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Refining trigonometric inequalities by using Padé approximant


A two-point Padé approximant method is presented for refining some remarkable trigonometric inequalities including the Jordan inequality, Kober inequality, Becker–Stark inequality, and Wu–Srivastava inequality. Simple proofs are provided. It shows to achieve better approximation results than those of prevailing methods.

1 Introduction

Trigonometric inequalities have caused interest of a lot of researchers, they analyzed the Wilker inequality [611, 14, 1619], Jordan inequality [3, 5, 15, 20, 21], Shafer–Fink inequality [12], Becker–Stark inequalities [13], and so on.

Recently, Bercu provided a Padé-approximant-based method and obtained the following inequalities [2].

$$\begin{aligned} &b_{1}(x)= \frac{-7 x^{2} + 60}{3 x^{2} + 60} < \frac{\sin(x)}{x} < \frac{11 x^{4} - 360 x^{2}+ 2520}{60 x^{2}+ 2520}= b_{2}(x),\quad \forall x \in (0,\pi/2); \end{aligned}$$
$$\begin{aligned} & \begin{aligned}[b] b_{3}(x) &=\frac{17x^{4} - 480x^{2}+ 1080}{2x^{4}+ 60x^{2}+ 1080} < \cos(x) < \frac{3x^{4} - 56x^{2}+ 120}{4x^{2}+ 120} \\ &= b_{4}(x),\quad \forall x \in (0,\pi/2); \end{aligned} \end{aligned}$$
$$\begin{aligned} & b_{5}(x) < \frac{\tan(x)}{x}< b_{6}(x),\quad \forall x \in (0,1.5701); \end{aligned}$$
$$\begin{aligned} & \biggl(\frac{x}{\sin(x)}\biggr)^{2}+\frac{x}{\tan(x)} > b_{7}(x),\quad \forall x \in (0,1.5701), \end{aligned}$$

where \(b_{5}(x)=\frac{-28x^{4} - 600x^{2}+ 7200}{9x^{6}+ 12x^{4} - 3000x^{2}+ 7200}\), \(b_{6}(x)=\frac{22x^{8} - 60x^{6} - 4680x^{4}- 237\mbox{,}600x^{2}+ 2\mbox{,}721\mbox{,}600}{1020x^{6}+ 14\mbox{,}040x^{4}- 1\mbox{,}144\mbox{,}800x^{2}+ 2\mbox{,}721\mbox{,}600}\) and \(b_{7}(x)=\frac{11\mbox{,}220x^{10}-205\mbox{,}560x^{8}-14\mbox{,}256\mbox{,}000x^{6}+512\mbox{,}179\mbox{,}200x^{4}- 3\mbox{,}157\mbox{,}056\mbox{,}000x^{2}+13\mbox{,}716\mbox{,}864\mbox{,}000}{242x^{12}-8580x^{10} +25\mbox{,}560x^{8}-1\mbox{,}080\mbox{,}000x^{6}+103\mbox{,}680\mbox{,}000x^{4}-1\mbox{,}578\mbox{,}528\mbox{,}000x^{2}+6\mbox{,}858\mbox{,}432\mbox{,}000}\).

In this paper, we present a two-point Padé-approximant-based method [1] for refining the rational bounds of several trigonometric inequalities, and also provide a method for proving the refined bounds. By applying the new method to \(\frac{\sin (x)}{x}\) and \(\cos(x)\), we refine the bounds of Eq. (1) (2), for \(\forall x \in [0,\pi/2]\), see also Theorems 3.1 and 3.2. Applied to \(\frac{\tan (x)}{x}\) and \((\frac{x}{\sin(x)})^{2}+\frac{x}{\tan(x)}\), it not only provides refined two-sided bounds with better approximation effect for Eq. (3) (4), but also extends the interval \((0,1.5701)\) to the interval \([0,\pi/2]\), see also the theorems and remarks in Sect. 3.

2 Find bounds by using two-point Padé approximant

Given a bounded smooth function \(f(x)\), \(x \in [x_{0}, x_{1}]\), let \(R(x)=\frac{\sum^{n}_{i=0} c_{i} x^{i}}{1+\sum^{m}_{i=1} d_{i} x^{i}}\) be a rational polynomial interpolating derivatives of \(f(x)\) at two points \(x_{0}\) and \(x_{1}\) such that

$$ E^{(i)}(x_{0})=0, \qquad E^{(j)}(x_{1})=0,\quad i=0,1, \ldots,k, j=0,1,\ldots,l, $$

where \(E(x)=(1+\sum^{m}_{i=1}d_{i} x^{i}) \cdot f(x) -(\sum^{n}_{i=1}c_{i} x^{i})\). There are \(m+n+2\) unknowns in Eq. (5). By selecting suitable values of k and l, we have that Eq. (5) consists of \(m+n+2\) linear equations in the unknown variables \(c_{i}\) and \(d_{j}\), and the interpolation polynomial \(R(x)\) can be determined by solving Eq. (5).

We give two examples. Without loss of generality, let \(\Gamma=[0,\pi/2]\).

Example 1

Let \(f_{1}(x)=\sin(x)\). By setting \(n_{1}=13\), \(m_{1}=0\), \(n_{2}=11\), and \(m_{2}=0\) and introducing the following constraints

$$ f_{1}^{(i)}(0)=R_{j}^{(i)}(0),\qquad f_{1}(\pi/2)=R_{j}(\pi/2),\quad j=1,2, i=0,1,\ldots,14-2j, $$

we obtain that

$$ R_{1}(x)= \beta_{1}(x)+ \alpha_{1} \cdot x^{13}, \qquad R_{2}(x)= \beta_{2}(x)- \alpha_{2} \cdot x^{11}, $$

where \(\alpha_{1}=\frac{\pi^{11}-440 \pi^{9}+126\mbox{,}720 \pi^{7}-21\mbox{,}288\mbox{,}960 \pi^{5}+1\mbox{,}703\mbox{,}116\mbox{,}800 \pi^{3}-40\mbox{,}874\mbox{,}803\mbox{,}200 \pi+81\mbox{,}749\mbox{,}606\mbox{,}400}{9\mbox{,}979\mbox{,}200 \pi^{13}}\), \(\beta_{1}(x)=t-\frac{t^{3}}{6}+ \frac{t^{5}}{120} - \frac{t^{7}}{5040} + \frac{t^{9}}{362\mbox{,}880}-\frac{x^{11}}{39\mbox{,}916\mbox{,}800}\), \(\alpha_{2}=\frac{\pi^{9}-288 \pi^{7}+48\mbox{,}384 \pi^{5}-3\mbox{,}870\mbox{,}720 \pi^{3} +92\mbox{,}897\mbox{,}280 \pi-185\mbox{,}794\mbox{,}560}{90\mbox{,}720 \pi^{11}} \), \(\beta_{2}(x)=t-\frac{t^{3}}{6}+ \frac{t^{5}}{120} - \frac{t^{7}}{5040} + \frac{t^{9}}{362\mbox{,}880}\). It can be verified that \(R_{j}(x) \geq 0, \forall x \in \Gamma, j=1,2\). From Eq. (6), \(\forall x \in \Gamma\), there exists \(\xi_{j}(x) \in \Gamma\) such that [4]

$$ f_{1}(x) - R_{j}(x) = \frac{f_{1}^{(16-2j)}(\xi_{j}(x))}{(16-2j)!} \cdot (x-\pi/2) \cdot x^{15-2j}, \quad x \in \Gamma, j=1,2. $$

Note that \(f_{1}^{(14)}(x) = -\sin(x) \leq 0\) and \(f_{1}^{(12)}(x) = \sin(x) \geq 0\), \(\forall x \in \Gamma\). Combining with Eq. (8), one obtains that

$$ 0 \leq R_{1}(x) \leq \sin(x) \leq R_{2}(x), \quad x \in \Gamma. $$

Example 2

Let \(f_{2}(x)=\cos(x)\). By setting \(n_{3}=12\), \(m_{3}=0\), \(n_{4}=10\), and \(m_{4}=0\) and introducing the following constraints

$$ f_{2}^{(i)}(0)=R_{j}^{(i)}(0),\qquad f_{2}(\pi/2)=R_{j}(\pi/2),\quad j=3,4, i=0,1,\ldots,17-2j, $$

we obtain that

$$ R_{3}(x)= \beta_{3}(x)+ \alpha_{3} \cdot x^{12} , \qquad R_{4}(x)= \beta_{4}(x)- \alpha_{4} \cdot x^{10}, $$

where \(\alpha_{3}=\frac{\pi^{10}-360 \pi^{8}+80\mbox{,}640 \pi^{6}-9\mbox{,}676\mbox{,}800 \pi^{4}+464\mbox{,}486\mbox{,}400 \pi^{2}-3\mbox{,}715\mbox{,}891\mbox{,}200}{907\mbox{,}200 \pi^{12}}\), \(\beta_{3}(x)=1-\frac{x^{2}}{2} +\frac{x^{4}}{24} -\frac{x^{6}}{720} +\frac{x^{8}}{40\mbox{,}320} -\frac{x^{10}}{3\mbox{,}628\mbox{,}800}\), \(\alpha_{4}=\frac{10\mbox{,}321\mbox{,}920-1\mbox{,}290\mbox{,}240 \pi^{2}+26\mbox{,}880 \pi^{4}-224 \pi^{6}+\pi^{8}}{10\mbox{,}080 \pi^{10}} \), \(\beta_{4}(x)=1-\frac{x^{2}}{2} +\frac{x^{4}}{24} -\frac{x^{6}}{720} +\frac{x^{8}}{40\mbox{,}320} \). It can be verified that \(R_{j}(x) \geq 0, \forall x \in \Gamma, j=3,4\). From Eq. (10), \(\forall x \in \Gamma\), there exists \(\xi_{j}(x) \in \Gamma, j=3,4\), such that [4]

$$ f_{2}(x) - R_{j}(x) = \frac{f_{2}^{(19-2j)}(\xi_{j}(x))}{(19-2j)!} \cdot (x-\pi/2) \cdot x^{18-2j}, \quad x \in \Gamma, j=3,4. $$

Note that \(f_{2}^{(13)}(x) = -\sin(x) \leq 0\) and \(f_{2}^{(11)}(x) = \sin(x) \geq 0\), \(\forall x \in \Gamma\). Combining with Eq. (12), one obtains that

$$ 0 \leq R_{3}(x) \leq \cos(x) \leq R_{4}(x), \quad x \in \Gamma. $$

3 Main results

The main results are as follows.

Theorem 3.1

For all \(\forall x \in \Gamma=[0,\pi/2]\), we have that

$$\begin{aligned}{} [b] c_{1}(x) &= \frac{60\mbox{,}480-9240 x^{2}+364 x^{4}-5 x^{6} }{840 (72+x^{2})} \leq \frac{\sin(x)}{x} \\ &\leq \frac{ (166\mbox{,}320-22\mbox{,}260 x^{2}+551 x^{4}) }{15 (11\mbox{,}088+364 x^{2}+5 x^{4})} =c_{2}(x). \end{aligned}$$


Eq. (14) is equivalent to

$$ \textstyle\begin{cases} (60\mbox{,}480-9240 x^{2}+364 x^{4}-5 x^{6}) x - 840 (72+x^{2}) \sin(x) \leq 0, \\ (166\mbox{,}320-22\mbox{,}260 x^{2}+551 x^{4}) x - 15 (11\mbox{,}088+364 x^{2}+5 x^{4}) \sin(x) \geq 0, \end{cases}\displaystyle \forall x \in \Gamma. $$

It is well known that \(\forall x \in \Gamma\),

$$\begin{aligned} \beta_{1}(x)&=t-\frac{t^{3}}{6}+ \frac{t^{5}}{120} - \frac{t^{7}}{5040} + \frac{t^{9}}{362\mbox{,}880}-\frac{x^{11}}{39\mbox{,}916\mbox{,}800} \\ &\leq \sin(x) \leq \beta_{1}(x) + \frac{x^{13}}{6\mbox{,}227\mbox{,}020\mbox{,}800}. \end{aligned}$$

Combining with Eq. (16), we have that

$$\begin{aligned} &\bigl(60\mbox{,}480-9240 x^{2}+364 x^{4}-5 x^{6}\bigr) x - 840 \bigl(72+x^{2}\bigr) \sin(x) \\ &\quad \leq \bigl(60\mbox{,}480-9240 x^{2}+364 x^{4}-5 x^{6}\bigr) x - 840 \bigl(72+x^{2}\bigr) \beta_{1}(x) \\ &\quad =\frac{x^{11} \cdot (-38+x^{2})}{39\mbox{,}916\mbox{,}800} \leq 0, \quad \forall x \in \Gamma, \\ & \bigl(166\mbox{,}320-22\mbox{,}260 x^{2}+551 x^{4}\bigr) x - 15 \bigl(11\mbox{,}088+364 x^{2}+5 x^{4}\bigr) \sin(x) \\ &\quad \geq \bigl(166\mbox{,}320-22\mbox{,}260 x^{2}+551 x^{4}\bigr) x\\ &\qquad {} - 15 \bigl(11\mbox{,}088+364 x^{2}+5 x^{4}\bigr) \biggl( \beta_{1}(x) + \frac{x^{13}}{6\mbox{,}227\mbox{,}020\mbox{,}800}\biggr) \\ &\quad = \frac{x^{11}}{6\mbox{,}227\mbox{,}020\mbox{,}800} \bigl(1\mbox{,}661\mbox{,}088-40\mbox{,}104 x^{2}+ 416 x^{4}- 5 x^{6}\bigr) \\ &\quad \geq \frac{x^{11}}{6\mbox{,}227\mbox{,}020\mbox{,}800} \bigl(1\mbox{,}661\mbox{,}088-40\mbox{,}104 \cdot 2^{2} - 5 \cdot 2^{6}\bigr) \geq 0,\quad \forall x \in \Gamma, \end{aligned}$$

which is just Eq. (15). So we have completed the proof of Eq. (14). □

Theorem 3.2

For all \(\forall x \in [0,\pi/2]\), we have that

$$ \begin{aligned}[b] c_{3}(x)&= \frac{ 20\mbox{,}160-9720 x^{2}+ 660 x^{4}-13 x^{6}}{360 (x^{2}+56)} \leq \cos(x) \\ & \leq \frac{15\mbox{,}120-6900 x^{2}+313 x^{4}}{15\mbox{,}120+660 x^{2}+13 x^{4}} =c_{4}(x). \end{aligned} $$


Eq. (17) is equivalent to

$$ \textstyle\begin{cases} (20\mbox{,}160-9720 x^{2}+ 660 x^{4}-13 x^{6})- 360 (x^{2}+56) \cos(x) \leq 0, \\ (15\mbox{,}120-6900 x^{2}+313 x^{4}) - (15\mbox{,}120+660 x^{2}+13 x^{4}) \cos(x) \geq 0, \end{cases}\displaystyle \forall x \in \Gamma. $$

It is well known that

$$ \begin{aligned}[b] \beta_{3}(x)&=1-\frac{x^{2}}{2} + \frac{x^{4}}{24} -\frac{x^{6}}{720} +\frac{x^{8}}{40\mbox{,}320} -\frac{x^{10}}{3\mbox{,}628\mbox{,}800} \leq \cos(x) \\ & \leq 1-\frac{x^{2}}{2} +\frac{x^{4}}{24} -\frac{x^{6}}{720} + \frac{x^{8}}{40\mbox{,}320} = \beta_{4}(x), \quad \forall x \in \Gamma. \end{aligned} $$

Combining with Eq. (19), we have that

$$ \textstyle\begin{cases} (20\mbox{,}160-9720 x^{2}+ 660 x^{4}-13 x^{6})- 360 (x^{2}+56) \cos(x) \\ \quad \leq (20\mbox{,}160-9720 x^{2}+ 660 x^{4}-13 x^{6})- 360 (x^{2}+56) \beta_{3}(x) \\ \quad = \frac{x^{10}}{3\mbox{,}628\mbox{,}800} (-34+x^{2}) \leq 0, \quad \forall x \in [0,\pi/2], \\ (15\mbox{,}120-6900 x^{2}+313 x^{4}) - (15\mbox{,}120+660 x^{2}+13 x^{4}) \cos(x) \\ \quad \geq (15\mbox{,}120-6900 x^{2}+313 x^{4}) - (15\mbox{,}120+660 x^{2}+13 x^{4}) \beta_{4}(x) \\ \quad =\frac{x^{10}}{40\mbox{,}320} (68 -13 x^{2}) \geq 0, \quad \forall x \in [0,\pi/2]. \end{cases} $$

Thus, we have completed the proof of both Eq. (18) and Eq. (17). □

Theorem 3.3

For all \(\forall x \in \Gamma\), we have that

$$ \begin{aligned}[b] c_{5}(x)&= \frac{ 21(495-60 x^{2}+x^{4})}{10\mbox{,}395-4725 x^{2}+210 x^{4}-x^{6}} \leq \frac{\tan(x)}{x} \\ &\leq \frac{T_{1}(x)}{105 (\pi^{2}-4 x^{2}) \cdot T_{2}(x)} =c_{6}(x), \end{aligned} $$

where \(T_{1}(x)=(\pi^{6}-840 \pi^{4}+75\mbox{,}600 \pi^{2}-665\mbox{,}280) x^{6} + (210 \pi^{6}+52\mbox{,}920 \pi^{4}-7\mbox{,}620\mbox{,}480 \pi^{2}+69\mbox{,}854\mbox{,}400) x^{4} + (-17\mbox{,}955 \pi^{6}+1\mbox{,}323\mbox{,}000 \pi^{4}+52\mbox{,}390\mbox{,}800 \pi^{2}-628\mbox{,}689\mbox{,}600) x^{2} + (155\mbox{,}925 (\pi^{4}-112 \pi^{2}+1008)) \pi^{2}\) and \(T_{2}(x) = (26 \pi^{4}-2664 \pi^{2}+23\mbox{,}760) x^{4} + (-666 \pi^{4}+73\mbox{,}980 \pi^{2}-665\mbox{,}280) x^{2} + (1485 \pi^{4}-166\mbox{,}320 \pi^{2}+1\mbox{,}496\mbox{,}880)\).


Eq. (21) is equivalent to

$$ \textstyle\begin{cases} \begin{aligned} H_{5}(x) ={}& 21(495-60 x^{2}+x^{4}) \cdot x \cos(x) \\ &{} - (10\mbox{,}395-4725 x^{2}+210 x^{4}-x^{6}) \cdot \sin(x) \leq 0; \end{aligned} \\ H_{6}(x) =105 (\pi^{2}-4 x^{2}) \cdot T_{2}(x) \cdot \sin(x) -T_{1}(x) \cdot x \cos(x) \leq 0, \end{cases}\displaystyle \forall x \in \Gamma. $$

It can be verified that

$$ \textstyle\begin{cases} \cos(x) \leq 1-\frac{x^{2}}{2} +\frac{x^{4}}{24} -\frac{x^{6}}{720} +\frac{x^{8}}{40\mbox{,}320} -\frac{x^{10}}{3\mbox{,}628\mbox{,}800} + \frac{x^{12}}{479\mbox{,}001\mbox{,}600}= \beta_{5}(x), \\ \beta_{1}(x)=t-\frac{t^{3}}{6}+ \frac{t^{5}}{120} - \frac{t^{7}}{5040} + \frac{t^{9}}{362\mbox{,}880}-\frac{x^{11}}{39\mbox{,}916\mbox{,}800} \leq \sin(x), \\ 495-60 x^{2}+x^{4}>0,\qquad 10\mbox{,}395-4725 x^{2}+210 x^{4}-x^{6}>0, \end{cases}\displaystyle \forall x \in \Gamma. $$

Combining with Eq. (23), we have that

$$ \begin{aligned}[b] H_{5}(x) &\leq 21\bigl(495-60 x^{2}+x^{4} \bigr) \cdot x \beta_{5}(x) - \bigl(10\mbox{,}395-4725 x^{2}+210 x^{4}-x^{6}\bigr) \cdot \beta_{1}(x) \\ &=\frac{x^{13}}{159\mbox{,}667\mbox{,}200} \bigl(-915-64 x^{2}+3 x^{4}\bigr) \leq 0, \quad \forall x \in \Gamma. \end{aligned} $$

Let \(\beta_{6}(x)=T_{1}(x)+105 (\pi^{2}-4 x^{2}) \cdot T_{2}'(x) -840 x \cdot T_{2}(x)\), \(\beta_{7}(x)=105 (\pi^{2}-4 x^{2}) \cdot T_{2}(x)-T_{1}'(x)\). On the other hand, it can be verified that, \(\forall x \in \Gamma\),

$$ \begin{aligned} &H_{6}'(x)= \beta_{6}(x) \cdot \sin(x) + \beta_{7}(x) \cdot \cos(x), \\ &\beta_{6}(x) \leq 0,\qquad \beta_{7}(x) \geq 0,\qquad T_{2}(x) \geq 0,\qquad T_{1}(x) \geq 0, \\ & \begin{aligned} \cos(x) \geq{}& 1-\frac{x^{2}}{2} +\frac{x^{4}}{24} -\frac{x^{6}}{720} + \frac{x^{8}}{40\mbox{,}320} -\frac{x^{10}}{3\mbox{,}628\mbox{,}800} + \frac{x^{12}}{479\mbox{,}001\mbox{,}600} \\ &{}-\frac{x^{14}}{87\mbox{,}178\mbox{,}291\mbox{,}200}= \beta_{8}(x), \end{aligned}\\ &\beta_{9}(x)=t-\frac{t^{3}}{6}+ \frac{t^{5}}{120} - \frac{t^{7}}{5040} + \frac{t^{9}}{362\mbox{,}880}-\frac{x^{11}}{39\mbox{,}916\mbox{,}800}+\frac{x^{13}}{6\mbox{,}227\mbox{,}020\mbox{,}800}\geq \sin(x). \end{aligned} $$

Combining Eq. (23) with Eq. (25), we have that

$$ \begin{aligned} H_{6}(x) &\leq 105 \bigl(\pi^{2}-4 x^{2}\bigr) \cdot T_{2}(x) \cdot \beta_{9}(x) -T_{1}(x) \cdot x \beta_{8}(x) \\ &= \frac{x^{13}}{9\mbox{,}153\mbox{,}720\mbox{,}576\mbox{,}000} \beta_{10}(x) \leq 0, \quad \forall x \in \biggl[0,\frac{31 \pi}{64}\biggr], \\ H_{6}'(x) &\geq \beta_{6}(x) \cdot \beta_{1}(x) + \beta_{7}(x) \cdot \beta_{5}(x) \\ &= \frac{x^{12}}{50\mbox{,}295\mbox{,}168\mbox{,}000} \beta_{11}(x) \geq 0, \quad \forall x \in \biggl[\frac{31 \pi}{64},\frac{\pi}{2}\biggr], \end{aligned} $$

where \(\beta_{10}(x)=(18\mbox{,}063\mbox{,}360 \pi^{6}-8\mbox{,}128\mbox{,}512\mbox{,}000 \pi^{4}+ 643\mbox{,}778\mbox{,}150\mbox{,}400 \pi^{2}- 5\mbox{,}579\mbox{,}410\mbox{,}636\mbox{,}800)+( -634\mbox{,}725 \pi^{6}+305\mbox{,}912\mbox{,}880 \pi^{4}-24\mbox{,}700\mbox{,}198\mbox{,}320 \pi^{2}+ 214\mbox{,}592\mbox{,}716\mbox{,}800) x^{2} +(6069 \pi^{6}-4\mbox{,}639\mbox{,}320 \pi^{4}+411\mbox{,}823\mbox{,}440 \pi^{2}-3\mbox{,}618\mbox{,}457\mbox{,}920) x^{4} + (28 \pi^{6}+52\mbox{,}920 \pi^{4}-5\mbox{,}715\mbox{,}360 \pi^{2}+51\mbox{,}226\mbox{,}560) x^{6} + (\pi^{6}-840 \pi^{4}+75\mbox{,}600 \pi^{2}-665\mbox{,}280) x^{8} \leq 0, \forall x \in [0,\frac{31 \pi}{64}]\), \(\beta_{11}(x)=(-1\mbox{,}290\mbox{,}240 \pi^{6}+580\mbox{,}608\mbox{,}000 \pi^{4}-45\mbox{,}984\mbox{,}153\mbox{,}600 \pi^{2}+ 398\mbox{,}529\mbox{,}331\mbox{,}200)+ (54\mbox{,}405 \pi^{6}-25\mbox{,}552\mbox{,}800 \pi^{4}+2\mbox{,}048\mbox{,}684\mbox{,}400 \pi^{2}- 17\mbox{,}782\mbox{,}934\mbox{,}400) x^{2}+ ( -1404 \pi^{6}+556\mbox{,}920 \pi^{4}-42\mbox{,}366\mbox{,}240 \pi^{2}+365\mbox{,}238\mbox{,}720) x^{4}+ ( 19 \pi^{6}-5040 \pi^{4}+317\mbox{,}520 \pi^{2}-2\mbox{,}661\mbox{,}120) x^{6}\geq 0, \forall x \in [\frac{31 \pi}{64},\frac{\pi}{2}]\). Combining Eq. (26) with \(H_{6}(\pi/2)=0\), we obtain that

$$ H_{6}(x) \leq 0,\quad \forall x \in \biggl[0,\frac{\pi}{2} \biggr]. $$

Combining Eq. (24) with Eq. (27), we have completed the proof of both Eq. (22) and Eq. (21). □

From Theorems 3.1, 3.2, and 3.3, we directly obtain the following theorem.

Theorem 3.4

We have that

$$\frac{1}{c_{2}(x)^{2}} + \frac{1}{c_{6}(x)} \leq \biggl(\frac{x}{\sin(x)} \biggr)^{2} + \frac{x}{\tan(x)} \leq \frac{1}{c_{1}(x)^{2}} + \frac{1}{c_{5}(x)},\quad \forall x \in [0,\pi/2]. $$

4 Discussion and conclusions

Firstly, we compare the results of \(\frac{\sin(x)}{x}\) between \(b_{i}(x)\) in [2] and \(c_{i}(x)\) in this paper, \(i=1,2\). It can be verified that \(c_{1}(x)-b_{1}(x)=\frac{x^{6} (264-5x^{2})}{840(72+x^{2})(x^{2}+20)} \geq 0\) and \(c_{2}(x)-b_{2}(x)=\frac{ -11 x^{8}}{12(11\mbox{,}088+364 x^{2}+5 x^{4})(x^{2}+42)} \leq 0\), \(\forall x \in [0,\pi/2]\), we have that

$$b_{1}(x) \leq c_{1}(x) \leq \frac{\sin(x)}{x} \leq c_{2}(x) \leq b_{2}(x), \quad \forall x \in [0,\pi/2]. $$

Secondly, we compare the approximation results of \(\cos(x)\) between previous \(b_{i}(x)\) and present \(c_{i}(x)\), \(i=3,4\). It can be verified that \(c_{3}(x)-b_{3}(x)=\frac{x^{8}(270-13 x^{2})}{360 (56+x^{2})(x^{4}+30 x^{2}+540)} \geq 0\) and \(c_{2}(x)-b_{2}(x)=\frac{ -39 x^{8}}{4 (15\mbox{,}120+660 x^{2}+13 x^{4})(x^{2}+30)} \leq 0\), \(\forall x \in [0,\pi/2]\), we have that

$$b_{3}(x) \leq c_{3}(x) \leq \cos(x) \leq c_{4}(x) \leq b_{4}(x), \quad \forall x \in [0,\pi/2]. $$

Thirdly, we compare the approximation results of \(\frac{\tan(x)}{x}\), which also shows that this paper achieves a much better result. It can be verified that \(\forall x \in [0,\pi/2]\),

$$c_{5}(x)-b_{5}(x)=\frac{x^{6}(161 x^{2}-495) (x^{2}-33)}{3(10\mbox{,}395-4725 x^{2}+210 x^{4}-x^{6}) (x^{2}+20) (3 x^{4}-56 x^{2}+120)} \geq 0. $$

However, note that the denominator of \(b_{6}(x)\) is \(T_{3}(x) = 1020x^{6}+ 14\mbox{,}040x^{4}- 1\mbox{,}144\mbox{,}800x^{2}+ 2\mbox{,}721\mbox{,}600 = 30(17 x^{4}-480 x^{2}+1080)(x^{2}+42)\), which has a real root ≈1.5701 within the interval Γ, and we have \(T_{3}(x) >0, \forall x \in [0,1.5701]\). It can be verified that \(c_{6}(x) - b_{6}(x)= \frac{-x^{8} H_{7}(x)}{210 T_{2}(x) T_{3}(x) (\pi^{2}-4 x^{2})}\), where \(H_{7}(x)=378\mbox{,}675 (\pi^{4}-112 \pi^{2}+1008) \pi^{2} + (-64\mbox{,}350 \pi^{6}+5\mbox{,}536\mbox{,}440 \pi^{4}+106\mbox{,}323\mbox{,}840 \pi^{2}-1\mbox{,}526\mbox{,}817\mbox{,}600)x^{2} + (1968 \pi^{6}+50\mbox{,}400 \pi^{4}-25\mbox{,}764\mbox{,}480 \pi^{2}+ 247\mbox{,}484\mbox{,}160) x^{4} + (-8008 \pi^{4}+820\mbox{,}512 \pi^{2}-7\mbox{,}318\mbox{,}080)x^{6}\). By using the Maple software, \(H_{7}(x)\) has six real roots \(\approx -9.16,-4.97 ,-2.76, 2.76, 4.97, 9.16\), and \(H_{7}(x), T_{2}(x), T_{3}(x)> 0, \forall x \in (0,1.5701)\), we have that

$$c_{6}(x) - b_{6}(x) \leq 0,\quad \forall x \in [0,1.5701]. $$


  1. Baker, G.A. Jr., Graves-Morris, P.: Padé Approximants. Cambridge University Press, New York (1996)

    Book  MATH  Google Scholar 

  2. Bercu, G.: The natural approach of trigonometric inequalities—Padé approximant. J. Math. Inequal. 11(1), 181–191 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, C.P., Debnath, L.: Sharpness and generalization of Jordan’s inequality and its application. Appl. Math. Lett. 25(3), 594–599 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Davis, P.J.: Interpolation and Approximation. Dover Publications, New York (1975)

    MATH  Google Scholar 

  5. Debnath, L., Mortici, C., Zhu, L.: Refinements of Jordan–Steckin and Becker–Stark inequalities. Results Math. 67(1–2), 207–215 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Jiang, W.D., Luo, Q.M., Qi, F.: Refinements and sharpening of some Huygens and Wilker type inequalities. Math. Inequal. Appl. 6(1), 19–22 (2014)

    Google Scholar 

  7. Lutovac, T., Malešsević, B., Mortici, C.: The natural algorithmic approach of mixed trigonometric-polynomial problems. J. Inequal. Appl. 2017, 116 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Malešević, B., Makragic, M.: A method for proving some inequalities on mixed trigonometric polynomial functions. J. Math. Inequal. 10, 849–876 (2015)

    MathSciNet  MATH  Google Scholar 

  9. Mortici, C.: The natural approach of Wilker–Cusa–Huygens inequalities. Math. Inequal. Appl. 14, 535–541 (2011)

    MathSciNet  MATH  Google Scholar 

  10. Mortici, C.: A subtly analysis of Wilker inequality. Appl. Math. Comput. 231, 516–520 (2014)

    MathSciNet  MATH  Google Scholar 

  11. Nenezić, M., Malesević, B., Mortici, C.: New approximations of some expressions involving trigonometric functions. Appl. Math. Comput. 283, 299–315 (2016)

    MathSciNet  Google Scholar 

  12. Nishizawa, Y.: Sharpening of Jordan’s type and Shafer–Fink’s type inequalities with exponential approximations. Appl. Math. Comput. 269, 146–154 (2015)

    MathSciNet  Google Scholar 

  13. Sun, Z.J., Zhu, L.: Simple proofs of the Cusa–Huygens-type and Becker–Stark-type inequalities. J. Math. Inequal. 7(4), 563–567 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Wilker, N.E.: Huygens-type inequalities for the generalized trigonometric and for the generalized hyperbolic functions. Appl. Math. Comput. 230(3), 211–217 (2014)

    MathSciNet  Google Scholar 

  15. Wu, S., Debnath, L.: A new generalized and sharp version of Jordan’s inequality and its applications to the improvement of the Yang Le inequality. Appl. Math. Lett. 20(5), 532–538 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wu, S.-H., Srivastava, H.M.: A weighted and exponential generalization of Wilker’s inequality and its applications. Integral Transforms Spec. Funct. 18(7–8), 529–535 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Wu, S.H., Li, S.G., Bencze, M.: Sharpened versions of Mitrinovic–Adamovic, Lazarevic and Wilker’s inequalities for trigonometric and hyperbolic functions. J. Nonlinear Sci. Appl. 9(5), 2688–2696 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wu, S.H., Yu, H.P., Deng, Y.P., et al.: Several improvements of Mitrinovic–Adamovic and Lazarevic’s inequalities with applications to the sharpening of Wilker-type inequalities. J. Nonlinear Sci. Appl. 9(4), 1755–1765 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Yang, Z.H., Chu, Y.M., Zhang, X.H.: Sharp Cusa type inequalities with two parameters and their applications. Appl. Math. Comput. 268, 1177–1198 (2015)

    MathSciNet  Google Scholar 

  20. Zhu, L.: Sharpening Jordan’s inequality and the Yang Le’s inequality. Appl. Math. Lett. 19(3), 240–243 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Zhu, L.: Sharpening Jordan’s inequality and Yang Le’s inequality. II. Appl. Math. Lett. 19(9), 990–994 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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This research work was partly supported by the National Science Foundation of China (61672009, 61502130, 61761136010) and the Open Project Program of the National Laboratory of Pattern Recognition (NLPR).

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Correspondence to Huaqing Shan.

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Zhang, Z., Shan, H. & Chen, L. Refining trigonometric inequalities by using Padé approximant. J Inequal Appl 2018, 149 (2018).

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