}\hfill \\ & & +\frac{1}{6}\sum _{n=2}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-4},\hfill \\ \hfill \frac{1}{{cos}^{6}x}& =& \frac{8}{15}\sum _{n=1}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-2}\hfill \\ & & +\frac{1}{6}\sum _{n=2}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-4}\hfill \\ & & +\frac{1}{120}\sum _{n=3}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)\left(2n-4\right)\left(2n-5\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-6},\hfill \\ \hfill \frac{1}{{cos}^{8}x}& =& \frac{16}{35}\sum _{n=1}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-2}\hfill \\ & & +\frac{7}{45}\sum _{n=2}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-4}\hfill \\ & & +\frac{1}{90}\sum _{n=3}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)\left(2n-4\right)\left(2n-5\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-6}\hfill \\ & & +\frac{1}{5040}\sum _{n=4}^{\infty }\frac{\begin{array}{c}{2}^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)\\ \left(2n-4\right)\left(2n-5\right)\left(2n-6\right)\left(2n-7\right)\end{array}}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-8},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \frac{1}{{cos}^{10}x}& =& \frac{128}{315}\sum _{n=1}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-2}\hfill \\ & & +\frac{82}{567}\sum _{n=2}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-4}\hfill \\ & & +\frac{13}{1080}\sum _{n=3}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)\left(2n-4\right)\left(2n-5\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-6}\hfill \\ & & +\frac{1}{3024}\sum _{n=4}^{\infty }\frac{\left[\begin{array}{c}{2}^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)\\ \left(2n-4\right)\left(2n-5\right)\left(2n-6\right)\left(2n-7\right)\end{array}\right]}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-8}\hfill \\ & & +\frac{1}{362880}\sum _{n=5}^{\infty }\frac{\left[\begin{array}{c}{2}^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)\left(2n-4\right)\\ \left(2n-5\right)\left(2n-6\right)\left(2n-7\right)\left(2n-8\right)\left(2n-9\right)\end{array}\right]}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-10},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \frac{1}{{cos}^{12}x}& =& \frac{256}{693}\sum _{n=1}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-2}\hfill \\ & & +\frac{1916}{14175}\sum _{n=2}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-4}\hfill \\ & & +\frac{139}{11340}\sum _{n=3}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)\left(2n-4\right)\left(2n-5\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-6}\hfill \\ & & +\frac{31}{75600}\sum _{n=4}^{\infty }\frac{\left[\begin{array}{c}{2}^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)\\ \left(2n-4\right)\left(2n-5\right)\left(2n-6\right)\left(2n-7\right)\end{array}\right]}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-8}\hfill \\ & & +\frac{1}{181440}\sum _{n=5}^{\infty }\frac{\left[\begin{array}{c}{2}^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)\\ \left(2n-4\right)\left(2n-5\right)\left(2n-6\right)\\ \left(2n-7\right)\left(2n-8\right)\left(2n-9\right)\end{array}\right]}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-10}\hfill \\ & & +\frac{1}{39916800}\sum _{n=6}^{\infty }\frac{\left[\begin{array}{c}{2}^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)\\ \left(2n-4\right)\left(2n-5\right)\left(2n-6\right)\left(2n-7\right)\\ \left(2n-8\right)\left(2n-9\right)\left(2n-10\right)\left(2n-11\right)\end{array}\right]}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-12},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \frac{sinx}{cosx}& =& tanx=\sum _{n=1}^{\infty }\frac{\left(2^{2n}-1\right)}{\left(2n\right)!}{2}^{2n}\left|B_{2n}\right|x^{2n-1},\hfill \\ \hfill \frac{sinx}{{cos}^{3}x}& =& \frac{1}{2}\sum _{n=2}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-3},\hfill \\ \hfill \frac{sinx}{{cos}^{5}x}& =& \frac{1}{6}\sum _{n=2}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-3}\hfill \\ & & +\frac{1}{24}\sum _{n=3}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)\left(2n-4\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-5},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \frac{sinx}{{cos}^{7}x}& =& \frac{4}{45}\sum _{n=2}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-3}\hfill \\ & & +\frac{1}{36}\sum _{n=3}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)\left(2n-4\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-5}\hfill \\ & & +\frac{1}{720}\sum _{n=4}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)\left(2n-4\right)\left(2n-5\right)\left(2n-6\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-7},\hfill \\ \hfill \frac{sinx}{{cos}^{9}x}& =& \frac{2}{35}\sum _{n=2}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-3}\hfill \\ & & +\frac{7}{360}\sum _{n=3}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)\left(2n-4\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-5}\hfill \\ & & +\frac{1}{720}\sum _{n=4}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)\left(2n-4\right)\left(2n-5\right)\left(2n-6\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-7}\hfill \\ & & +\frac{1}{40320}\sum _{n=5}^{\infty }\frac{\left[\begin{array}{c}{2}^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)\\ \left(2n-4\right)\left(2n-5\right)\left(2n-6\right)\left(2n-7\right)\left(2n-8\right)\end{array}\right]}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-9},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \frac{sinx}{{cos}^{11}x}& =& \frac{64}{1575}\sum _{n=2}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-3}\hfill \\ & & +\frac{41}{2835}\sum _{n=3}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)\left(2n-4\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-5}\hfill \\ & & +\frac{13}{10800}\sum _{n=4}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)\left(2n-4\right)\left(2n-5\right)\left(2n-6\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-7}\hfill \\ & & +\frac{1}{30240}\sum _{n=5}^{\infty }\frac{\left[\begin{array}{c}{2}^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)\left(2n-4\right)\\ \left(2n-5\right)\left(2n-6\right)\left(2n-7\right)\left(2n-8\right)\end{array}\right]}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-9}\hfill \\ & & +\frac{1}{3628800}\sum _{n=6}^{\infty }\frac{\left[\begin{array}{c}{2}^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)\left(2n-4\right)\\ \left(2n-5\right)\left(2n-6\right)\left(2n-7\right)\left(2n-8\right)\left(2n-9\right)\left(2n-10\right)\end{array}\right]}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-11},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \frac{sinx}{{cos}^{13}x}& =& \frac{64}{2079}\sum _{n=2}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-3}\hfill \\ & & +\frac{479}{42525}\sum _{n=3}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)\left(2n-4\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-5}\hfill \\ & & +\frac{139}{136080}\sum _{n=4}^{\infty }\frac{\left[\begin{array}{c}{2}^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\\ \left(2n-3\right)\left(2n-4\right)\left(2n-5\right)\left(2n-6\right)\end{array}\right]}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-7}\hfill \\ & & +\frac{31}{907200}\sum _{n=5}^{\infty }\frac{\left[\begin{array}{c}{2}^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)\\ \left(2n-4\right)\left(2n-5\right)\left(2n-6\right)\left(2n-7\right)\left(2n-8\right)\end{array}\right]}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-9}\hfill \\ & & +\frac{1}{2177280}\sum _{n=6}^{\infty }\frac{\left[\begin{array}{c}{2}^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)\left(2n-4\right)\\ \left(2n-5\right)\left(2n-6\right)\left(2n-7\right)\left(2n-8\right)\left(2n-9\right)\left(2n-10\right)\end{array}\right]}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-11}\hfill \\ & & +\frac{1}{479001600}\sum _{n=7}^{\infty }\frac{\left[\begin{array}{c}{2}^{2n}(2^{2n}-1)\left(2n-1\right)\left(2n-2\right)\left(2n-3\right)\left(2n-4\right)\\ \left(2n-5\right)\left(2n-6\right)\left(2n-7\right)\left(2n-8\right)\\ \left(2n-9\right)\left(2n-10\right)\left(2n-11\right)\left(2n-12\right)\end{array}\right]}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-13}\hfill \end{array}$$

hold for all $x\in (-\pi /2,\pi /2).$

Proof. 

By the power series expansion of $tanx$ (see [53,54]):

$$tanx=\sum _{n=1}^{\infty }\frac{2^{2n}\left(2^{2n}-1\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-1},\left|x\right|<\frac{\pi }{2}$$

(12)

we can obtain

$$\frac{1}{{cos}^{2}x}={sec}^{2}x={\left(tanx\right)}^{\prime }=\sum _{n=1}^{\infty }\frac{2^{2n}(2^{2n}-1)\left(2n-1\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-2},\left|x\right|<\frac{\pi }{2}.$$

Via

$$\begin{array}{ccc}\hfill \frac{1}{{cos}^{n+1}x}& =& \frac{1}{n}\left[\frac{n-1}{{cos}^{n-1}x}+\frac{1}{n-1}{\left(\frac{1}{{cos}^{n-1}x}\right)}^{″}\right],\hfill \\ \hfill \frac{sinx}{{cos}^{n+1}x}& =& \frac{1}{n}{\left(\frac{1}{{cos}^{n}x}\right)}^{\prime }\hfill \end{array}$$

we can get the desired power series expansions of other functions. □

Lemma 2

([12,55,56]). Let $B_{2n}$ be the even-indexed Bernoulli numbers, we have

$$\frac{2^{2n-1}-1}{2^{2n+1}-1}\frac{(2n+2)(2n+1)}{{\pi }^2}<\frac{|B_{2n+2}|}{|B_{2n}|}<\frac{2^{2n}-1}{2^{2n+2}-1}\frac{(2n+2)(2n+1)}{{\pi }^2}.$$

(13)

Lemma 3.

Let $B_{2n}$ be the even-indexed Bernoulli numbers, $n\ge 10$ and $n\in \mathbb{N}$. Then

$$\begin{array}{ccc}\hfill a_n& =& \frac{128}{51975}\frac{\left\{\begin{array}{c}{2}^{2n+6}\left(2^{2n+6}-1\right)n\left(2n+1\right)\left(2n+3\right)\left(2n+5\right)\left(n+1\right)\left(n+2\right)\left(n+3\right)\\ \left[\begin{array}{c}14130836279n+2610109397n^2+252157346n^3\\ +13120940n^4+273800n^5+29051019788\end{array}\right]\end{array}\right\}}{\left(2n+6\right)!}\left|B_{2n+6}\right|\hfill \\ & & -\frac{256}{31185}\frac{\left\{\begin{array}{c}{2}^{2n+4}\left(2^{2n+4}-1\right)n\left(2n+1\right)\left(2n+3\right)\left(n+1\right)\\ \left[\begin{array}{c}1007457799854n+448538582689n^2+91600811650n^3\\ +10838033881n^4+868613338n^5\\ +83295436n^6+4983160n^7+724428764568\end{array}\right]\end{array}\right\}}{\left(2n+4\right)!}\left|B_{2n+4}\right|\hfill \\ & & -\frac{512}{1575}\frac{\left\{2^{2n+2}\left(2^{2n+2}-1\right)n\left(2n+1\right)\left[\begin{array}{c}-324988669975n-74432045925n^2\\ -33240865835n^3-11124715381n^4\\ -922530750n^5-44831420n^6\\ +5749800n^7-5273077314\end{array}\right]\right\}}{\left(2n+2\right)!}\left|B_{2n+2}\right|\hfill \\ & & -\frac{1024}{945}\frac{\left\{2^{2n}\left(2^{2n}-1\right)\left[\begin{array}{c}470063451466n-274339736559n^2\\ +108583995240n^3+12695894043n^4-3857965770n^5\\ +610797924n^6+45615080n^7-154356228096\end{array}\right]\right\}}{\left(2n\right)!}\left|B_{2n}\right|\hfill \\ & & -\frac{2048}{14175}\frac{\left\{2^{2n-2}\left(2^{2n-2}-1\right)\left[\begin{array}{c}-2587181560206n+1213011374071n^2\\ -180257338355n^3-1918656460n^4\\ +2580701900n^5+1889134753248\end{array}\right]\right\}}{\left(2n-2\right)!}\left|B_{2n-2}\right|\hfill \\ & & -\frac{16384}{10395}\frac{\left(2n-5\right)\left(-1277114903n+345227575n^2+551290272\right)2^{2n-4}\left(2^{2n-4}-1\right)}{\left(2n-4\right)!}\left|B_{2n-4}\right|\hfill \\ & & -\frac{12560629760}{33}\frac{\left(2n-7\right)2^{2n-6}\left(2^{2n-6}-1\right)}{\left(2n-6\right)!}\left|B_{2n-6}\right|\hfill \\ & >& 0.\hfill \end{array}$$

Proof. 

Since

$$\begin{array}{ccc}\hfill \frac{a_n}{|B_{2n}|}& =& \frac{128}{51975}\frac{\left\{\begin{array}{c}{2}^{2n+6}\left(2^{2n+6}-1\right)n\left(2n+1\right)\left(2n+3\right)\left(2n+5\right)\left(n+1\right)\left(n+2\right)\left(n+3\right)\\ \left[\begin{array}{c}14130836279n+2610109397n^2+252157346n^3\\ +13120940n^4+273800n^5+29051019788\end{array}\right]\end{array}\right\}}{\left(2n+6\right)!}\hfill \\ & & \times \frac{|B_{2n+6}|}{|B_{2n+4}|}\frac{|B_{2n+4}|}{|B_{2n+2}|}\frac{|B_{2n+2}|}{|B_{2n}|}\hfill \\ & & -\frac{256}{31185}\frac{\left\{\begin{array}{c}{2}^{2n+4}\left(2^{2n+4}-1\right)n\left(2n+1\right)\left(2n+3\right)\left(n+1\right)\\ \left[\begin{array}{c}1007457799854n+448538582689n^2\\ +91600811650n^3+10838033881n^4\\ +868613338n^5+83295436n^6\\ +4983160n^7+724428764568\end{array}\right]\end{array}\right\}}{\left(2n+4\right)!}\frac{|B_{2n+4}|}{|B_{2n}|}\frac{|B_{2n+2}|}{|B_{2n}|}\hfill \\ & & -\frac{512}{1575}\frac{\left\{\begin{array}{c}{2}^{2n+2}\left(2^{2n+2}-1\right)n\left(2n+1\right)\\ \left[\begin{array}{c}-324988669975n-74432045925n^2-33240865835n^3\\ -11124715381n^4-922530750n^5-44831420n^6\\ +5749800n^7-5273077314\end{array}\right]\end{array}\right\}}{\left(2n+2\right)!}\frac{|B_{2n+2}|}{|B_{2n}|}\hfill \\ & & -\frac{1024}{945}\frac{\left\{2^{2n}\left(2^{2n}-1\right)\left[\begin{array}{c}470063451466n-274339736559n^2+108583995240n^3\\ +12695894043n^4-3857965770n^5\\ +610797924n^6+45615080n^7-154356228096\end{array}\right]\right\}}{\left(2n\right)!}\hfill \\ & & -\frac{2048}{14175}\frac{\left\{2^{2n-2}\left(2^{2n-2}-1\right)\left[\begin{array}{c}-2587181560206n+1213011374071n^2\\ -180257338355n^3-1918656460n^4\\ +2580701900n^5+1889134753248\end{array}\right]\right\}}{\left(2n-2\right)!}\frac{|B_{2n-2}|}{|B_{2n}|}\hfill \\ & & -\frac{16384}{10395}\frac{\left(2n-5\right)\left(-1277114903n+345227575n^2+551290272\right)2^{2n-4}\left(2^{2n-4}-1\right)}{\left(2n-4\right)!}\hfill \\ & & \times \frac{|B_{2n-4}|}{|B_{2n-2}|}\frac{|B_{2n-2}|}{|B_{2n}|}\hfill \\ & & -\frac{12560629760}{33}\frac{\left(2n-7\right)2^{2n-6}\left(2^{2n-6}-1\right)}{\left(2n-6\right)!}\frac{|B_{2n-6}|}{|B_{2n-4}|}\frac{|B_{2n-4}|}{|B_{2n-2}|}\frac{|B_{2n-2}|}{|B_{2n}|},\hfill \end{array}$$

by Lemma 2 we have

$$\begin{array}{ccc}& & \frac{a_n}{|B_{2n}|}\hfill \\ & >& \\ & & \frac{128}{51975}\frac{\left\{\begin{array}{c}{2}^{2n+6}\left(2^{2n+6}-1\right)n\left(2n+1\right)\left(2n+3\right)\left(2n+5\right)\left(n+1\right)\left(n+2\right)\left(n+3\right)\\ \left[\begin{array}{c}14130836279n+2610109397n^2+252157346n^3\\ +13120940n^4+273800n^5+29051019788\end{array}\right]\end{array}\right\}}{\left(2n+6\right)!}\hfill \\ & & \times \frac{2^{2n+3}-1}{2^{2n+5}-1}\frac{(2n+6)(2n+5)}{{\pi }^2}\frac{2^{2n+1}-1}{2^{2n+3}-1}\frac{(2n+4)(2n+3)}{{\pi }^2}\frac{2^{2n-1}-1}{2^{2n+1}-1}\frac{(2n+2)(2n+1)}{{\pi }^2}\hfill \\ & & -\frac{256}{31185}\frac{\left\{\begin{array}{c}{2}^{2n+4}\left(2^{2n+4}-1\right)n\left(2n+1\right)\left(2n+3\right)\left(n+1\right)\\ \left[\begin{array}{c}1007457799854n+448538582689n^2\\ +91600811650n^3+10838033881n^4+868613338n^5\\ +83295436n^6+4983160n^7+724428764568\end{array}\right]\end{array}\right\}}{\left(2n+4\right)!}\hfill \\ & & \times \frac{2^{2n+2}-1}{2^{2n+4}-1}\frac{(2n+4)(2n+3)}{{\pi }^2}\frac{2^{2n}-1}{2^{2n+2}-1}\frac{(2n+2)(2n+1)}{{\pi }^2}\hfill \\ & & -\frac{512}{1575}\frac{\left\{2^{2n+2}\left(2^{2n+2}-1\right)n\left(2n+1\right)\left[\begin{array}{c}-324988669975n-74432045925n^2\\ -33240865835n^3-11124715381n^4\\ -922530750n^5-44831420n^6\\ +5749800n^7-5273077314\end{array}\right]\right\}}{\left(2n+2\right)!}\hfill \\ & & \times \frac{2^{2n}-1}{2^{2n+2}-1}\frac{(2n+2)(2n+1)}{{\pi }^2}\hfill \\ & & -\frac{1024}{945}\frac{\left\{2^{2n}\left(2^{2n}-1\right)\left[\begin{array}{c}470063451466n-274339736559n^2\\ +108583995240n^3+12695894043n^4-3857965770n^5\\ +610797924n^6+45615080n^7-154356228096\end{array}\right]\right\}}{\left(2n\right)!}\hfill \\ & & -\frac{2048}{14175}\frac{\left\{2^{2n-2}\left(2^{2n-2}-1\right)\left[\begin{array}{c}-2587181560206n+1213011374071n^2\\ -180257338355n^3-1918656460n^4\\ +2580701900n^5+1889134753248\end{array}\right]\right\}}{\left(2n-2\right)!}\hfill \\ & & \times \frac{2^{2n-1}-1}{2^{2n-3}-1}\frac{{\pi }^2}{\left(2n\right)(2n-1)}\hfill \\ & & -\frac{16384}{10395}\frac{\left(2n-5\right)\left(-1277114903n+345227575n^2+551290272\right)2^{2n-4}\left(2^{2n-4}-1\right)}{\left(2n-4\right)!}\hfill \\ & & \times \frac{2^{2n-3}-1}{2^{2n-5}-1}\frac{{\pi }^2}{(2n-2)(2n-3)}\frac{2^{2n-1}-1}{2^{2n-3}-1}\frac{{\pi }^2}{\left(2n\right)(2n-1)}\hfill \\ & & -\frac{12560629760}{33}\frac{\left(2n-7\right)2^{2n-6}\left(2^{2n-6}-1\right)}{\left(2n-6\right)!}\frac{2^{2n-5}-1}{2^{2n-7}-1}\frac{{\pi }^2}{(2n-4)(2n-5)}\frac{2^{2n-3}-1}{2^{2n-5}-1}\hfill \\ & & \times \frac{{\pi }^2}{(2n-2)(2n-3)}\frac{2^{2n-1}-1}{2^{2n-3}-1}\frac{{\pi }^2}{\left(2n\right)(2n-1)}\hfill \end{array}$$

$$\begin{array}{ccc}& =& \frac{128}{51975}\frac{\left\{\begin{array}{c}{2}^{2n+6}\left(2^{2n+6}-1\right)n\left(2n+1\right)\left(2n+3\right)\left(2n+5\right)\left(n+1\right)\left(n+2\right)\left(n+3\right)\\ \left[\begin{array}{c}14130836279n+2610109397n^2+252157346n^3\\ +13120940n^4+273800n^5+29051019788\end{array}\right]\end{array}\right\}}{\left(2n\right)!}\hfill \\ & & \times \frac{2^{2n+3}-1}{2^{2n+5}-1}\frac{1}{{\pi }^2}\frac{2^{2n+1}-1}{2^{2n+3}-1}\frac{1}{{\pi }^2}\frac{2^{2n-1}-1}{2^{2n+1}-1}\frac{1}{{\pi }^2}\hfill \\ & & -\frac{256}{31185}\frac{\left\{\begin{array}{c}{2}^{2n+4}\left(2^{2n+4}-1\right)n\left(2n+1\right)\left(2n+3\right)\left(n+1\right)\\ \left[\begin{array}{c}1007457799854n+448538582689n^2\\ +91600811650n^3+10838033881n^4+868613338n^5\\ +83295436n^6+4983160n^7+724428764568\end{array}\right]\end{array}\right\}}{\left(2n\right)!}\hfill \\ & & \times \frac{2^{2n+2}-1}{2^{2n+4}-1}\frac{1}{{\pi }^2}\frac{2^{2n}-1}{2^{2n+2}-1}\frac{1}{{\pi }^2}\hfill \\ & & -\frac{512}{1575}\frac{\left\{\begin{array}{c}{2}^{2n+2}\left(2^{2n+2}-1\right)n\left(2n+1\right)\\ \left[\begin{array}{c}-324988669975n-74432045925n^2-33240865835n^3\\ -11124715381n^4-922530750n^5\\ -44831420n^6+5749800n^7-5273077314\end{array}\right]\end{array}\right\}}{\left(2n\right)!}\frac{2^{2n}-1}{2^{2n+2}-1}\frac{1}{{\pi }^2}\hfill \\ & & -\frac{1024}{945}\frac{\left\{2^{2n}\left(2^{2n}-1\right)\left[\begin{array}{c}470063451466n-274339736559n^2+108583995240n^3\\ +12695894043n^4-3857965770n^5+610797924n^6\\ +45615080n^7-154356228096\end{array}\right]\right\}}{\left(2n\right)!}\hfill \\ & & -\frac{2048}{14175}\frac{\left\{2^{2n-2}\left(2^{2n-2}-1\right)\left[\begin{array}{c}-2587181560206n+1213011374071n^2\\ -180257338355n^3-1918656460n^4\\ +2580701900n^5+1889134753248\end{array}\right]\right\}}{\left(2n\right)!}\frac{2^{2n-1}-1}{2^{2n-3}-1}\frac{{\pi }^2}{1}\hfill \\ & & -\frac{16384}{10395}\frac{\left(2n-5\right)\left(-1277114903n+345227575n^2+551290272\right)2^{2n-4}\left(2^{2n-4}-1\right)}{\left(2n\right)!}\hfill \\ & & \times \frac{2^{2n-3}-1}{2^{2n-5}-1}\frac{{\pi }^2}{1}\frac{2^{2n-1}-1}{2^{2n-3}-1}\frac{{\pi }^2}{1}\hfill \\ & & -\frac{12560629760}{33}\frac{\left(2n-7\right)2^{2n-6}\left(2^{2n-6}-1\right)}{\left(2n\right)!}\frac{2^{2n-5}-1}{2^{2n-7}-1}\frac{{\pi }^2}{1}\frac{2^{2n-3}-1}{2^{2n-5}-1}\frac{{\pi }^2}{1}\frac{2^{2n-1}-1}{2^{2n-3}-1}\frac{{\pi }^2}{1}\hfill \end{array}$$

$$:=\frac{512}{155925}\frac{2^{2n}h\left(n\right)}{\left(2n\right)!{\pi }^6\left(32\times 2^{2n}-1\right)\left(2^{2n}-8\right)\left(2^{2n}-32\right)\left(2^{2n}-128\right)},$$

where

$$h\left(n\right)=\left[32u_1\left(n\right)2^{2n}-3v_1\left(n\right)\right]2^{8n}+2\left[u_2\left(n\right)2^{2n}-4v_2\left(n\right)\right]2^{4n}+\left[768u_3\left(n\right)2^{2n}-v_3\left(n\right)\right]$$

with

$$\begin{array}{ccc}\hfill u_1\left(n\right)& =& 50937555271680{\pi }^6-20780482285728{\pi }^8+82693540800{\pi }^{10}+12678309000{\pi }^{12}\hfill \\ & & -n\left[\begin{array}{c}86931451748160{\pi }^2-2088138616344{\pi }^4+155120938983780{\pi }^6\\ -28458997162266{\pi }^8+224644651770{\pi }^{10}\\ +3622374000{\pi }^{12}-125500405484160\end{array}\right]\hfill \\ & & +n^2\left[\begin{array}{c}132871790542788{\pi }^4-439643592392400{\pi }^2+90532113064470{\pi }^6\\ -13343125114781{\pi }^8+128411030430{\pi }^{10}+675997199597664\end{array}\right]\hfill \\ & & +n^3\left[\begin{array}{c}286866116806500{\pi }^4-844831868851080{\pi }^2-35832718429200{\pi }^6\\ +1982830721905{\pi }^8-20713654500{\pi }^{10}+1442689762206000\end{array}\right]\hfill \\ & & +n^4\left[\begin{array}{c}72113563243260{\pi }^4-807837420041960{\pi }^2-4189645034190{\pi }^6\\ +21105221060{\pi }^8+1632023568381360\end{array}\right]\hfill \\ & & +n^5\left[\begin{array}{c}30732153032196{\pi }^4-418096688859080{\pi }^2+1273128704100{\pi }^6\\ -28387720900{\pi }^8+1093719009379920\end{array}\right]\hfill \\ & & -n^6\left(120607531330440{\pi }^2-9176096758752{\pi }^4+201563314920{\pi }^6-456723432390192\right)\hfill \\ & & -n^7\left(20250571447920{\pi }^2-748397596320{\pi }^4+15052976400{\pi }^6-121394270067840\right)\hfill \\ & & +n^8\left(33229563840{\pi }^4-2188267794240{\pi }^2+20660024024640\right)\hfill \\ & & -n^9\left(181152533760{\pi }^2+4553841600{\pi }^4-2249071883520\right)\hfill \\ & & -n^{10}\left(15719186560{\pi }^2-154331890944\right)-n^{11}\left(797305600{\pi }^2-6142402560\right)\hfill \\ & & +105139200n^{12},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill v_1\left(n\right)& =& 91840412154839040{\pi }^6-36802234128024288{\pi }^8+135865487534400{\pi }^{10}\hfill \\ & & +14339167479000{\pi }^{12}\hfill \\ & & -n\left[\begin{array}{c}156737407501932480{\pi }^2-3764913925268232{\pi }^4+279683052987755340{\pi }^6\\ -50400883974373086{\pi }^8+369091162858110{\pi }^{10}\\ +4096904994000{\pi }^{12}-227594985345524160\end{array}\right]\hfill \\ & & +n^2\left[\begin{array}{c}239567838348646764{\pi }^4-792677397083497200{\pi }^2+163229399855239410{\pi }^6\\ -23630674578277151{\pi }^8+210979322996490{\pi }^{10}+1225920921470363664\end{array}\right]\hfill \\ & & -n^3\left[\begin{array}{c}1523231859538497240{\pi }^2-517219608602119500{\pi }^4+64606391327847600{\pi }^6\\ -3511593208493755{\pi }^8+34032534343500{\pi }^{10}-2616317883760581000\end{array}\right]\hfill \\ & & +n^4\left[\begin{array}{c}130020754527597780{\pi }^4-1456530868335653880{\pi }^2-7553929996644570{\pi }^6\\ +37377346497260{\pi }^8+2959674741259596360\end{array}\right]\hfill \\ & & +n^5\left[\begin{array}{c}55410071917049388{\pi }^4-753828330012921240{\pi }^2+2295451053492300{\pi }^6\\ -50274653713900{\pi }^8+1983459423510484920\end{array}\right]\hfill \\ & & -n^6\left[\begin{array}{c}217455378988783320{\pi }^2-16544502456029856{\pi }^4+363418656800760{\pi }^6\\ -828267944639613192\end{array}\right]\hfill \\ & & -n^7\left[\begin{array}{c}36511780320599760{\pi }^2-1349360866164960{\pi }^4+27140516449200{\pi }^6\\ -220148508768027840\end{array}\right]\hfill \\ & & +n^8\left(59912903603520{\pi }^4-3945446833014720{\pi }^2+37466953568684640\right)\hfill \\ & & -n^9\left(326618018369280{\pi }^2+8210576404800{\pi }^4-4078691860763520\right)\hfill \\ & & -n^{10}\left(28341693367680{\pi }^2-279880884226944\right)\hfill \\ & & -n^{11}\left(1437541996800{\pi }^2-11139247042560\right)+190669939200n^{12},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill u_2\left(n\right)& =& 4522669126239559680{\pi }^6-1685442576748540896{\pi }^8+4642498074052800{\pi }^{10}\hfill \\ & & +614099253033000{\pi }^{12}\hfill \\ & & -n\left[\begin{array}{c}7718513203541504160{\pi }^2-185402695537259244{\pi }^4\\ +13772955490961350530{\pi }^6\\ -2308223882839908462{\pi }^8+12611775395019570{\pi }^{10}\\ +175456929438000{\pi }^{12}-11475066825241427520\end{array}\right]\hfill \\ & & +n^2\left[\begin{array}{c}11797486974608332338{\pi }^4-39035295103132607400{\pi }^2\\ +8038210520824694595{\pi }^6-1082220848684542567{\pi }^8\\ +7209123659370630{\pi }^{10}+61809465946612609008\end{array}\right]\hfill \\ & & -n^3\left[\begin{array}{c}75011354387484116580{\pi }^2-25470412212073925250{\pi }^4\\ +3181533320251024200{\pi }^6-160821451361548835{\pi }^8\\ +1162885277284500{\pi }^{10}-131911617062424507000\end{array}\right]\hfill \\ & & +n^4\left[\begin{array}{c}6402855110024190510{\pi }^4-71726672769395565460{\pi }^2\\ -371992298118178815{\pi }^6+1711781164513420{\pi }^8+149223258963165460920\end{array}\right]\hfill \\ & & +n^5\left[\begin{array}{c}2728661769499134546{\pi }^4-37122177858764424580{\pi }^2\\ +113039187943982850{\pi }^6-2302442879036300{\pi }^8+100003650763148295240\end{array}\right]\hfill \\ & & -n^6\left[\begin{array}{c}10708561795532771940{\pi }^2-814731867064451952{\pi }^4\\ +17896504386774420{\pi }^6-41760278678881010424\end{array}\right]\hfill \\ & & +n^7\left[\begin{array}{c}66449099980858320{\pi }^4-1798017863003644920{\pi }^2\\ -1336531195091400{\pi }^6+11099624386517916480\end{array}\right]\hfill \\ & & +n^8\left(2950403129007840{\pi }^4-194293015048878240{\pi }^2+1889038966680946080\right)\hfill \\ & & -n^9\left(16084261743749760{\pi }^2+404328764901600{\pi }^4-205642763133709440\right)\hfill \\ & & -n^{10}\left(1395682995882560{\pi }^2-14111259282519168\right)\hfill \\ & & -n^{11}\left(70791568265600{\pi }^2-561627506872320\right)+9613350182400n^{12},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill v_2\left(n\right)& =& 7807148158935121920{\pi }^6-2162354645205998496{\pi }^8+7572578305219200{\pi }^{10}\hfill \\ & & +1121295004578000{\pi }^{12}\hfill \\ & & -n\left[\begin{array}{c}13323896677988735040{\pi }^2-320046917588428536{\pi }^4\\ +23775231197104976820{\pi }^6-2961357867713913162{\pi }^8\\ +20571609341185980{\pi }^{10}+320370001308000{\pi }^{12}-21891914231440417920\end{array}\right]\hfill \\ & & +n^2\left[\begin{array}{c}20365126464702573972{\pi }^4-67383733762390755600{\pi }^2\\ +13875766437278252430{\pi }^6-1388445570068766517{\pi }^8\\ +11759111700596820{\pi }^{10}+117918923506217715168\end{array}\right]\hfill \\ & & -n^3\left[\begin{array}{c}129486535706936180520{\pi }^2-43967682856815448500{\pi }^4\\ +5492044920925054800{\pi }^6-206327416429268585{\pi }^8\\ +1896832197183000{\pi }^{10}-251658474049928022000\end{array}\right]\hfill \\ & & +n^4\left[\begin{array}{c}11052773724731216940{\pi }^4-123816433532411167240{\pi }^2\\ -642142704745267110{\pi }^6+2196145987840420{\pi }^8+284685295197739294320\end{array}\right]\hfill \\ & & +n^5\left[\begin{array}{c}4710286363091648724{\pi }^4-64081261404742332520{\pi }^2\\ +195131163348702900{\pi }^6-2953941073691300{\pi }^8+190785062839205105040\end{array}\right]\hfill \\ & & +n^6\left[\begin{array}{c}1406411174117160288{\pi }^4-18485395719485208360{\pi }^2\\ -30893407714473480{\pi }^6+79669465375847921904\end{array}\right]\hfill \\ & & -n^7\left[\begin{array}{c}3103784835251250480{\pi }^2-114706151190370080{\pi }^4\\ +2307154639851600\pi -21175652287823806080\end{array}\right]\hfill \\ & & +n^8\left(5093062020192960{\pi }^4-335393616555370560{\pi }^2+3603872610786127680\right)\hfill \\ & & -n^9\left(27765067696861440{\pi }^2+697962748190400{\pi }^4-392321352145578240\right)\hfill \\ & & -n^{10}\left(2409264004864640{\pi }^2-26921192060598528\right)\hfill \\ & & -n^{11}\left(122202232006400{\pi }^2-1071462275358720\right)+18340166630400n^{12},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill u_3\left(n\right)& =& 72076640709427200{\pi }^6-29071894717733472{\pi }^8+115357489416000{\pi }^{10}\hfill \\ & & +17673562746000{\pi }^{12}\hfill \\ & & -n\left[\begin{array}{c}123008004223646400{\pi }^2-2954716142126760{\pi }^4\\ +219496128662048700{\pi }^6-39814137030010134{\pi }^8\\ +313379289219150{\pi }^{10}+5049589356000{\pi }^{12}\\ -346255618730797440\end{array}\right]\hfill \\ & & +n^2\left[\begin{array}{c}188013583618045020{\pi }^4-622095683235246000{\pi }^2\\ +128102939986225050{\pi }^6-18667032035578619{\pi }^8\\ +179133387449850{\pi }^{10}+1865076273689954976\end{array}\right]\hfill \\ & & -n^3\left[\begin{array}{c}1195437094424278200{\pi }^2-405915555281197500{\pi }^4\\ +50703296577318000{\pi }^6-2773980179945095{\pi }^8\\ +28895548027500{\pi }^{10}-3980381053926354000\end{array}\right]\hfill \\ & & +n^4\left[\begin{array}{c}102040691989212900{\pi }^4-1143089949359373400{\pi }^2\\ -5928347723378850{\pi }^6+29526204262940{\pi }^8+4502753025164172240\end{array}\right]\hfill \\ & & +n^5\left[\begin{array}{c}43485996540557340{\pi }^4-591606814735598200{\pi }^2\\ +1801477116301500{\pi }^6-39714421539100{\pi }^8+3017570746879199280\end{array}\right]\hfill \\ & & -n^6\left[\begin{array}{c}170659656832572600{\pi }^2-12984176913634080{\pi }^4\\ +285212090611800{\pi }^6-1260099949964539728\end{array}\right]\hfill \\ & & -n^7\left[\begin{array}{c}28654558598806800{\pi }^2-1058982598792800{\pi }^4\\ +21299961606000{\pi }^6-334926791117170560\end{array}\right]\hfill \\ & & +n^8\left(47019832833600{\pi }^4-3096398928849600{\pi }^2+57001006283981760\right)\hfill \\ & & -n^9\left(256330835270400{\pi }^2+6443685864000{\pi }^4-6205189326631680\right)\hfill \\ & & -n^{10}\left(22242648982400{\pi }^2-425801687114496\right)\hfill \\ & & -n^{11}\left(1128187424000{\pi }^2-16946888663040\right)+290079052800n^{12},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill v_3\left(n\right)& =& 1669121811142410240{\pi }^6-680934843538735104{\pi }^8+2709701944934400{\pi }^{10}\hfill \\ & & +415442829312000{\pi }^{12}\hfill \\ & & -n\left[\begin{array}{c}2848569810883706880{\pi }^2-68424126180360192{\pi }^4\\ +5083002928620503040{\pi }^6-932544419013132288{\pi }^8\\ +7361155949199360{\pi }^{10}+118697951232000{\pi }^{12}-4112397286904954880\end{array}\right]\hfill \\ & & +n^2\left[\begin{array}{c}4353942832506077184{\pi }^4-14406241235514163200{\pi }^2\\ +2966556280896552960{\pi }^6-437227523761143808{\pi }^8\\ +4207772645130240{\pi }^{10}+22151076236416253952\end{array}\right]\hfill \\ & & -n^3\left[\begin{array}{c}27683450678512189440{\pi }^2-9400028915515392000{\pi }^4\\ +1174166517488025600{\pi }^6-64973397095383040{\pi }^8\\ +678745030656000{\pi }^{10}-47274058127966208000\end{array}\right]\hfill \\ & & +n^4\left[\begin{array}{c}2363017240355143680{\pi }^4-26471216579934945280{\pi }^2\\ -137286288480337920{\pi }^6+691575883694080{\pi }^8+53478148288720404480\end{array}\right]\hfill \\ & & +n^5\left[\begin{array}{c}1007031190558998528{\pi }^4-13700192300534333440{\pi }^2\\ +41717881375948800{\pi }^6-930208838451200{\pi }^8+35838984499361218560\end{array}\right]\hfill \\ & & -n^6\left[\begin{array}{c}3952067586635857920{\pi }^2-300682338590785536{\pi }^4\\ +6604826703298560{\pi }^6-14965913432561811456\end{array}\right]\hfill \\ & & -n^7\left[\begin{array}{c}663570725205442560{\pi }^2-24523492436213760{\pi }^4\\ +493255930675200{\pi }^6-3977847441582981120\end{array}\right]\hfill \\ & & +n^8\left(1088866347909120{\pi }^4-71705159081656320{\pi }^2+676987667239403520\right)\hfill \\ & & -n^9\left(5936006226247680{\pi }^2+149220281548800{\pi }^4-73697587479183360\right)\hfill \\ & & -n^{10}\left(515086305198080{\pi }^2-5057147402452992\right)\hfill \\ & & -n^{11}\left(26126109900800{\pi }^2-201274247086080\right)+3445201305600n^{12}\hfill \end{array}$$

In order to prove the fact $h\left(n\right)>0$ for all $n\ge 10,$ we only need to prove

$$\begin{array}{ccc}\hfill 2^{2n}& >& \frac{3v_1\left(n\right)}{32u_1\left(n\right)},\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill 2^{2n}& >& \frac{4v_2\left(n\right)}{u_2\left(n\right)},\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill 2^{2n}& >& \frac{v_3\left(n\right)}{768u_3\left(n\right)}.\hfill \end{array}$$

(16)

By mathematical induction we can prove the inequality (14). First, the inequality (14) is obviously true for $n=10$. Let’s assume that (14) holds for $n=m\ge 10$, that is,

$$2^{2m}>\frac{3v_1\left(m\right)}{32u_1\left(m\right)}$$

holds. In the following we shall prove that (14) holds for $n=m+1$. Since

$$2^{2m+2}=4\times 2^{2m}>4\times \frac{3v_1\left(m\right)}{32u_1\left(m\right)},$$

we can complete the proof of (14) when showing that

$$\frac{A}{B}:=4\times \frac{v_1\left(m\right)}{u_1\left(m\right)}>\frac{v_1(m+1)}{u_1(m+1)}:=\frac{C}{D}.$$

In fact,

$$AD-BC=:\sum _{k=0}^{24}{c}_k{\left(m-10\right)}^k>0$$

due to

$$\begin{array}{ccc}\hfill c_0& =& 432\left(\begin{array}{c}-477072562034020909738451869462850857670400000{\pi }^2\\ +36333844528776677085760862383653680730240000{\pi }^4\\ -1096660044119881173039606612032659496064000{\pi }^6\\ +22603628786951665158157246354946557617600{\pi }^8\\ -177081693090107277506917029813967593600{\pi }^{10}\\ +1011386182849460538599746017498399960{\pi }^{12}\\ +15735694892973599257277408544518940{\pi }^{14}\\ +164925177170381951496845513459706{\pi }^{16}\\ +833787102925329026898819379985{\pi }^{18}\\ +2735552245409728360495799625{\pi }^{20}\\ +5698553429932956286470000{\pi }^{22}+5024135096342834062500{\pi }^{24}\\ +2024606215325539212345231880222924263936000000\end{array}\right),\hfill \\ \hfill c_1& =& 36\left(\begin{array}{c}-9259092928588108213738157014254288967190784000{\pi }^2\\ +666781730666771801641859416284587094918566400{\pi }^4\\ -17704367485003756899701481245215081272940800{\pi }^6\\ +349609143706987180240936965504219114769920{\pi }^8\\ -2292786547993054490216376074770576241280{\pi }^{10}\\ +16985127930125982303297749052630548352{\pi }^{12}\\ +238405063890342212113531066361567640{\pi }^{14}\\ +2056775089232944789108129630992300{\pi }^{16}\\ +8718404153959968277275906816475{\pi }^{18}\\ +23361306381577748319512467950{\pi }^{20}\\ +34836030692240124305070000{\pi }^{22}+17313942485858382000000{\pi }^{24}\\ +40333285172958894033508595153841536752427520000\end{array}\right),\hfill \\ \hfill c_2& =& 30\left(\begin{array}{c}-8600405454695740217297202993698878218858777600{\pi }^2\\ +585851470657954967766279287854352057781666816{\pi }^4\\ -13420979649772297096274328067880241801173760{\pi }^6\\ +256664843747929290271675413219724391636448{\pi }^8\\ -1318191158762232633137332745334314955072{\pi }^{10}\\ +13361650095808114588512739955776612716{\pi }^{12}\\ +164357094665669167870687737363349914{\pi }^{14}\\ +1153393325332854286640222711239803{\pi }^{16}\\ +4011026782712020246552983942479{\pi }^{18}\\ +8301133858901607698296592250{\pi }^{20}\\ +7938314104745805365718000{\pi }^{22}\\ +1484052213073575600000{\pi }^{24}\\ +38392471102288701428746670282674730459011522560\end{array}\right),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill c_3& =& 3\left(\begin{array}{c}-42406595192953714239088120732295910168515013120{\pi }^2\\ +2735291565968835971679114298266471264749820288{\pi }^4\\ -52741670654846269236778650448648201209274560{\pi }^6\\ +993185948970726744747604136369383526789664{\pi }^8\\ -3555997306232206297388095624723723524000{\pi }^{10}\\ +54062406959239828489920937977121874508{\pi }^{12}\\ +566940597003240042350635238785548750{\pi }^{14}\\ +3187649409611680011798317927299165{\pi }^{16}\\ +8804865616400119163677040780860{\pi }^{18}\\ +13053672727689667172707162800{\pi }^{20}\\ +6659045394142410022320000{\pi }^{22}\\ +193654905206289056409840361351472752548845829120\end{array}\right),\hfill \\ \hfill c_4& =& 3\left(\begin{array}{c}-14955148798977736804103569755902321698803321600{\pi }^2\\ +915015308159362839386515084161748493858700288{\pi }^4\\ -14378012135982618092933419839668648672422080{\pi }^6\\ +273349592197748554354724042991362976034704{\pi }^8\\ -529593588850577904103989040346522699600{\pi }^{10}\\ +15009655217430104938947296447083454608{\pi }^{12}\\ +130599893877570912079553672496420350{\pi }^{14}\\ +575949029053184203059736727251211{\pi }^{16}\\ +1205272758255181883607894742430{\pi }^{18}\\ +1145672964077124655646345700{\pi }^{20}\\ +208140442125042042000000{\pi }^{22}\\ +69735333795425763249955129532712165446253789696\end{array}\right),\hfill \\ \hfill c_5& =& 33\left(\begin{array}{c}-364971673005972619541275786034588814464538240{\pi }^2\\ +21235642834673600746560066559308986044337760{\pi }^4\\ -260272720784174227696306689234131950815600{\pi }^6\\ +5185831288296271982221695502649477464680{\pi }^8\\ -1517338330433370798176501636490316880{\pi }^{10}\\ +273963079484107283718813263942928076{\pi }^{12}\\ +1922123655099304273410851708056570{\pi }^{14}\\ +6447000532486188914092761900225{\pi }^{16}\\ +9531698924412844816314398900{\pi }^{18}\\ +4816843709968452110760000{\pi }^{20}\\ +1734512417114596725446236927443944851936945920\end{array}\right),\hfill \\ \hfill c_6& =& 165\left(\begin{array}{c}-15498197959547487369746919409739057432616768{\pi }^2\\ +860371404359307357300924677133234543516864{\pi }^4\\ -7722153657761174093282743885994122481136{\pi }^6\\ +170744152763792675021128653134791925376{\pi }^8\\ +210214671864247349408272379185208784{\pi }^{10}\\ +8181588172460856740957493021360356{\pi }^{12}\\ +44801413876306265313159782036522{\pi }^{14}\\ +109236316154736833622723890241{\pi }^{16}\\ +102352853202229552240678640{\pi }^{18}\\ +18242791857206659890000{\pi }^{20}\\ +74932250091965019743988691424419505579968128\end{array}\right),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill c_7& =& 132\left(\begin{array}{c}-3336483320624550277082804821179295650189120{\pi }^2\\ +177423483418269848617086554241647021814744{\pi }^4\\ -1054313446926034572685791686663751857520{\pi }^6\\ +28435416737882498941385585576136821888{\pi }^8\\ +70884370014078824758070811714152740{\pi }^{10}\\ +1151438819679557596009052527626068{\pi }^{12}\\ +4727935854230453222127552968555{\pi }^{14}\\ +7829247889431961426897274700{\pi }^{16}\\ +3850065296847814425825500{\pi }^{18}\\ +16384197077544879182425462583508872540455680\end{array}\right),\hfill \\ \hfill c_8& =& 132\left(\begin{array}{c}-474242872953666237583921762124208248951760{\pi }^2\\ +24264409040721159948864583225281930027504{\pi }^4\\ -78402560531987172021983078009066260920{\pi }^6\\ +3114682925218861740201767089397817568{\pi }^8\\ +10489929857119997818261682505350280{\pi }^{10}\\ +98088835968824635053831799950513{\pi }^{12}\\ +286554414340316246294571853510{\pi }^{14}\\ +289431461545719136085136750{\pi }^{16}\\ +49260491111811084525000{\pi }^{18}\\ +2362007630426151607945908367872419621154144\end{array}\right),\hfill \\ \hfill c_9& =& 1320\left(\begin{array}{c}-5629877288570319416997038146969516556832{\pi }^2\\ +278442060045992946080707207531074123396{\pi }^4\\ -253292501283486387462068625056270430{\pi }^6\\ +28270385849322427748766607916610979{\pi }^8\\ +107046143835094422774951890666596{\pi }^{10}\\ +628570656348882755440623147684{\pi }^{12}\\ +1213867590548404875329886800{\pi }^{14}\\ +619538560057063391389750{\pi }^{16}\\ +28411534488704240841370947744349251289824\end{array}\right),\hfill \\ \hfill c_{10}& =& 2640\left(\begin{array}{c}-281331583582014611449980637041769931436{\pi }^2\\ +13510570740017929393709981880756475284{\pi }^4\\ +13087642759352321650198647418555857{\pi }^6\\ +1064155448978148256227645686806861{\pi }^8\\ +3997026049205628330034092483580{\pi }^{10}\\ +14904436197642393080361285029{\pi }^{12}\\ +17024945059305125246624430{\pi }^{14}\\ +2897785783916157035125{\pi }^{16}\\ +1437953217456186736325001905444225594136\end{array}\right),\hfill \\ \hfill c_{11}& =& 96\left(\begin{array}{c}-654142166808161291482752284899283952120{\pi }^2\\ +30622350854820437346460584814635483927{\pi }^4\\ +72647894779938686582882719829927595{\pi }^6\\ +1817767836009184023588819061194702{\pi }^8\\ +6102125876861689812210796065620{\pi }^{10}\\ +13944810933176636180593961600{\pi }^{12}\\ +7793968097768424002835250{\pi }^{14}\\ +3387419372637281269675203706863313900560\end{array}\right),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill c_{12}& =& 192\left(\begin{array}{c}-23448273103738305639345475800400670145{\pi }^2\\ +1073000636718656551244165531387881431{\pi }^4\\ +3537234644266436071857010425022050{\pi }^6\\ +46060244426147238364390717134786{\pi }^8\\ +125088476859641817643384089420{\pi }^{10}\\ +160886574944092901528877625{\pi }^{12}+29074440554757759078750{\pi }^{14}\\ +123179632897044306386027545261215114294\end{array}\right),\hfill \\ \hfill c_{13}& =& 1920\left(\begin{array}{c}-142512239971185783671360961222010803{\pi }^2\\ +6381902474805353580527764618726671{\pi }^4\\ +24025453176262392534800825080200{\pi }^6\\ +187106225790007458438906057128{\pi }^8\\ +368869412666063388113272660{\pi }^{10}+225472297847063882873300{\pi }^{12}\\ +761394621013098460862455421333712000\end{array}\right),\hfill \\ \hfill c_{14}& =& 3840\left(\begin{array}{c}-3661949524532302117959247029761739{\pi }^2\\ +160288394718581152122551619251667{\pi }^4\\ +613223170928199896028344483994{\pi }^6\\ +2968196572317252224510234552{\pi }^8\\ +3705409180239129445272060{\pi }^{10}+724592963261234153625{\pi }^{12}\\ +19979501787561082580106002588049444\end{array}\right),\hfill \\ \hfill c_{15}& =& 16896\left(\begin{array}{c}-35978247257137216758065815515375{\pi }^2\\ +1532726528655032867385038505066{\pi }^4\\ +5441621946243633291744787950{\pi }^6\\ +16080285251594580782740150{\pi }^8\\ +10324423800377796581500{\pi }^{10}\\ +201696230866728389067234661444560\end{array}\right),\hfill \\ \hfill c_{16}& =& 67584\left(\begin{array}{c}-324384222357330712997593014615{\pi }^2\\ +13337410857823397206189989309{\pi }^4\\ +40326199296784791667944120{\pi }^6+67638374749826629884125{\pi }^8\\ +14563211542748706250{\pi }^{10}+1885127045277192968136557300967\end{array}\right),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill c_{17}& =& 1013760\left(\begin{array}{c}-642785638178810588749900158{\pi }^2\\ +25150395742996351629888498{\pi }^4\\ +58845770244082231841370{\pi }^6\\ +47711575106631655975{\pi }^8\\ +3920991972850346056633649952\end{array}\right),\hfill \\ \hfill c_{18}& =& 6082560\left(\begin{array}{c}-2578876374474054585889758{\pi }^2\\ +93892348055183622493182{\pi }^4\\ +149964428634529802305{\pi }^6+39786653594457875{\pi }^8\\ +16807275050343112745119008\end{array}\right),\hfill \\ \hfill c_{19}& =& 4055040\left(\begin{array}{c}-73520039656935692234682{\pi }^2+2403072677327868116495{\pi }^4\\ +2150159715906956775{\pi }^6+525199288403727204596784\end{array}\right),\hfill \\ \hfill c_{20}& =& 1622016\left(\begin{array}{c}-2654056221052595612805{\pi }^2+73331459824530209275{\pi }^4\\ +24215563398048750{\pi }^6+21598998856016744526858\end{array}\right),\hfill \\ \hfill c_{21}& =& 600145920\left(-73933454572793875{\pi }^2+1540752654978775{\pi }^4+729733152472938432\right),\hfill \\ \hfill c_{22}& =& 20186726400\left(-14400815186065{\pi }^2+170333752225{\pi }^4+193626789624684\right),\hfill \\ \hfill c_{23}& =& 22182474098824445952000-909492677357322240000{\pi }^2,\hfill \\ \hfill c_{24}& =& 60140654614609920000\hfill \end{array}$$

are all positive.

We can prove (15) and (16) in a same way. This completes the proof of Lemma 3. □

Lemma 4.

The function

$$\begin{array}{ccc}& & q\left(x\right)=86048790528x+19300663296x^3\hfill \\ & & +\left[-344139832320x+107890544640x^3-7832862720x^5\right]\frac{1}{{cos}^{2}x}\hfill \\ & & +\left[516126753792x-462262103040x^3+44561252352x^5\right]\frac{1}{{cos}^{4}x}\hfill \\ & & +\left[-344029172736x+594343956480x^3-96517152768x^5\right]\frac{1}{{cos}^{6}x}\hfill \\ & & +\left[85993460736x-329967098880x^3+114528079872x^5-1570078720x^7\right]\frac{1}{{cos}^{8}x}\hfill \\ & & +\left[70694037504x^3-61197136896x^5+3728936960x^7\right]\frac{1}{{cos}^{10}x}\hfill \\ & & +\left[6457820160x^5-3189222400x^7\right]\frac{1}{{cos}^{12}x}\hfill \end{array}$$

$$\begin{array}{ccc}& & -\left[28548897792+20104857600x^2\right]\frac{sinx}{cosx}\hfill \\ & & +\left[85646693376-150623247360x^2-9010511872x^4\right]\frac{sinx}{{cos}^{3}x}\hfill \\ & & +\left[-85646693376+467203461120x^2-132699697152x^4+3491758080x^6\right]\frac{sinx}{{cos}^{5}x}\hfill \\ & & +\left[28548897792-402117749760x^2+259354011648x^4-19975766016x^6\right]\frac{sinx}{{cos}^{7}x}\hfill \\ & & +\left[105642393600x^2-145966994432x^4+20627970048x^6\right]\frac{sinx}{{cos}^{9}x}\hfill \\ & & +\left[28323191808x^4-18861972480x^6\right]\frac{sinx}{{cos}^{11}x}\hfill \\ & & +630835200x^6\frac{sinx}{{cos}^{13}x}\hfill \\ & >& 0\hfill \end{array}$$

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

Proof. 

Substituting the power series expansions of the functions in Lemma 1 into the above function $q\left(x\right)$, we can obtain that

$$\begin{array}{ccc}\hfill q\left(x\right)& =& \sum _{n=13}^{\infty }\frac{128}{51975}\frac{\left\{\begin{array}{c}{2}^{2n}\left(2^{2n}-1\right)n\left(n-1\right)\left(n-2\right)\left(n-3\right)\left(2n-1\right)\left(2n-3\right)\left(2n-5\right)\\ \left[\begin{array}{c}3972255719n+975298043n^2+119348066n^3\\ +9013940n^4+273800n^5+4337509922\end{array}\right]\end{array}\right\}}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-7}\hfill \\ & & -\sum _{n=12}^{\infty }\frac{256}{31185}\frac{\left\{\begin{array}{c}{2}^{2n}\left(2^{2n}-1\right)\left(n-1\right)\left(n-2\right)\left(2n-1\right)\left(2n-3\right)\\ \left[\begin{array}{c}21424923714n+106199680013n^2\\ +29104373962n^3+5754341861n^4+287653546n^5\\ +13531196n^6+4983160n^7-78833018880\end{array}\right]\end{array}\right\}}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-5}\hfill \\ & & -\sum _{n=11}^{\infty }\frac{512}{1575}\frac{\left\{\begin{array}{c}{2}^{2n}\left(2^{2n}-1\right)\left(n-1\right)\left(2n-1\right)\\ \left[\begin{array}{c}-235651730736n-33025650306n^2\\ +3130559589n^3-7385775931n^4-532796430n^5\\ -85080020n^6+5749800n^7+268271646720\end{array}\right]\end{array}\right\}}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-3}\hfill \\ & & -\sum _{n=10}^{\infty }\frac{1024}{945}\frac{\left\{\begin{array}{c}{2}^{2n}\left(2^{2n}-1\right)\\ \left[\begin{array}{c}470063451466n-274339736559n^2+108583995240n^3\\ +12695894043n^4-3857965770n^5\\ +610797924n^6+45615080n^7-154356228096\end{array}\right]\end{array}\right\}}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-1}\hfill \\ & & -\sum _{n=9}^{\infty }\frac{2048}{14175}\frac{\left\{\begin{array}{c}{2}^{2n}\left(2^{2n}-1\right)\\ \left[\begin{array}{c}-696701943469n+686534439246n^2-162124945195n^3\\ +10984853040n^4+2580701900n^5+335369274198\end{array}\right]\end{array}\right\}}{\left(2n\right)!}\left|B_{2n}\right|x^{2n+1}\hfill \\ & & -\sum _{n=8}^{\infty }\frac{16384}{10395}\frac{2^{2n}\left(2n-1\right)\left(103795397n+345227575n^2-622029234\right)\left(2^{2n}-1\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n+3}\hfill \\ & & -\sum _{n=7}^{\infty }\frac{12560629760}{33}\frac{2^{2n}\left(2n-1\right)\left(2^{2n}-1\right)}{\left(2n\right)!}\left|B_{2n}\right|x^{2n+5}\hfill \end{array}$$

$$\begin{array}{ccc}& =& \sum _{n=10}^{\infty }\frac{128}{51975}\frac{\left\{\begin{array}{c}{2}^{2n+6}\left(2^{2n+6}-1\right)n\left(2n+1\right)\left(2n+3\right)\left(2n+5\right)\left(n+1\right)\left(n+2\right)\left(n+3\right)\\ \left[\begin{array}{c}14130836279n+2610109397n^2+252157346n^3\\ +13120940n^4+273800n^5+29051019788\end{array}\right]\end{array}\right\}}{\left(2n+6\right)!}\left|B_{2n+6}\right|x^{2n-1}\hfill \\ & & -\sum _{n=10}^{\infty }\frac{256}{31185}\frac{\left\{\begin{array}{c}{2}^{2n+4}\left(2^{2n+4}-1\right)n\left(2n+1\right)\left(2n+3\right)\left(n+1\right)\\ \left[\begin{array}{c}1007457799854n+448538582689n^2\\ +91600811650n^3+10838033881n^4\\ +868613338n^5+83295436n^6+4983160n^7+724428764568\end{array}\right]\end{array}\right\}}{\left(2n+4\right)!}\left|B_{2n+4}\right|x^{2n-1}\hfill \\ & & -\sum _{n=10}^{\infty }\frac{512}{1575}\frac{\left\{\begin{array}{c}{2}^{2n+2}\left(2^{2n+2}-1\right)n\left(2n+1\right)\\ \left[\begin{array}{c}-324988669975n-74432045925n^2-33240865835n^3\\ -11124715381n^4-922530750n^5\\ -44831420n^6+5749800n^7-5273077314\end{array}\right]\end{array}\right\}}{\left(2n+2\right)!}\left|B_{2n+2}\right|x^{2n-1}\hfill \\ & & -\sum _{n=10}^{\infty }\frac{1024}{945}\frac{\left\{\begin{array}{c}{2}^{2n}\left(2^{2n}-1\right)\\ \left[\begin{array}{c}470063451466n-274339736559n^2+108583995240n^3\\ +12695894043n^4-3857965770n^5\\ +610797924n^6+45615080n^7-154356228096\end{array}\right]\end{array}\right\}}{\left(2n\right)!}\left|B_{2n}\right|x^{2n-1}\hfill \\ & & -\sum _{n=10}^{\infty }\frac{2048}{14175}\frac{\left\{\begin{array}{c}{2}^{2n-2}\left(2^{2n-2}-1\right)\\ \left[\begin{array}{c}-2587181560206n+1213011374071n^2-180257338355n^3\\ -1918656460n^4+2580701900n^5+1889134753248\end{array}\right]\end{array}\right\}}{\left(2n-2\right)!}\left|B_{2n-2}\right|x^{2n-1}\hfill \\ & & -\sum _{n=10}^{\infty }\frac{16384}{10395}\frac{2^{2n-4}\left(2^{2n-4}-1\right)\left(2n-5\right)\left(-1277114903n+345227575n^2+551290272\right)}{\left(2n-4\right)!}\left|B_{2n-4}\right|x^{2n-1}\hfill \\ & & -\sum _{n=10}^{\infty }\frac{12560629760}{33}\frac{\left(2n-7\right)2^{2n-6}\left(2^{2n-6}-1\right)}{\left(2n-6\right)!}\left|B_{2n-6}\right|x^{2n-1}\hfill \\ & :& =\sum _{n=10}^{\infty }{a}_n{x}^{2n-1},\hfill \end{array}$$

where $a_n$ ($n\ge 10$) are defined as Lemma 3. From Lemma 3 we have $a_n>0$ for all $n\ge 10$, which completes the proof of this Lemma. □

Lemma 5.

Let $0<x<\pi /2$, and

$$\begin{array}{ccc}\hfill P_1\left(x\right)& =& 10\left[\begin{array}{c}-21x{sin}^{2}x+21x{sin}^{4}x-1577cosx{sin}^{3}x+1577cosx{sin}^{5}x+185x^3\\ +1413x^{2}cosxsinx-960x^{2}cosx{sin}^{3}x\end{array}\right],\hfill \\ \hfill P_2\left(x\right)& =& \left[\begin{array}{c}3201x{sin}^{2}x+3273x{sin}^{4}x-6474x{sin}^{6}x+5893cosx{sin}^{3}x\\ -5893cosx{sin}^{5}x+1554x^3{sin}^{2}x+1184x^3{sin}^{4}x-1813x^3\\ -7281x^{2}cosxsinx+15972x^{2}cosx{sin}^{3}x-3984x^{2}cosx{sin}^{5}x.\end{array}\right].\hfill \end{array}$$

Then

$${\left(\frac{x}{sinx}\right)}^{\frac{6}{5}}<\frac{P_1\left(x\right)}{P_2\left(x\right)}.$$

(17)

Proof. 

Let

$$P\left(x\right)=ln\frac{P_1\left(x\right)}{P_2\left(x\right)}-\frac{6}{5}ln\left(\frac{x}{sinx}\right),0<x<\frac{\pi }{2}.$$

Then

$$\begin{array}{ccc}\hfill P^{\prime }\left(x\right)& =& \frac{P_1^{\prime }\left(x\right)P_2\left(x\right)-P_1\left(x\right)P_2^{\prime }\left(x\right)}{P_1\left(x\right)P_2\left(x\right)}-\frac{6}{5}\frac{\left(sinx-xcosx\right)}{xsinx}\hfill \\ & =& \frac{5x\left(sinx\right)\left[P_1^{\prime }\left(x\right)P_2\left(x\right)-P_1\left(x\right)P_2^{\prime }\left(x\right)\right]-6P_1\left(x\right)P_2\left(x\right)\left(sinx-xcosx\right)}{5x\left(sinx\right)P_1\left(x\right)P_2\left(x\right)}\hfill \\ & :& =\frac{\left({cos}^{13}x\right)q\left(x\right)}{2560x\left(sinx\right)P_1\left(x\right)P_2\left(x\right)},\hfill \end{array}$$

where $q\left(x\right)$ is defined as Lemma 4. From Lemma 4 we have $P^{\prime }\left(x\right)>0$ for all $x\in \left(0,\pi /2\right)$. Considering the fact $P\left(0^{+}\right)=0$, we have completed the proof of Lemma 5. □

3. Proof of Theorem 1

Proof. 

Let $x\in \left(0,\pi /2\right)$ and

$$F\left(x\right)=ln\left[\frac{16}{7875}{x}^6{\left(\frac{tanx}{x}\right)}^{\frac{5}{4}}\right]-ln\left[2-\left({\left(\frac{x}{sinx}\right)}^{\frac{6}{5}}+{\left(\frac{x}{tanx}\right)}^{\frac{3}{5}}\right)\right].$$

Then

$$F^{\prime }\left(x\right):=\frac{f\left(x\right)}{20x\left(tanx\right)\left({sin}^{2}x\right)\left[2\left(sinxtanx\right){\left(\frac{x}{tanx}\right)}^{\frac{2}{5}}-x\left(tanx\right)\sqrt[5]{\frac{x}{sinx}}{\left(\frac{x}{tanx}\right)}^{\frac{2}{5}}-xsinx\right]},$$

where

$$f\left(x\right)=\left({tan}^{3}x\right){\left(\frac{x}{tanx}\right)}^{\frac{2}{5}}\left[\begin{array}{c}10\left(sinx\right)\left(5x+19cosxsinx\right)\\ -x{\left(\frac{x}{sinx}\right)}^{\frac{1}{5}}\left(25x+24x{cos}^{2}x+71cosxsinx\right)\\ -x\left(cosx\right)\left(37x+83cosxsinx\right){\left(\frac{tanx}{x}\right)}^{\frac{2}{5}}.\end{array}\right]$$

In the following we shall prove $f\left(x\right)>0$, which is

$$\begin{array}{ccc}& & 10\left(sinx\right)\left(5x+19cosxsinx\right)-x{\left(\frac{x}{sinx}\right)}^{\frac{1}{5}}\left(25x+24x{cos}^{2}x+71cosxsinx\right)\hfill \\ & >& x\left(cosx\right)\left(37x+83cosxsinx\right){\left(\frac{tanx}{x}\right)}^{\frac{2}{5}}.\hfill \end{array}$$

Let

$$\begin{array}{ccc}\hfill G\left(x\right)& =& ln\left[10\left(sinx\right)\left(5x+19cosxsinx\right)-x{\left(\frac{x}{sinx}\right)}^{\frac{1}{5}}\left(25x+24x{cos}^{2}x+71cosxsinx\right)\right]\hfill \\ & & -ln\left[x\left(cosx\right)\left(37x+83cosxsinx\right){\left(\frac{tanx}{x}\right)}^{\frac{2}{5}}\right].\hfill \end{array}$$

Then

$$G^{\prime }\left(x\right):=\frac{3\left(sinx\right){\left(\frac{x}{sinx}\right)}^{\frac{4}{5}}H\left(x\right)}{\left\{\begin{array}{c}5x\left(cosx\right)\left(37x+83cosxsinx\right)\\ \left[\begin{array}{c}50x\left({sin}^{2}x\right){\left(\frac{x}{sinx}\right)}^{\frac{4}{5}}+190\left(cosx{sin}^{3}x\right){\left(\frac{x}{sinx}\right)}^{\frac{4}{5}}\\ -\left(24x^3{cos}^{2}x+25x^3+71x^{2}cosxsinx\right)\end{array}\right]\end{array}\right\}},$$

where

$$H\left(x\right)=P_1\left(x\right)-{\left(\frac{x}{sinx}\right)}^{\frac{6}{5}}{P}_2\left(x\right)=P_2\left(x\right)\left[\frac{P_1\left(x\right)}{P_2\left(x\right)}-{\left(\frac{x}{sinx}\right)}^{\frac{6}{5}}\right],$$

$P_1\left(x\right)$ and $P_2\left(x\right)$ are defined as Lemma 5. From Lemma 5 we obtain that $H\left(x\right)>0$ for all $x\in \left(0,\pi /2\right)$, which leads to $G^{\prime }\left(x\right)>0$. Since $G\left(0^{+}\right)=0$ we get $G\left(x\right)>G\left(0^{+}\right)=0$ for all $x\in \left(0,\pi /2\right)$. So $f\left(x\right)>0$ and $F^{\prime }\left(x\right)>0$. Taking the fact $F\left(0^{+}\right)=0$ we have $F\left(x\right)>F\left(0^{+}\right)=0$ for all $x\in \left(0,\pi /2\right)$.

The proof of Theorem 1 is completed. □