1 Introduction
Trigonometric inequalities have caused interest of a lot of researchers, they analyzed the Wilker inequality [6‐11, 14, 16‐19], 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].
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 \(b_{7}(x)=\frac{11\mbox{,}220x^{10}-205\mbox{,}560x^{8}-14\mbox{,}256\mbox{,}000x^{6}+512\mbox{,}179\mbox{,}200x^{4}- 3\mbox{,}157\mbox{,}056\mbox{,}000x^{2}+13\mbox{,}716\mbox{,}864\mbox{,}000}{242x^{12}-8580x^{10} +25\mbox{,}560x^{8}-1\mbox{,}080\mbox{,}000x^{6}+103\mbox{,}680\mbox{,}000x^{4}-1\mbox{,}578\mbox{,}528\mbox{,}000x^{2}+6\mbox{,}858\mbox{,}432\mbox{,}000}\).
$$\begin{aligned} &b_{1}(x)= \frac{-7 x^{2} + 60}{3 x^{2} + 60} < \frac{\sin(x)}{x} < \frac{11 x^{4} - 360 x^{2}+ 2520}{60 x^{2}+ 2520}= b_{2}(x),\quad \forall x \in (0,\pi/2); \end{aligned}$$
(1)
$$\begin{aligned} & \begin{aligned}[b] b_{3}(x) &=\frac{17x^{4} - 480x^{2}+ 1080}{2x^{4}+ 60x^{2}+ 1080} < \cos(x) < \frac{3x^{4} - 56x^{2}+ 120}{4x^{2}+ 120} \\ &= b_{4}(x),\quad \forall x \in (0,\pi/2); \end{aligned} \end{aligned}$$
(2)
$$\begin{aligned} & b_{5}(x) < \frac{\tan(x)}{x}< b_{6}(x),\quad \forall x \in (0,1.5701); \end{aligned}$$
(3)
$$\begin{aligned} & \biggl(\frac{x}{\sin(x)}\biggr)^{2}+\frac{x}{\tan(x)} > b_{7}(x),\quad \forall x \in (0,1.5701), \end{aligned}$$
(4)
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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
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).
$$ E^{(i)}(x_{0})=0, \qquad E^{(j)}(x_{1})=0,\quad i=0,1, \ldots,k, j=0,1,\ldots,l, $$
(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
we obtain that
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]
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
$$ f_{1}^{(i)}(0)=R_{j}^{(i)}(0),\qquad f_{1}(\pi/2)=R_{j}(\pi/2),\quad j=1,2, i=0,1,\ldots,14-2j, $$
(6)
$$ R_{1}(x)= \beta_{1}(x)+ \alpha_{1} \cdot x^{13}, \qquad R_{2}(x)= \beta_{2}(x)- \alpha_{2} \cdot x^{11}, $$
(7)
$$ f_{1}(x) - R_{j}(x) = \frac{f_{1}^{(16-2j)}(\xi_{j}(x))}{(16-2j)!} \cdot (x-\pi/2) \cdot x^{15-2j}, \quad x \in \Gamma, j=1,2. $$
(8)
$$ 0 \leq R_{1}(x) \leq \sin(x) \leq R_{2}(x), \quad x \in \Gamma. $$
(9)
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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
we obtain that
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]
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
$$ f_{2}^{(i)}(0)=R_{j}^{(i)}(0),\qquad f_{2}(\pi/2)=R_{j}(\pi/2),\quad j=3,4, i=0,1,\ldots,17-2j, $$
(10)
$$ R_{3}(x)= \beta_{3}(x)+ \alpha_{3} \cdot x^{12} , \qquad R_{4}(x)= \beta_{4}(x)- \alpha_{4} \cdot x^{10}, $$
(11)
$$ f_{2}(x) - R_{j}(x) = \frac{f_{2}^{(19-2j)}(\xi_{j}(x))}{(19-2j)!} \cdot (x-\pi/2) \cdot x^{18-2j}, \quad x \in \Gamma, j=3,4. $$
(12)
$$ 0 \leq R_{3}(x) \leq \cos(x) \leq R_{4}(x), \quad x \in \Gamma. $$
(13)
3 Main results
The main results are as follows.
Theorem 3.1
For all
\(\forall x \in \Gamma=[0,\pi/2]\), we have that
$$\begin{aligned}{} [b] c_{1}(x) &= \frac{60\mbox{,}480-9240 x^{2}+364 x^{4}-5 x^{6} }{840 (72+x^{2})} \leq \frac{\sin(x)}{x} \\ &\leq \frac{ (166\mbox{,}320-22\mbox{,}260 x^{2}+551 x^{4}) }{15 (11\mbox{,}088+364 x^{2}+5 x^{4})} =c_{2}(x). \end{aligned}$$
(14)
Proof
Eq. (14) is equivalent to
It is well known that \(\forall x \in \Gamma\),
Combining with Eq. (16), we have that
which is just Eq. (15). So we have completed the proof of Eq. (14). □
$$ \textstyle\begin{cases} (60\mbox{,}480-9240 x^{2}+364 x^{4}-5 x^{6}) x - 840 (72+x^{2}) \sin(x) \leq 0, \\ (166\mbox{,}320-22\mbox{,}260 x^{2}+551 x^{4}) x - 15 (11\mbox{,}088+364 x^{2}+5 x^{4}) \sin(x) \geq 0, \end{cases}\displaystyle \forall x \in \Gamma. $$
(15)
$$\begin{aligned} \beta_{1}(x)&=t-\frac{t^{3}}{6}+ \frac{t^{5}}{120} - \frac{t^{7}}{5040} + \frac{t^{9}}{362\mbox{,}880}-\frac{x^{11}}{39\mbox{,}916\mbox{,}800} \\ &\leq \sin(x) \leq \beta_{1}(x) + \frac{x^{13}}{6\mbox{,}227\mbox{,}020\mbox{,}800}. \end{aligned}$$
(16)
$$\begin{aligned} &\bigl(60\mbox{,}480-9240 x^{2}+364 x^{4}-5 x^{6}\bigr) x - 840 \bigl(72+x^{2}\bigr) \sin(x) \\ &\quad \leq \bigl(60\mbox{,}480-9240 x^{2}+364 x^{4}-5 x^{6}\bigr) x - 840 \bigl(72+x^{2}\bigr) \beta_{1}(x) \\ &\quad =\frac{x^{11} \cdot (-38+x^{2})}{39\mbox{,}916\mbox{,}800} \leq 0, \quad \forall x \in \Gamma, \\ & \bigl(166\mbox{,}320-22\mbox{,}260 x^{2}+551 x^{4}\bigr) x - 15 \bigl(11\mbox{,}088+364 x^{2}+5 x^{4}\bigr) \sin(x) \\ &\quad \geq \bigl(166\mbox{,}320-22\mbox{,}260 x^{2}+551 x^{4}\bigr) x\\ &\qquad {} - 15 \bigl(11\mbox{,}088+364 x^{2}+5 x^{4}\bigr) \biggl( \beta_{1}(x) + \frac{x^{13}}{6\mbox{,}227\mbox{,}020\mbox{,}800}\biggr) \\ &\quad = \frac{x^{11}}{6\mbox{,}227\mbox{,}020\mbox{,}800} \bigl(1\mbox{,}661\mbox{,}088-40\mbox{,}104 x^{2}+ 416 x^{4}- 5 x^{6}\bigr) \\ &\quad \geq \frac{x^{11}}{6\mbox{,}227\mbox{,}020\mbox{,}800} \bigl(1\mbox{,}661\mbox{,}088-40\mbox{,}104 \cdot 2^{2} - 5 \cdot 2^{6}\bigr) \geq 0,\quad \forall x \in \Gamma, \end{aligned}$$
Theorem 3.2
For all
\(\forall x \in [0,\pi/2]\), we have that
$$ \begin{aligned}[b] c_{3}(x)&= \frac{ 20\mbox{,}160-9720 x^{2}+ 660 x^{4}-13 x^{6}}{360 (x^{2}+56)} \leq \cos(x) \\ & \leq \frac{15\mbox{,}120-6900 x^{2}+313 x^{4}}{15\mbox{,}120+660 x^{2}+13 x^{4}} =c_{4}(x). \end{aligned} $$
(17)
Proof
Eq. (17) is equivalent to
It is well known that
Combining with Eq. (19), we have that
Thus, we have completed the proof of both Eq. (18) and Eq. (17). □
$$ \textstyle\begin{cases} (20\mbox{,}160-9720 x^{2}+ 660 x^{4}-13 x^{6})- 360 (x^{2}+56) \cos(x) \leq 0, \\ (15\mbox{,}120-6900 x^{2}+313 x^{4}) - (15\mbox{,}120+660 x^{2}+13 x^{4}) \cos(x) \geq 0, \end{cases}\displaystyle \forall x \in \Gamma. $$
(18)
$$ \begin{aligned}[b] \beta_{3}(x)&=1-\frac{x^{2}}{2} + \frac{x^{4}}{24} -\frac{x^{6}}{720} +\frac{x^{8}}{40\mbox{,}320} -\frac{x^{10}}{3\mbox{,}628\mbox{,}800} \leq \cos(x) \\ & \leq 1-\frac{x^{2}}{2} +\frac{x^{4}}{24} -\frac{x^{6}}{720} + \frac{x^{8}}{40\mbox{,}320} = \beta_{4}(x), \quad \forall x \in \Gamma. \end{aligned} $$
(19)
$$ \textstyle\begin{cases} (20\mbox{,}160-9720 x^{2}+ 660 x^{4}-13 x^{6})- 360 (x^{2}+56) \cos(x) \\ \quad \leq (20\mbox{,}160-9720 x^{2}+ 660 x^{4}-13 x^{6})- 360 (x^{2}+56) \beta_{3}(x) \\ \quad = \frac{x^{10}}{3\mbox{,}628\mbox{,}800} (-34+x^{2}) \leq 0, \quad \forall x \in [0,\pi/2], \\ (15\mbox{,}120-6900 x^{2}+313 x^{4}) - (15\mbox{,}120+660 x^{2}+13 x^{4}) \cos(x) \\ \quad \geq (15\mbox{,}120-6900 x^{2}+313 x^{4}) - (15\mbox{,}120+660 x^{2}+13 x^{4}) \beta_{4}(x) \\ \quad =\frac{x^{10}}{40\mbox{,}320} (68 -13 x^{2}) \geq 0, \quad \forall x \in [0,\pi/2]. \end{cases} $$
(20)
Theorem 3.3
For all
\(\forall x \in \Gamma\), we have that
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
\(T_{2}(x) = (26 \pi^{4}-2664 \pi^{2}+23\mbox{,}760) x^{4} + (-666 \pi^{4}+73\mbox{,}980 \pi^{2}-665\mbox{,}280) x^{2} + (1485 \pi^{4}-166\mbox{,}320 \pi^{2}+1\mbox{,}496\mbox{,}880)\).
$$ \begin{aligned}[b] c_{5}(x)&= \frac{ 21(495-60 x^{2}+x^{4})}{10\mbox{,}395-4725 x^{2}+210 x^{4}-x^{6}} \leq \frac{\tan(x)}{x} \\ &\leq \frac{T_{1}(x)}{105 (\pi^{2}-4 x^{2}) \cdot T_{2}(x)} =c_{6}(x), \end{aligned} $$
(21)
Proof
Eq. (21) is equivalent to
It can be verified that
Combining with Eq. (23), we have that
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, \(\forall x \in \Gamma\),
Combining Eq. (23) with Eq. (25), we have that
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
Combining Eq. (24) with Eq. (27), we have completed the proof of both Eq. (22) and Eq. (21). □
$$ \textstyle\begin{cases} \begin{aligned} H_{5}(x) ={}& 21(495-60 x^{2}+x^{4}) \cdot x \cos(x) \\ &{} - (10\mbox{,}395-4725 x^{2}+210 x^{4}-x^{6}) \cdot \sin(x) \leq 0; \end{aligned} \\ H_{6}(x) =105 (\pi^{2}-4 x^{2}) \cdot T_{2}(x) \cdot \sin(x) -T_{1}(x) \cdot x \cos(x) \leq 0, \end{cases}\displaystyle \forall x \in \Gamma. $$
(22)
$$ \textstyle\begin{cases} \cos(x) \leq 1-\frac{x^{2}}{2} +\frac{x^{4}}{24} -\frac{x^{6}}{720} +\frac{x^{8}}{40\mbox{,}320} -\frac{x^{10}}{3\mbox{,}628\mbox{,}800} + \frac{x^{12}}{479\mbox{,}001\mbox{,}600}= \beta_{5}(x), \\ \beta_{1}(x)=t-\frac{t^{3}}{6}+ \frac{t^{5}}{120} - \frac{t^{7}}{5040} + \frac{t^{9}}{362\mbox{,}880}-\frac{x^{11}}{39\mbox{,}916\mbox{,}800} \leq \sin(x), \\ 495-60 x^{2}+x^{4}>0,\qquad 10\mbox{,}395-4725 x^{2}+210 x^{4}-x^{6}>0, \end{cases}\displaystyle \forall x \in \Gamma. $$
(23)
$$ \begin{aligned}[b] H_{5}(x) &\leq 21\bigl(495-60 x^{2}+x^{4} \bigr) \cdot x \beta_{5}(x) - \bigl(10\mbox{,}395-4725 x^{2}+210 x^{4}-x^{6}\bigr) \cdot \beta_{1}(x) \\ &=\frac{x^{13}}{159\mbox{,}667\mbox{,}200} \bigl(-915-64 x^{2}+3 x^{4}\bigr) \leq 0, \quad \forall x \in \Gamma. \end{aligned} $$
(24)
$$ \begin{aligned} &H_{6}'(x)= \beta_{6}(x) \cdot \sin(x) + \beta_{7}(x) \cdot \cos(x), \\ &\beta_{6}(x) \leq 0,\qquad \beta_{7}(x) \geq 0,\qquad T_{2}(x) \geq 0,\qquad T_{1}(x) \geq 0, \\ & \begin{aligned} \cos(x) \geq{}& 1-\frac{x^{2}}{2} +\frac{x^{4}}{24} -\frac{x^{6}}{720} + \frac{x^{8}}{40\mbox{,}320} -\frac{x^{10}}{3\mbox{,}628\mbox{,}800} + \frac{x^{12}}{479\mbox{,}001\mbox{,}600} \\ &{}-\frac{x^{14}}{87\mbox{,}178\mbox{,}291\mbox{,}200}= \beta_{8}(x), \end{aligned}\\ &\beta_{9}(x)=t-\frac{t^{3}}{6}+ \frac{t^{5}}{120} - \frac{t^{7}}{5040} + \frac{t^{9}}{362\mbox{,}880}-\frac{x^{11}}{39\mbox{,}916\mbox{,}800}+\frac{x^{13}}{6\mbox{,}227\mbox{,}020\mbox{,}800}\geq \sin(x). \end{aligned} $$
(25)
$$ \begin{aligned} H_{6}(x) &\leq 105 \bigl(\pi^{2}-4 x^{2}\bigr) \cdot T_{2}(x) \cdot \beta_{9}(x) -T_{1}(x) \cdot x \beta_{8}(x) \\ &= \frac{x^{13}}{9\mbox{,}153\mbox{,}720\mbox{,}576\mbox{,}000} \beta_{10}(x) \leq 0, \quad \forall x \in \biggl[0,\frac{31 \pi}{64}\biggr], \\ H_{6}'(x) &\geq \beta_{6}(x) \cdot \beta_{1}(x) + \beta_{7}(x) \cdot \beta_{5}(x) \\ &= \frac{x^{12}}{50\mbox{,}295\mbox{,}168\mbox{,}000} \beta_{11}(x) \geq 0, \quad \forall x \in \biggl[\frac{31 \pi}{64},\frac{\pi}{2}\biggr], \end{aligned} $$
(26)
$$ H_{6}(x) \leq 0,\quad \forall x \in \biggl[0,\frac{\pi}{2} \biggr]. $$
(27)
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 \(\forall x \in [0,\pi/2]\),
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
$$c_{5}(x)-b_{5}(x)=\frac{x^{6}(161 x^{2}-495) (x^{2}-33)}{3(10\mbox{,}395-4725 x^{2}+210 x^{4}-x^{6}) (x^{2}+20) (3 x^{4}-56 x^{2}+120)} \geq 0. $$
$$c_{6}(x) - b_{6}(x) \leq 0,\quad \forall x \in [0,1.5701]. $$
Competing interests
The authors declare that they have no competing interests.
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