Abstract
In this paper, some exponential inequalities are derived from the inequalities containing trigonometric functions. Numerical examples show that one can achieve much tighter bounds than those of prevailing methods, which are presented by Cusa, Huygens, Chen and Sándor.
Similar content being viewed by others
1 Introduction
Inequalities involving trigonometric and inverse trigonometric functions play an important role and have many applications in science and engineering [2, 8, 12, 17,18,19, 27]. The sinc function, defined as \(\frac{\sin (x)}{x}\), is often used in signal processing, optics, radio transmission, sound recording [12], has been studied in many references [1, 3,4,5, 7, 10, 13,14,15,16,17, 19, 21, 26, 28,29,30]. The study starts from the Jordan’s inequality [19], namely
Later, the sinc function is bounded by using polynomials [7, 10, 17, 24], or by using exponential bounds [3, 4, 25].
Cusa–Huygens’s inequality is studied in [3, 4, 11, 20, 22, 23, 25], and gives
where the constants \(\theta = \frac{\ln (\pi /2)}{\ln (3/2)} \approx 1.113\) and \(\vartheta =1\) are the best possible.
Becker–Stark’s inequality, namely
is studied in [3, 4, 10, 19, 29, 31]. In [32], Zhu provided improved bounds:
where \(t_{1}(x)=\frac{8}{\pi ^{2}-4 x^{2}}+\frac{2}{\pi ^{2}}-\frac{\pi ^{2}-9}{6 \pi ^{4}}(\pi ^{2}-4 x^{2})\) and \(t_{2}(x)=\frac{8}{\pi ^{2}-4 x^{2}}+\frac{2}{\pi ^{2}}-\frac{(10-\pi ^{2})}{\pi ^{4}}(\pi ^{2}-4 x^{2})\). Later, there in [6] further improved bounds were given, namely
where \(\bar{t}_{3}(x)=45 \pi ^{8}+(-2 \pi ^{8}-3660 \pi ^{6}+36000 \pi ^{4}) x^{2}+(16 \pi ^{7}+21000 \pi ^{5}-208800 \pi ^{3}) x^{3}+(-48 \pi ^{6}-49440 \pi ^{4}+492480 \pi ^{2}) x^{4}+(64 \pi ^{5}+54240 \pi ^{3}-541440 \pi ) x^{5}+(-32 \pi ^{4}-23040 \pi ^{2}+230400) x^{6}\) and \(\bar{t}_{4}(x)=3 \pi ^{8}+(-12 \pi ^{6}+\pi ^{8}) x^{2}+(5280 \pi ^{3}-456 \pi ^{5}-8 \pi ^{7}) x^{3}+(-24768 \pi ^{2}+2272 \pi ^{4}+24 \pi ^{6}) x ^{4}+(40704 \pi -3808 \pi ^{3}-32 \pi ^{5}) x^{5}+(-23040+2176 \pi ^{2}+16 \pi ^{4}) x^{6}\).
Chen and Cheng established the following exponential bounds [3]:
Recently, Nishizawa established [25]
where
Motivated by Eqs. (5), (8) and (9), we provide some inequalities with much tighter bounds by using power exponential functions, which are described in Theorems 1.1–1.2.
Theorem 1.1
For every \(0 < x < \pi /2\), we have
where \(a(x)= 1+\frac{\pi -4}{\pi } x -\frac{2 (\pi -2)}{\pi ^{2}} x ^{2}\), \(\theta _{1}(x)=-\frac{\pi }{2(\pi -4)} x- \frac{\pi ^{2}-4 \pi +8 }{4 (\pi -4)^{2}} x^{2} + \frac{\pi ^{4}+8 \pi ^{2}-128 \pi +256}{2 \pi ^{3} (\pi -4)^{2}} x^{3}\) and \(\theta _{2}(x)=- \frac{\pi }{2(\pi -4)} x +\frac{2 (\pi ^{2}+6 \pi -24)}{\pi ^{2} ( \pi -4)} x^{2}-\frac{ 2(\pi ^{2}+8 \pi -32)}{(\pi -4)\pi ^{3}}x^{3}\).
Theorem 1.2
For every \(0 < x < \pi /2\), we have
2 Proof of Theorem 1.1
Proof
Equation (10) is equivalent to
Step 1. Firstly, we prove that for every \(0 < x < \pi /2\),
which is equivalent to
Let \(x_{1}=0.9\) and \(x_{2}=\frac{111 \cdot \pi }{256} \approx 1.362\).
Case 1.1. Proof of \(D_{1}(x) \leq 0\), \(\forall x \in (0,x_{1}]\).
Combining with Eq. (6), for every \(0< x \leq x_{1}\), we have that
It can be verified that \(\theta _{1}'(x) = \alpha _{2} x^{2} + \alpha _{1} x + \alpha _{0} = \alpha _{2} (x+\frac{\alpha _{1}}{2 \alpha _{2}})^{2} + \alpha _{3}\), where \(\alpha _{0} \approx 1.82\), \(\alpha _{1} \approx -3.59\), \(\alpha _{2} \approx 1.98>0\) and \(\alpha _{3} \approx 0.19>0\), so we have \(\theta _{1}'(x)>0\), \(\forall x \in (0, \pi /2)\). Let \(D_{3}(x)=\frac{D_{2}(x)}{\theta _{1}'(x)}\). It can be verified that
where \(W_{1}(x)=((\pi x-2 x+\pi )\cdot (2 x+\pi )\cdot (-768 x^{2}-3 \pi ^{4} x^{2}-24 \pi ^{2} x^{2}+384 \pi x^{2}+\pi ^{5} x-4 \pi ^{4} x+8 \pi ^{3} x-4 \pi ^{4}+\pi ^{5})\cdot \pi )^{2} (\pi -2 x)\), \(B_{1,i}(x) = \frac{C_{9}^{i} \cdot x^{i} \cdot (x_{1}-x)^{9-i}}{(x_{1}-0)^{9}}\), and \(\gamma _{1,0} \approx -3.5 \cdot 10^{7}<0\), \(\gamma _{1,1} \approx -3.2 \cdot 10^{7}<0\), \(\gamma _{1,2} \approx -2.9 \cdot 10^{7}<0\), \(\gamma _{1,3} \approx -2.4 \cdot 10^{7}<0\), \(\gamma _{1,4} \approx -2.0 \cdot 10^{7}<0\), \(\gamma _{1,5} \approx -1.5 \cdot 10^{7}<0\), \(\gamma _{1,6} \approx -1.1 \cdot 10^{7}<0\), \(\gamma _{1,7} \approx -7.7 \cdot 10^{6}<0\), \(\gamma _{1,8} \approx -4.5 \cdot 10^{6}<0\), \(\gamma _{1,9} \approx -1.7 \cdot 10^{6}<0\). Note that \(W_{1}(x)>0\), \(\forall x \in [0,\pi /2)\) and \(B_{1,i}(x) \geq 0\), \(\forall x \in [0,x_{1}]\), and from Eq. (15) we have that \(D_{3}'(x)\leq 0\), \(\forall x \in [0,x_{1}]\). So
Combining Eq. (16) with \(\theta _{1}'(x)>0\), we have that \(D_{2}(x) \leq 0\), \(\forall x \in (0,x_{1}]\). Combining with Eq. (14) yields
From Eq. (17), we have that
Case 1.2. Proof of \(D_{1}(x) \leq 0\), \(\forall x \in (x_{1},x_{2}]\).
For every \(x_{1}< x \leq x_{2}\), we have \(\theta _{1}'(x) \geq 0\), \(\ln (a(x))\leq 0\), \(\theta _{1}(x) \leq \theta _{1}(x_{2})\) and
Combining with Eq. (5) gives
where \(\gamma _{2,6} \approx 1.95>0\), \(\gamma _{2,5} \approx 5.36>0\), \(\gamma _{2,4} \approx 56.7>0\), \(\gamma _{2,3} \approx 156.2>0\), \(\gamma _{2,2} \approx -265.4<0\), \(\gamma _{2,1} \approx -1050<0\), \(\gamma _{2,0} \approx 502.8>0\). So for every \(x_{1} < x \leq x_{2}\),
Combining with Eq. (20) yields
Combining with Eqs. (19)–(21), we have that
Case 1.3. Proof of \(D_{1}(x) \leq 0\), \(\forall x \in (x_{2},\pi /2)\).
Combining with Eq. (5), for every \(x_{2}< x < \pi /2\), we have
Let \(D_{7}(x)=\frac{D_{6}(x)}{\theta _{1}'(x)}\). It can be verified that
where \(B_{2,i}(x) = \frac{C_{7}^{i} \cdot (x-x_{2})^{i} \cdot (\pi /2-x)^{7-i}}{( \pi /2-x_{2})^{7}} \geq 0\), \(\forall x \in [x_{2},\pi /2)\), \(\gamma _{3,0} \approx 8.8 \cdot 10^{6}\), \(\gamma _{3,1} \approx 9.7 \cdot 10^{6}\), \(\gamma _{3,2} \approx 1.0 \cdot 10^{7}\), \(\gamma _{3,3} \approx 1.1 \cdot 10^{7}\), \(\gamma _{3,4} \approx 1.3 \cdot 10^{7}\), \(\gamma _{3,5} \approx 1.4 \cdot 10^{7}\), \(\gamma _{3,6} \approx 1.6 \cdot 10^{7}\) and \(\gamma _{3,7} \approx 1.8 \cdot 10^{7}\), such that \(\gamma _{3,i}>0\), \(i=0,1, \dots ,7\), and \(D_{7}'(x) \geq 0\), \(\forall x \in (x_{2},\pi /2)\). So we have
Note that \(\theta _{1}'(x)>0\), \(\forall x \in (0,\pi /2)\), and, combining with Eqs. (23)–(25), we have that
From Eq. (26), we obtain
Using Eqs. (18), (22) and (27), we have completed the proof of Eq. (13).
Step 2. Now we prove that for every \(0 < x < \pi /2\),
which is equivalent to
Let \(x_{3}=\frac{98 \cdot \pi }{256} \approx 1.202\).
Case 2.1. Proof of \(E_{1}(x) \geq 0\), \(\forall x \in (0,1]\).
It can be verified that for every \(x \in (0,\pi /2)\),
Combining Eq. (29) with Eq. (5), for every \(0< x \leq 1\), we have that
where \(B_{3,i}(x)=C_{8}^{i} x^{i} (1-x)^{8-i}\), \(\gamma _{4,0} \approx 5.54\), \(\gamma _{4,1} \approx 5.28\), \(\gamma _{4,2} \approx 4.91\), \(\gamma _{4,3} \approx 4.43\), \(\gamma _{4,4} \approx 3.84\), \(\gamma _{4,5} \approx 3.16\), \(\gamma _{4,6} \approx 2.38\), \(\gamma _{4,7} \approx 1.52\) and \(\gamma _{4,8} \approx 0.632\). In Eq. (30), for every \(x \in (0,1)\), we have \(B_{3,i}(x) >0\), \(\gamma _{4,i}>0\) and \(((\pi -2)x+\pi )(2 x+\pi )>0\), which means that
From Eq. (31), we obtain
Case 2.2. Proof of \(E_{1}(x) \geq 0\), \(\forall x \in (1,x_{3}]\).
For every \(1< x \leq x_{3}\), note that \(\theta _{2}'(x) >0\) and \(\ln (a(x)) < 0\), hence
Combining with Eq. (5), we have that
where \(B_{4,i}(x)= \frac{C_{6}^{i} (x-1)^{i} (x_{3}-x)^{6-i}}{(x_{3}-1)^{6}}\), \(\gamma _{5,0} \approx -3.98\), \(\gamma _{5,1} \approx -4.23\), \(\gamma _{5,2} \approx -4.45\), \(\gamma _{5,3} \approx -4.65\), \(\gamma _{5,4} \approx -4.81\), \(\gamma _{5,5} \approx -4.94\) and \(\gamma _{5,6} \approx -5.03\). In Eq. (34), for every \(1< x< x_{3}\), we have \(\gamma _{5,i}<0\), \(B_{4,i}(x) >0\) and \((\pi x-2 x+\pi ) (-2 x+\pi ) (2 x+\pi )>0\), which means that \(E_{3}(x) \leq 0\), \(\forall x \in (1,x_{3})\). Combining with Eq. (33), we get
Combining Eq. (35) with Eq. (33) yields
Case 2.3. Proof of \(E_{1}(x) \geq 0\), \(\forall x \in (x_{3},\pi /2)\).
Combining with Eq. (5), for every \(x_{3}< x < \pi /2\), we have
Let \(E_{5}(x)=\frac{E_{4}(x)}{\theta _{2}'(x)}\). It can be verified that
where \(W_{2}(x)=((\pi x-2 x+\pi ) (-2 x+\pi ) (2 x+\pi ) (-48 \pi x+192 x-6 \pi ^{2} x+\pi ^{3}))^{2} \geq 0\), \(B_{5,i}(x) = \frac{C_{8}^{i} \cdot (x-x_{3})^{i} \cdot (\pi /2-x)^{8-i}}{(\pi /2-x_{3})^{8}} \geq 0\), \(\forall x \in [x_{3},\pi /2)\), \(\gamma _{6,0} \approx -1.34 \cdot 10^{5}\), \(\gamma _{6,1} \approx -1.46 \cdot 10^{5}\), \(\gamma _{6,2} \approx -1.54 \cdot 10^{5}\), \(\gamma _{6,3} \approx -1.5 \cdot 10^{5}\), \(\gamma _{6,4} \approx -1.5 \cdot 10^{5}\), \(\gamma _{6,5} \approx -1.4 \cdot 10^{5}\), \(\gamma _{6,6} \approx -1.25 \cdot 10^{5}\), \(\gamma _{6,7} \approx -98875\) and \(\gamma _{6,8} \approx -64163\). In Eq. (38), for every \(x_{3}< x<\pi /2\), noting that \(W_{2}(x)>0\), \(\gamma _{6,i}<0\) and \(B_{5,i}(x) >0\), we have
From Eq. (39), we obtain
Combining Eq. (40) with Eq. (37), for every \(x \in (x_{3},\pi /2)\), and noting that \(\theta _{2}'(x)>0\), we get
Using Eqs. (32), (36) and (41), we have completed the proof of Eq. (28).
Now combining Eq. (13) with Eq. (28), we have completed the proof of Eq. (10), and hence of Theorem 1.1. □
3 Proof of Theorem 1.2
3.1 Lemmas
We recall Theorem 3.5.1 in [9, Chap. 3.5, p. 67] as follows.
Theorem 3.1
Let \(w_{0}, w_{1}, \dots , w_{r}\) be \(r+1\) distinct points in \([a,b]\), and \(n_{0}, \dots , n_{r}\) be \(r+1\) integers ≥0. Let \(N=n_{0}+ \cdots + n_{r} + r\). Suppose that \(g(t)\) is a polynomial of degree N such that \(g^{(i)}(w_{j})=f^{(i)}(w_{j})\), \(i=0,\dots , n_{j}\), \(j=0,\dots , r \). Then there exists \(\xi _{1}(t) \in [a,b]\) such that \(f(t)-g(t)= \frac{f^{(N+1)}(\xi _{1}(t))}{(N+1)!} \overset{r}{\underset{i=0}{\prod }} (t-w_{i})^{n_{i}+1} \).
Lemma 3.2
For every \(0< x<\pi /2\), we have that
Proof
Let \(H_{1}(x)=\sin (x) -( \frac{1}{120} x^{5}- \frac{1}{6} x^{3}+x)\). It can be verified that \(H_{1}(0)=H_{1}'(0)= \cdots = H^{(6)}(0)=0\) and \(H^{(7)}(0)=-1 \neq 0\). For every \(0< x<\pi /2\), using Theorem 3.1, there exists \(\psi (x) \in (0,\pi /2)\) such that
completing the proof. □
Lemma 3.3
For every \(0< x < \pi /2\), we have
where \(\ln ^{(7)}(\cos (x))\) denotes the seventh derivative.
Proof
For every \(0< x < \pi /2\), it can be verified that
This completes the proof. □
Lemma 3.4
For every \(0< x<\pi /2\), we have
Proof
For every \(0< x<\pi /2\), it can be verified that \(\bar{\kappa }_{1}^{(i)}(0)=0\), \(i=0,1,\dots ,15\) and \(\bar{\kappa } _{1}^{(16)}(0)=32768>0\), where \(\bar{\kappa }_{1}(x)=\kappa _{1}(x)- \varphi _{1}(x)\). Employing Theorem 3.1, for every \(0< x<\pi /2\), there exists \(\xi _{2}(x) \in (0,\pi /2)\) such that
Note that \(\bar{\kappa }_{1}(\pi /2) = \frac{32768 \cos (2\xi _{2}( \pi /2))}{16!} (\pi /2)^{16} \approx 0.0000020>0\), and, on the other hand, \(\cos (2x)>0\), \(\forall x \in (0,\pi /4)\) and \(\cos (2x)<0\), \(\forall x \in (\pi /4,\pi /2)\), hence we have that \(\xi _{2}(\pi /2) \in (0,\pi /4)\) and then \(\xi _{2}(x) \in (\xi _{2}(0), \xi _{2}(\pi /2)) \in (0,\pi /4)\). Combining with Eq. (42), we get \(\kappa _{1}(x)-\varphi _{1}(x)>0\), \(\forall x \in (0,\pi /2)\), completing the proof. □
Lemma 3.5
For every \(0< x<\pi /2\), we have
Proof
It can be verified that
where \(\kappa _{2}(x)= (2 \cos (x)^{4}+11 \cos (x)^{2}+2)+ \cos (x)^{2} (45-45 \cos (x)^{2}+15 \cos (x)^{4}) -15\). Combining with Lemma 3.4, we have
where \(\kappa _{3}(x)=\frac{x^{10}}{5142140516927060521875} \bar{ \kappa }_{3}(x)\), and
Combining with Eq. (43), for every \(0< x<\pi /2\), and noting that \(\sin (x)^{6} x^{6}>0\), we have \(\ln ^{(6)}(\frac{x}{\sin (x)})>0\), which completes the proof. □
Lemma 3.6
For every \(0< x<\pi /3\), we have
where \(\varphi _{2}(x)= \frac{-x^{2}}{2}-\frac{x^{4}}{12}+\frac{(162 \sqrt{3} \pi +108 \pi ^{2}+\pi ^{4}-2916 \ln (2))\cdot x^{5}}{2 \pi ^{5}} -\frac{3 (324 \sqrt{3} \pi +162 \pi ^{2}+\pi ^{4}-4860 \ln (2))\cdot x^{6}}{4 \pi ^{6}}\).
Proof
Let \(\kappa _{4}(x) = \ln (\cos (x))-\varphi _{2}(x)\). It can be verified that
Using Theorem 3.1 and Lemma 3.3, for \(0< x<\pi /3\), there exists \(\xi _{3}(x) \in (0,\pi /3)\) such that
which means that \(\ln (\cos (x))-\varphi _{2}(x) \leq 0\), and we complete the proof. □
Lemma 3.7
For every \(0< x<\pi /3\), we have
Proof
Let \(\kappa _{5}(x) = \ln (\frac{x}{\sin (x)})-\varphi _{3}(x)\). It can be verified that
Now by Theorem 3.1 and Lemma 3.5, for \(0< x<\pi /3\), there exists \(\xi _{4}(x) \in (0,\pi /3)\) such that
which means that \(\ln (\frac{x}{\sin (x)})-\varphi _{3}(x) < 0\), \(\forall x \in (0,\pi /3)\), and we complete the proof. □
3.2 Proof of Theorem 1.2
Proof of Theorem 1.2
Step 1. Firstly, we prove that \(\frac{\sin (x)}{x} \leq \cos (x)^{ \theta _{3}(x)}\), \(\forall x \in (0,\pi /2)\), where \(\theta _{3}(x)= \frac{1}{3}-\frac{4}{3 \pi ^{2}} x^{2}\). This is equivalent to
Combining with Lemma 3.2, we have that
For every \(0< x < \pi /2\), noting that \(\theta _{3}'(x)=-\frac{8 x}{3 \pi ^{2}} <0\) and \(\theta _{3}(x) >0\), and combining with Eq. (4), we have
Combining Eq. (45) with Eq. (46), we obtain
This completes the proof of Eq. (44), and hence proves \(\frac{\sin (x)}{x} \leq \cos (x)^{\theta _{3}(x)}\).
Step 2. Secondly, we prove that \(\cos (x)^{\theta _{4}(x)} \leq \frac{ \sin (x)}{x}\), \(\forall x \in (0,\pi /2)\), where \(\theta _{4}(x)= \frac{1}{3}- \frac{2}{45} x^{2}+\frac{5}{124} x^{3}-\frac{41}{1000}x ^{4}\). This is equivalent to
Case 2.1. \(0< x<\pi /3\).
Noting that \(\theta _{4}(x) \geq 0\), and combining with Lemmas 3.6 and 3.7, we have that
where \(B_{6,i}=\frac{C_{5}^{i} \cdot x^{i}\cdot (\pi /3-x)^{5-i}}{( \pi /3)^{5}}\), and \(\gamma _{7,0} \approx -0.0068\), \(\gamma _{7,1} \approx -0.0067\), \(\gamma _{7,2} \approx -0.0071\), \(\gamma _{7,3} \approx -0.0072\), \(\gamma _{7,4} \approx -0.0070\), \(\gamma _{7,5} \approx -0.0041\). Noting that \(B_{6,i} \geq 0\) and \(\gamma _{7,i}<0\), \(i=0,1,\dots ,5\), and combining with Eq. (49), we obtain
Case 2.2. \(\pi /3< x<1.36\).
Let \(\varphi _{4}(x)\) be the sextic interpolation polynomial such that
and \(\kappa _{6}(x) = \ln (\cos (x))-\varphi _{4}(x)\). We have that
Similar as in the proof of Lemma 3.6, by Theorem 3.1 and Lemma 3.3, for \(\pi /3< x<1.36\), there exists \(\xi _{5}(x) \in (\pi /3,1.36)\) such that
which means that \(\ln (\cos (x))-\varphi _{4}(x) \leq 0\).
On the other hand, let \(\varphi _{5}(x)\) be the quintic interpolation polynomial such that
Similarly, let \(\kappa _{7}(x)=\ln (\frac{x}{\sin (x)})-\varphi _{5}(x)\), and then, for every \(\pi /3< x<1.36\), there exists \(\xi _{6}(x) \in ( \pi /3,1.36)\) such that
which means that \(\ln (\frac{x}{\sin (x)}) \leq \varphi _{5}(x)\). Finally, for every \(\pi /3< x<1.36\), we have
where \(B_{7,i}(x) \approx \frac{C_{10}^{i} \cdot (x-\pi /3)^{i} \cdot (1.36-x)^{10-i}}{(1.36-\pi /3)^{10}}\), \(\gamma _{8,0} \approx -0.0052\), \(\gamma _{8,1} \approx -0.0054\), \(\gamma _{8,2} \approx -0.0055\), \(\gamma _{8,3} \approx -0.0056\), \(\gamma _{8,4} \approx -0.0055\), \(\gamma _{8,5} \approx -0.0052\), \(\gamma _{8,6} \approx -0.0048\), \(\gamma _{8,7} \approx -0.0041\), \(\gamma _{8,8} \approx -0.0033\), \(\gamma _{8,9} \approx -0.0026\) and \(\gamma _{8,10} \approx -0.0022\). Noting that \(B_{7,i}(x)\geq 0\) and \(\gamma _{8,i}<0\), and combining with Eq. (51), we have
Case 2.3. \(1.36< x<1.54\).
Let \(\varphi _{6}(x)\) be the sextic interpolation polynomial such that
and \(\kappa _{8}(x) = \ln (\cos (x))-\varphi _{6}(x)\). We have that
Similar as in the proof of Lemma 3.6, by Theorem 3.1 and Lemma 3.3, for \(1.36< x<1.54\), there exists \(\xi _{7}(x) \in (1.36,1.54)\) such that
which means that \(\ln (\cos (x)) \leq \varphi _{6}(x)\).
On the other hand, let \(\varphi _{7}(x)\) be the quintic interpolation polynomial such that
Similarly, letting \(\kappa _{9}(x)=\ln (\frac{x}{\sin (x)})-\varphi _{7}(x)\), for every \(1.36< x<1.54\), there exists \(\xi _{8}(x) \in (1.36,1.54)\) such that
which means that \(\ln (\frac{x}{\sin (x)}) \leq \varphi _{7}(x)\). Finally, for every \(1.36< x<1.54\), we have
where \(B_{8,i}(x)= \frac{C_{10}^{i} \cdot (x-1.36)^{i} \cdot (1.54-x)^{10-i}}{(1.54-1.36)^{10}}\), \(\gamma _{9,0} \approx -0.0022\), \(\gamma _{9,1} \approx -0.0019\), \(\gamma _{9,2} \approx -0.0018\), \(\gamma _{9,3} \approx -0.0020\), \(\gamma _{9,4} \approx -0.0025\), \(\gamma _{9,5} \approx -0.0029\), \(\gamma _{9,6} \approx -0.0024\), \(\gamma _{9,7} \approx -0.0021\), \(\gamma _{9,8} \approx -0.0068\), \(\gamma _{9,9} \approx -0.025\) and \(\gamma _{9,10} \approx -0.071\). Noting that \(B_{8,i}(x)> 0\), \(\forall x \in (1.36,1.54)\), and \(\gamma _{9,i}<0\), and combining with Eq. (53), we have that
Case 2.3. \(1.54< x<\pi /2\).
Let \(\varphi _{8}(x)\) be the quintic interpolation polynomial such that
Similarly, letting \(\kappa _{10}(x)=\ln (\frac{x}{\sin (x)})-\varphi _{8}(x)\), for every \(1.54< x<\pi /2\), there exists \(\xi _{9}(x) \in (1.54, \pi /2)\) such that
which means that \(\ln (\frac{x}{\sin (x)}) \leq \varphi _{8}(x)\). Finally, for every \(1.54< x<\pi /2\), we have
where \(B_{9,i}(x)=\frac{C_{5}^{i} \cdot (x-1.54)^{i} \cdot (\pi /2-x)^{5-i}}{( \pi /2-1.54)^{5}}\), \(\gamma _{10,0} = -0.071\), \(\gamma _{10,1} = -0.057\), \(\gamma _{10,2} = -0.043\), \(\gamma _{10,3} = -0.030\), \(\gamma _{10,4} = -0.01\) and \(\gamma _{10,5} = -0.0020\). Noting that \(B_{9,i}(x)> 0\), \(\forall x \in (1.54,\pi /2)\), and \(\gamma _{10,i}<0\), and combining with Eq. (55), we have
Using Eqs. (50), (52), (54) and (56), we have completed the proof of Eq. (48).
Combining Eqs. (44) and (48), we have completed the proof of Theorem 1.2. □
4 Comparisons and conclusion
Let \(f_{1}(x)=(\frac{4 \pi }{\pi ^{2}-4 x^{2}})^{(2x/\pi )^{2}}\), \(f_{2}(x)= a(x)^{-\theta _{1}(x)}\), \(f_{3}(x)= a(x)^{-\theta _{2}(x)}\) and \(f_{4}(x)=(\frac{4 \pi }{\pi ^{2}-4 x^{2}})^{\bar{\theta }_{3}(x)}\), and \(F_{i}(x)=\frac{1}{f_{i}(x)}\), \(i=1,2,3,4\). As shown in Fig. 1(a), \(F_{1}(x) \geq F_{2}(x) \geq F_{3}(x) \geq F_{4}(x)\), which means that Theorem 1.1 achieves much tighter bounds than those of Eq. (9).
Let \(p_{1}(x)=\bar{\theta }_{1}(x)/3\), \(p_{2}(x)=\frac{1}{3}- \frac{2}{45} x^{2}+\frac{5}{124} x^{3}-\frac{41}{1000}x^{4}\), \(p_{3}(x)=\frac{1}{3}-\frac{4}{3 \pi ^{2}} x^{2}\) and \(p_{4}(x)=\bar{ \theta }_{2}(x)/3\). As shown in Fig. 1(b), we have that \(p_{1}(x) \geq p_{2}(x) \geq p_{3}(x) \geq p_{4}(x)\), \(\forall x \in (0, \pi /2)\), combining with \(\cos (x) < 1\), \(\forall x \in (0,\pi /2)\), we have that
which means that the bounds in Theorem 1.2 are tighter than in Eq. (8).
References
Agarwal, R.P., Kim, Y.H., Sen, S.K.: A new refined Jordan’s inequality and its application. Math. Inequal. Appl. 12(2), 255–264 (2009)
Alirezaei, G., Mathar, R.: Scrutinizing the average error probability for Nakagami fading channels. In: The IEEE International Symposium on Information Theory (ISIT’14), Honolulu, pp. 2884–2888 (2014)
Chen, C.-P., Cheung, W.-S.: Sharp Cusa and Becker–Stark inequalities. J. Inequal. Appl. 2011, 136 (2011)
Chen, C.-P., Cheung, W.-S.: Sharpness of Wilker and Huygens type inequalities. J. Inequal. Appl. 2012, 72 (2012)
Chen, C.-P., Sandor, J.: Sharp inequalities for trigonometric and hyperbolic functions. J. Math. Inequal. 9(1), 203–217 (2015)
Chen, X.-D., Ma, J.Y., Jin, J.P., Wang, Y.G.: A two-point-Padé-approximant-based method for bounding some trigonometric functions. J. Inequal. Appl. 2018, 140 (2018) 1–15
Chen, X.-D., Shi, J., Wang, Y., Xiang, P.: A new method for sharpening the bounds of several special functions. Results Math. 72(1–2), 695–702 (2017)
Cloud, M.J., Drachman, B.C., Lebedev, L.P.: Inequalities with Applications to Engineering. Springer, Berlin (2014)
Davis, P.: Interpolation and Approximation. Dover, New York (1975)
Debnath, L., Mortici, C., Zhu, L.: Refinements of Jordan–Stečkin and Becker–Stark inequalities. Results Math. 67(1–2), 207–215 (2015)
Lutovac, T., Malešević, B., Mortici, C.: The natural algorithmic approach of mixed trigonometric-polynomial problems. J. Inequal. Appl. 2017(116), 1 (2017)
Lutovac, T., Malešević, B., Rašajski, M.: A new method for proving some inequalities related to several special functions. Results Math. 73(3), 100 (2018)
Makragic, M.: A method for proving some inequalities on mixed hyperbolic-trigonometric polynomial functions. J. Math. Inequal. 11(3), 817–829 (2017)
Malešević, B., Lutovac, T., Banjac, B.: A proof of an open problem of Yusuke Nishizawa for a power-exponential function. J. Math. Inequal. 12(2), 473–485 (2018)
Malešević, B., Lutovac, T., Rašajski, M., Mortici, C.: Extensions of the natural approach to refinements and generalizations of some trigonometric inequalities. Adv. Differ. Equ. 2018(90), 1 (2018)
Malešević, B., Makragic, M.: A method for proving some inequalities on mixed trigonometric polynomial functions. J. Math. Inequal. 10(3), 849–876 (2016)
Malešević, B., Rašajski, M., Lutovac, T.: Refinements and generalizations of some inequalities of Shafer–Fink’s type for the inverse sine function. J. Inequal. Appl. 2017, 275 (2017)
Malešević, B., Rašajski, M., Lutovac, T.: Refined estimates and generalizations of inequalities related to the arctangent function and Shafer’s inequality. arXiv:1711.03786
Mitrinović, D.S.: Analytic Inequalities. Springer, Berlin (1970)
Mortici, C.: The natural approach of Wilker–Cusa–Huygens inequalities. Math. Inequal. Appl. 14(3), 535–541 (2011)
Nenezić, M., Malešević, B., Mortici, C.: New approximations of some expressions involving trigonometric functions. Appl. Math. Comput. 283, 299–315 (2016)
Neuman, E.: On Wilker and Huygens type inequalities. Math. Inequal. Appl. 15(2), 271–279 (2012)
Neuman, E., Sandor, J.: On some inequalities involving trigonometric and hyperbolic functions with emphasis on the Cusa–Huygens, Wilker and Huygens inequalities. Math. Inequal. Appl. 13(4), 715–723 (2010)
Nishizawa, Y.: Sharpening of Jordan’s type and Shafer–Fink’s type inequalities with exponential approximations. Appl. Math. Comput. 269, 146–154 (2015)
Nishizawa, Y.: Sharp exponential approximate inequalities for trigonometric functions. Results Math. 71(3–4), 609–621 (2017)
Qi, F.: Extensions and sharpenings of Jordan’s and Kober’s inequality. J. Math. Technol. 12(4), 98–102 (1996)
Rahmatollahi, G., De Abreu, G.T.F.: Closed-form hop-count distributions in random networks with arbitrary routing. IEEE Trans. Commun. 60(2), 429–444 (2012)
Rašajski, M., Lutovac, T., Malešević, B.: Sharpening and generalizations of Shafer–Fink and Wilker type inequalities: a new approach. J. Nonlinear Sci. Appl. 11(7), 885–893 (2018)
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)
Wu, S., Debnath, L.: A generalization of L’Hospital-type rules for monotonicity and its application. Appl. Math. Lett. 22, 284–290 (2009)
Zhu, L.: Sharp Becker–Stark-type inequalities for Bessel functions. J. Inequal. Appl. 2010, Article ID 838740 (2010)
Zhu, L.: A refinement of the Becker–Stark inequalities. Math. Notes 93(3–4), 421–425 (2013)
Funding
This research work was partially supported by the National Science Foundation of China (61672009, 61761136010), Zhejiang Key Research and Development Project of China (LY19F020041, 2018C01030) and the Open Project Program of the National Laboratory of Pattern Recognition (201800006).
Author information
Authors and Affiliations
Contributions
All authors contributed equally to the manuscript and read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Chen, XD., Ma, J. & Li, Y. Approximating trigonometric functions by using exponential inequalities. J Inequal Appl 2019, 53 (2019). https://doi.org/10.1186/s13660-019-1992-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-019-1992-z