# New inequalities between the inverse hyperbolic tangent and the analogue for corresponding functions

## Abstract

In this paper, we present new inequalities which reveal further relationship for both the inverse tangent function $$\arctan (x)$$ and the inverse hyperbolic function $$\operatorname{arctanh}(x)$$. At the same time, we give the analogue for inverse hyperbolic tangent and other corresponding functions.

## 1 Introduction

Masjed-Jamei [1] obtained the following inequality for $$|x|<1$$:

$$(\arctan x)^{2} \leq \frac{x \ln (x+\sqrt{1+x^{2}})}{\sqrt{1+x^{2}}}.$$
(1)

Many similar or relative inequalities are discussed in references [2â€“14]. Recently, Zhu and Malesevic [15] affirmed inequality (1) for the large interval $$(-\infty ,\infty )$$, pointed out that $$\sinh ^{-1}(x) = \ln (x+\sqrt{1+x^{2}})$$, and provided the following TheoremsÂ 1â€“6, which (or relative results) can be also found in [11, 12].

### Theorem 1

([15])

The inequality

$$( \arctan x)^{2} \leq \frac{x \ln (x+\sqrt{1+x^{2}})}{\sqrt{1+x^{2}}}$$
(2)

holds for all$$x \in (-\infty ,\infty )$$, and the power number 2 is the best in (2).

### Theorem 2

([15])

Let$$0 < r < \infty$$, $$\lambda = 1$$, and$$\mu = r \ln (r + \sqrt{r^{2} + 1})/(\sqrt{r^{2} + 1}(\arctan r)^{2})$$. Then the double inequality

$$\lambda (\arctan x)^{2} \leq \frac{x \ln (x+\sqrt{1+x^{2}})}{\sqrt{1+x^{2}}} \leq \mu (\arctan x)^{2}$$
(3)

holds for all$$x \in (--r, r)$$, whereÎ»andÎ¼are the best constants in (3).

### Theorem 3

([15])

We have

\begin{aligned}& -\frac{1}{45} x^{6} \leq (\arctan x)^{2} - \frac{x \ln (x+\sqrt{1+x^{2}})}{\sqrt{1+x^{2}}} \leq -\frac{1}{45} x^{6} + \frac{4}{105} x^{8}, \end{aligned}
(4)
\begin{aligned}& \begin{aligned}[b] -\frac{1}{45} x^{6} + \frac{4}{105} x^{8}- \frac{11}{225} x^{10}& \leq ( \arctan x)^{2} - \frac{x \ln (x+\sqrt{1+x^{2}})}{\sqrt{1+x^{2}}} \\ &\leq -\frac{1}{45} x^{6} \leq + \frac{4}{105} x^{8}- \frac{11}{225} x^{10} + \frac{586}{10\text{,}395} x^{12}. \end{aligned} \end{aligned}
(5)

### Theorem 4

([15])

The inequality

$$(\operatorname{arctanh} x)^{2} \leq \frac{x \arcsin x}{\sqrt{1-x^{2}}}$$
(6)

holds for all$$x \in (-1, 1)$$, and the power number 2 is the best in (6).

### Theorem 5

([15])

Let$$0 < r < 1$$, $$\alpha _{1} = 1$$, and$$\beta _{1} = r(\arcsin r)/(\sqrt{1 -- r^{2}}(\operatorname{arctanh} r)^{2})$$. Then the double inequality

$$\alpha _{1} (\operatorname{arctanh} x)^{2} \leq \frac{x \arcsin x}{\sqrt{1-x^{2}}} \leq \beta _{1} (\operatorname{arctanh} x)^{2}$$
(7)

holds for all$$x \in (-r, r)$$, where$$\alpha _{1}$$and$$\beta _{1}$$are the best constants in (7).

Recently, Chen and MaleÅ¡eviÄ‡ [14] proposed the following results:

\begin{aligned}& \frac{ x \operatorname{arcsinh} x }{\sqrt{1+x^{2}+\alpha _{2} x^{4}}} \leq ( \arctan x)^{2} \leq \frac{ x \operatorname{arcsinh} x }{\sqrt{1+x^{2}+\beta _{2} x^{4}}},\quad x>0, \end{aligned}
(8)
\begin{aligned}& \frac{x \arcsin x}{1-\alpha _{3} x^{2}} < (\operatorname{arctanh} x)^{2},\quad 0< x< 1, \end{aligned}
(9)

where $$\alpha _{2}=\frac{2}{45}$$, $$\beta _{2}=0$$, and $$\alpha _{3}=\frac{1}{2}$$ are the best possible constants.

In 2020, Zhu and MaleÅ¡eviÄ‡ [13] proposed natural approximation of Masjed-Jameiâ€™s inequality and provided two-sided bounds in a polynomial form of $$(\arctan x)^{2}-\frac{x \ln (x+\sqrt{1+x^{2}})}{\sqrt{1+x^{2}}}$$, which consists of explicit formulae of different degrees.

The values of Î¼ in TheoremÂ 2 and $$\beta _{1}$$ in TheoremÂ 5 tend to be +âˆž for r tends to be Â±âˆž and Â±1, respectively. In this paper, we obtain the following new inequalities, which improve the approximation effect of the inequalities in [15]. The main results are as follows.

### Theorem 6

The inequality

$$( \arctan x)^{2} \geq \frac{3(8+9 x^{2}- 8 \sqrt{1+x^{2}} )}{(4+11 \sqrt{1+x^{2}})\sqrt{1+x^{2}}} \triangleq F(x)$$
(10)

holds for all$$x \in (- \infty , \infty )$$.

### Theorem 7

Let$$\kappa _{1}=\frac{108}{11 \pi ^{2}} \approx 0.9947$$and$$\kappa _{2}=1$$. The inequality

$$\kappa _{1} (\arctan x)^{2} \leq F(x) \leq \kappa _{2} (\arctan x)^{2}$$
(11)

holds for all$$x \in (-\infty , \infty )$$, where$$\kappa _{1}$$and$$\kappa _{2}$$are the best constants in (11).

### Theorem 8

The inequality

$$\frac{23}{75\text{,}600} x^{8} \geq (\arctan x)^{2} - F(x) \geq \frac{23}{75\text{,}600} x^{8} - \frac{899}{1\text{,}134\text{,}000} x^{10}$$
(12)

holds for all$$x \in (-\infty , \infty )$$.

### Theorem 9

The inequality

$$G_{1}(x) \triangleq (\operatorname{arctanh} x)^{2} \leq \biggl( \frac{-\ln (1-x^{2})}{\arcsin x}\biggr)^{2} \triangleq G_{2}(x)$$
(13)

holds for all$$x \in (-1, 1)$$.

### Theorem 10

Let$$\kappa _{3}=1$$and$$\kappa _{4}=\frac{16}{\pi ^{2}} \approx 1.6211$$. The inequality

$$\kappa _{3} (\operatorname{arctanh} x)^{2} \leq \biggl( \frac{-\ln (1-x^{2})}{\arcsin x}\biggr)^{2} \leq \kappa _{4} ( \operatorname{arctanh} x)^{2}$$
(14)

holds for all$$x \in (-1, 1)$$, where$$\kappa _{3}$$and$$\kappa _{4}$$are the best constants in (14).

## 2 Proofs of TheoremsÂ 6â€“10

Let $$\arctan x=t$$, then one has that $$x=\tan (t)$$ and $$\sqrt{1+x^{2}}=\sec (t)$$, where $$x \in (- \infty ,\infty )$$ and $$t \in (-\pi /2,\pi /2)$$. It can be verified that

$$\begin{gathered} (\arctan x)^{2} = t^{2}, \\ F(x)= -\frac{3}{4} \cos (t) - \frac{63}{16} + \frac{1125}{16 (4\cos (t) + 11)}=f_{1}(t), \\ (\arctan x)^{2} - F(x) = \bigl(t^{2} - f_{1}(t) \bigr) = \delta _{1}(t), \\ \delta _{1}^{\prime \prime \prime }(t) = \frac{ (12 (16 \cos (t)^{2}+208 \cos (t)+1501)) (\cos (t)-1)^{2} \sin (t)}{(4 \cos (t)+11)^{4}}. \end{gathered}$$
(15)

### 2.1 Proof of TheoremÂ 6

From Eq.Â (15), one has that

$$\delta _{1}^{\prime \prime \prime }(t) > 0, \quad t \in (0,\pi /2), \qquad \delta _{1}^{\prime \prime }(0)= \delta _{1}^{\prime }(0)= \delta _{1}(0) =0,$$
(16)

$$\delta _{1}^{\prime \prime }(t) > 0,\qquad \delta _{1}^{\prime }(t) > 0, \quad t \in (0,\pi /2), \qquad \delta _{1}(t) \geq \delta _{1}(0) = 0, \quad t \in [0,\pi /2).$$
(17)

Note that $$\delta _{1}(t)=\delta _{1}(-t)$$, combining Eq.Â (15) with Eq.Â (17), one has that

$$\delta _{1}(t) \geq 0, \quad t \in (-\pi /2,\pi /2), \quad \text{and} \quad (\arctan x)^{2}-F(x) \geq 0, \quad x\in (-\infty , \infty ).$$
(18)

And we complete the proof.

### 2.2 Proof of TheoremÂ 7

From TheoremÂ 6, one has that

$$F(x) \leq \kappa _{2} (\arctan x)^{2}.$$

Now we prove that $$\kappa _{1} (\arctan x)^{2} \leq F(x)$$. From Eq.Â (15), one has that

$$\begin{gathered} \kappa _{1} (\arctan x)^{2} - F(x) = \kappa _{1} t^{2} -f_{1}(t) = \delta _{2}(t), \\ \delta _{2}^{\prime \prime \prime }(t) = -f_{1}^{\prime \prime \prime }(t)= \delta _{1}^{\prime \prime \prime }(t)> 0, \quad t \in (0,\pi /2), \\ \delta _{2}^{\prime \prime }(0)= \frac{216-22 \pi ^{2}}{ 11 \pi ^{2}} \approx -0.01< 0 , \qquad \delta _{2}^{\prime \prime }(\pi /2)= \frac{26136-2250 \pi ^{2}}{1331 \pi ^{2}} \approx 0.2>0. \end{gathered}$$
(19)

From Eq.Â (19), there exists a unique root $$t_{1} \in (0,\pi /2)$$ such that

$$\begin{gathered} \delta _{2}^{\prime \prime }(t)< 0 , \quad t \in (0,t_{1}), \qquad \delta _{2}^{\prime }(0)= 0, \\ \delta _{2}^{\prime \prime }(t)>0 , \quad t \in (t_{1},\pi /2), \qquad \delta _{2}^{\prime }(\pi /2)= \frac{1188-372 \pi }{121 \pi } \approx 0.05>0. \end{gathered}$$
(20)

From Eq.Â (19), there exists a unique root $$t_{2} \in (t_{1},\pi /2)$$ such that

$$\begin{gathered} \delta _{2}^{\prime }(t)< 0 , \quad t \in (0,t_{2}), \qquad \delta _{2}(0)= 0, \\ \delta _{2}^{\prime }(t)>0 , \quad t \in (t_{2},\pi /2), \qquad \delta _{2}(\pi /2)=0. \end{gathered}$$
(21)

From Eq.Â (21), one has that

$$\delta _{2}(t) \leq 0 , \quad t \in [0,t_{2}] \cup [t_{2},\pi /2)=[0, \pi /2).$$
(22)

Note that $$\delta _{2}(t)=\delta _{2}(-t)$$, combining Eq.Â (19) with Eq.Â (22), one has that

$$\delta _{2}(t) \leq 0 , \quad t \in (-\pi /2,\pi /2),\quad \text{and}\quad \kappa _{1} (\arctan x)^{2} \leq F(x), \quad x\in (-\infty , \infty ).$$
(23)

Note that

$$\lim_{x \rightarrow \infty } \frac{F(x)}{(\arctan x)^{2}} = \kappa _{1},\qquad \lim_{x \rightarrow 0} \frac{F(x)}{(\arctan x)^{2}} =\kappa _{2},$$

both $$\kappa _{1}$$ and $$\kappa _{2}$$ are the best constants. And the proof is completed.

### 2.3 Proof of TheoremÂ 8

Let $$f_{2}(t)=\frac{23}{75\text{,}600} (\tan t)^{8}$$ and $$f_{3}(t)=\frac{23}{75\text{,}600} (\tan t)^{8} - \frac{899}{1\text{,}134\text{,}000} (\tan t)^{10}$$. EquationÂ (12) in TheoremÂ 8 is equivalent to

$$\delta _{3}(t) = \delta _{1}(t)-f_{2}(t) \leq 0, \qquad \delta _{4}(t) = \delta _{1}(t)-f_{3}(t) \geq 0, \quad t \in (-\pi /2,\pi /2).$$
(24)

It can be verified that

$$\begin{gathered} f_{2}^{\prime \prime \prime }(t)= \frac{23 \sin (t)^{5} (2 \cos (t)^{4}-26 \cos (t)^{2}+45)}{4725 (\cos t)^{11} }, \\ f_{3}^{\prime \prime \prime }(t)= \frac{\sin (t)^{5} (1175 \cos (t)^{6}-18871 \cos (t)^{4}+50\text{,}261 \cos (t)^{2}-29\text{,}667)}{28\text{,}350 (\cos t)^{13}}. \end{gathered}$$
(25)

Let $$\phi _{1}(t)=907\text{,}200 \cos (t)^{12}+12\text{,}700\text{,}800 \cos (t)^{11}+97\text{,}807\text{,}500 \cos (t)^{10}+97\text{,}795\text{,}724 \cos (t)^{9}+97\text{,}642\text{,}636 \cos (t)^{8}+96\text{,}990\text{,}540 \cos (t)^{7}+96\text{,}802\text{,}860 \cos (t)^{6}+103\text{,}838\text{,}238 \cos (t)^{5} + 126\text{,}378\text{,}882 \cos (t)^{4}+148\text{,}760\text{,}458 \cos (t)^{3}+130\text{,}005\text{,}062 \cos (t)^{2}+67\text{,}501\text{,}665 \cos (t)+ 15\text{,}153\text{,}435$$ and $$\phi _{2}(t)=5\text{,}443\text{,}200 \cos (t)^{13}+81\text{,}648\text{,}000 \cos (t)^{12}+668\text{,}493\text{,}000 \cos (t)^{11}+1\text{,}255\text{,}037\text{,}200 \cos (t)^{10}+1\text{,}837\text{,}671\text{,}000 \cos (t)^{9}+2\text{,}404\text{,}568\text{,}576 \cos (t)^{8}+ 2\text{,}978\text{,}639\text{,}640 \cos (t)^{7}+3\text{,}789\text{,}264\text{,}297 \cos (t)^{6}+5\text{,}266\text{,}619\text{,}820 \cos (t)^{5}+ 7\text{,}153\text{,}847\text{,}855 \cos (t)^{4}+7\text{,}714\text{,}708\text{,}320 \cos (t)^{3}+5\text{,}610\text{,}730\text{,}675 \cos (t)^{2}+2\text{,}369\text{,}206\text{,}620 \cos (t)+434\text{,}354\text{,}547$$. Combining Eq.Â (24) with Eq.Â (25), one has that

$$\begin{gathered} \delta _{3}^{\prime \prime \prime }(t)= \frac{\sin (t)(\cos (t)-1)^{3} }{(4 \cos (t)+11)^{4} (\cos t)^{11}} \phi _{1}(t) < 0,\quad \forall t \in (0,\pi /2), \\ \delta _{4}^{\prime \prime \prime }(t)= \frac{\sin (t) (\cos (t)-1)^{4} }{28\text{,}350 (4 \cos (t)+11)^{4} (\cos t)^{13}} \phi _{2}(t) > 0, \quad \forall t \in (0,\pi /2), \\ \delta _{3}^{\prime \prime }(0)=0, \qquad \delta _{4}^{\prime \prime }(0)=0. \end{gathered}$$
(26)

From Eq.Â (25), one has that

$$\delta _{3}^{\prime \prime }(t) < 0, \qquad \delta _{4}^{\prime \prime }(t) > 0,\quad \forall t \in (0, \pi /2), \qquad \delta _{3}^{\prime }(0)=0, \qquad \delta _{4}^{\prime }(0)=0.$$
(27)

From Eq.Â (27), one obtains that

$$\delta _{3}^{\prime }(t) < 0, \qquad \delta _{4}^{\prime }(t) > 0,\quad \forall t \in (0, \pi /2), \qquad \delta _{3}(0)=0, \qquad \delta _{4}(0)=0,$$
(28)

$$\delta _{3}(t) \leq 0, \qquad \delta _{4}(t) \geq 0,\quad \forall t \in [0,\pi /2).$$
(29)

Note that $$\delta _{i}(t)=\delta _{i}(-t)$$, $$i=3,4$$, combining with Eq.Â (29), both Eq.Â (24) and TheoremÂ 8 are proved.

### 2.4 Proof of TheoremÂ 9

Let $$\arcsin (x)=s$$, then one has that $$x=\sin (s)$$, where $$x \in (-1,1)$$, $$s \in (-\pi /2,\pi /2)$$. It can be verified that

$$\begin{gathered} (\operatorname{arctanh} x) = \frac{1}{2} \ln \biggl( \frac{1+\sin (s)}{1-\sin (s)}\biggr)>0, \\ \biggl(\frac{-\ln (1-x^{2})}{\arcsin x}\biggr) = \frac{-\ln (1-(\sin s)^{2})}{s} >0, \quad s \in (0,\pi /2). \end{gathered}$$
(30)

Let

$$\begin{gathered} (\operatorname{arctanh} x) - \biggl( \frac{-\ln (1-x^{2})}{\arcsin x}\biggr)= \frac{1}{2} \ln \biggl(\frac{1+\sin (s)}{1-\sin (s)} \biggr)- \frac{-\ln (1-(\sin s)^{2})}{s}= \delta _{5}(s), \\ \delta _{6}(s)=\delta _{5}^{\prime }(s) \cdot s^{2}, \qquad \phi _{3}(s)=-2+ \sin (s) s+2 \cos (s). \end{gathered}$$
(31)

It can be verified that

$$\phi _{3}^{\prime \prime }(s)= -\sin (s) s < 0, \quad s \in (0,\pi /2), \qquad \phi _{3}^{\prime }(0)= \phi _{3}(0)=0,$$

$$\phi _{3}(s)\leq 0, \qquad \delta _{6}^{\prime }(s)= \frac{s }{(\cos s)^{2} } \phi _{3}(s) \leq 0, \qquad \delta _{6}(0)=0, \quad s \in [0,\pi /2).$$
(32)

Combining Eq.Â (31) with Eq.Â (32), one obtains that

$$\delta _{6}(s)\leq 0, \qquad \delta _{5}^{\prime }(s) \leq 0, \qquad \delta _{5}(0)=0, \quad s \in [0,\pi /2).$$
(33)

Combining Eq.Â (31) with Eq.Â (33), we have that

$$\delta _{5}(s) \leq 0, \quad s \in [0,\pi /2), \qquad 0 \leq (\operatorname{arctanh} x)^{2} \leq \biggl(\frac{-\ln (1-x^{2})}{\arcsin x} \biggr)^{2}, \quad x \in [0,1).$$
(34)

Note that $$G_{i}(-x)=G_{i}(x)$$, $$i=1,2$$, combining with Eq.Â (34), we have proved both Eq.Â (13) and TheoremÂ 9.

### 2.5 Proof of TheoremÂ 10

Directly from TheoremÂ 9, we have proved the left-hand side in Eq.Â (14) in TheoremÂ 10.

$$\kappa _{3} (\operatorname{arctanh} x)^{2} \leq \biggl( \frac{-\ln (1-x^{2})}{\arcsin x}\biggr)^{2}.$$

Now, we will prove the right-hand side of Eq.Â (14). Combining with Eq.Â (30), let

$$\begin{gathered} \frac{4}{\pi } (\operatorname{arctanh} x) - \biggl(\frac{-\ln (1-x^{2})}{\arcsin x}\biggr)= \frac{4}{2 \pi } \ln \biggl(\frac{1+\sin (s)}{1-\sin (s)} \biggr)- \frac{-\ln (1-(\sin s)^{2})}{s} \triangleq \delta _{7}(s), \\ \delta _{8}(s)=\delta _{7}^{\prime }(s) \cdot s^{2}, \qquad \phi _{4}(s)= \frac{2(2 \sin (s) s+4 \cos (s)-\pi )}{\pi }. \end{gathered}$$
(35)

It can be verified that

$$\phi _{4}^{\prime \prime }(s)= \frac{-4 \sin (s) s}{\pi } < 0, \quad s \in (0,\pi /2),\qquad \phi _{4}^{\prime }(0)=\phi _{4}(\pi /2)=0,$$

$$\phi _{4}(s)\geq 0, \qquad \delta _{8}^{\prime }(s)= \frac{s }{(\cos s)^{2} } \phi _{4}(s) \geq 0,\qquad \delta _{8}(0)=0, \quad s \in [0,\pi /2).$$
(36)

Combining Eq.Â (35) with Eq.Â (36), one obtains that

$$\delta _{8}(s)\geq 0, \qquad \delta _{7}^{\prime }(s) \geq 0, \qquad \delta _{7}(0)=0, \quad s \in [0,\pi /2).$$
(37)

Combining Eq.Â (35) with Eq.Â (37), we have that

$$\begin{gathered} \delta _{7}(s) \geq 0, \quad s \in [0,\pi /2), \\ 0 \leq \biggl( \frac{-\ln (1-x^{2})}{\arcsin x}\biggr)^{2} \leq \biggl( \frac{4}{\pi } \operatorname{arctanh} x\biggr)^{2}, \quad x \in [0,1). \end{gathered}$$
(38)

Note that $$G_{i}(-x)=G_{i}(x)$$, $$i=1,2$$, combining with Eq.Â (38), one obtains that

$$\biggl(\frac{-\ln (1-x^{2})}{\arcsin x}\biggr)^{2} \leq \kappa _{2} ( \operatorname{arctanh} x)^{2}, \quad x \in (-1,1).$$
(39)

Combining TheoremÂ 9 with Eq.Â (39), we have completed the proofs of both Eq.Â (14) and TheoremÂ 10.

## 3 Discussions and conclusions

The values of Î¼ in TheoremÂ 2 and $$\beta _{1}$$ in TheoremÂ 5 tend to be +âˆž for r tends to be Â±âˆž and Â±1, respectively, while the values of $$\kappa _{i}$$ in TheoremsÂ 7 and 10 are constant. The error plots of the bounds from Eq.Â (2) and Eq.Â (6) in [15], from Eq.Â (8) and Eq.Â (9) in [14], and from Eq.Â (6) and Eq.Â (13) are plotted in Fig.Â 1. It shows that the results of Eq.Â (11) and Eq.Â (13) in this paper achieve better approximation effect than those of Eq.Â (2), Eq.Â (6), Eq.Â (8), and Eq.Â (9).

## References

1. Masjed-Jamei, M.: A main inequality for several special functions. Comput. Math. Appl. 60, 1280â€“1289 (2010)

2. MaleÅ¡evic, B., Lutovac, T., RaÅ¡ajski, M., Mortici, C.: Extensions of the natural approach to refinements and generalizationsâ€™ of some trigonometric inequalities. Adv. Differ. Equ. 2018, 90 (2018). https://doi.org/10.1186/s13662-018-1545-7

3. Zhu, L., Hua, J.K.: Sharpening the Beckerâ€“Stark inequalities. J. Inequal. Appl. 2010, Article ID 931275 (2010). https://doi.org/10.1155/2010/931275

4. Sun, Z.J., Zhu, L.: Simple proofs of the Cusaâ€“Huygens-type and Beckerâ€“Stark-type inequalities. J. Math. Inequal. 7(4), 563â€“567 (2013). https://doi.org/10.7153/jmi-07-52

5. Debnath, L., Mortici, C., Zhu, L.: Refinements of Jordanâ€“Steckin and Beckerâ€“Stark inequalities. Results Math. 67, 207â€“215 (2015). https://doi.org/10.1007/s00025-014-0405-3

6. Lv, H.L., Yang, Z.H., Luo, T.Q., Zheng, S.Z.: Sharp inequalities for tangent function with applications. J. Inequal. Appl. 2017, 94 (2017). https://doi.org/10.1186/s13660-017-1372-5

7. Zhu, L.: Sharpening Redhefferâ€“type inequalities for circular functions. Appl. Math. Lett. 22, 743â€“748 (2009). https://doi.org/10.1016/j.aml.2008.08.012

8. Wu, S., Debnath, L.: A generalization of lâ€™Hospitalâ€“type rules for monotonicity and its application. Appl. Math. Lett. 22, 284â€“290 (2009). https://doi.org/10.1016/j.aml.2008.06.001

9. Yang, Z.H., Chu, Y.M., Wang, M.K.: Monotonicity criterion for the quotient of power series with applications. J. Math. Anal. Appl. 428(1), 587â€“604 (2015). https://doi.org/10.1016/j.jmaa.2015.03.043

10. Zhu, L.: New bounds for the exponential function with cotangent. J. Inequal. Appl. 2018, Article ID 106 (2018). https://doi.org/10.1186/s13660-018-1697-8

11. Banjac, B.D.: System for automatic proving of some classes of analytic inequalities. School of Electrical Engineering (2019). Available on: http://baig.etf.bg.ac.rs/

12. Malesevic, B., Makragic, M.: A method for proving some inequalities on mixed trigonometric polynomial functions. J. Math. Inequal. 3, 849â€“876 (2016)

13. Zhu, L., MaleÅ¡eviÄ‡, B.: Natural approximation of Masjedâ€“Jameiâ€™s inequality. Rev. R. Acad. Cienc. Exactas FÃ­s. Nat., Ser. A Mat. 114, 25 (2020)

14. Chen, C.P., MaleÅ¡eviÄ‡, B.: Inequalities related to certain inverse trigonometric and inverse hyperbolic functions. Rev. R. Acad. Cienc. Exactas FÃ­s. Nat., Ser. A Mat. 114, 105 (2020)

15. Zhu, L., MaleÅ¡evic, B.: Inequalities between the inverse hyperbolic tangent and the inverse sine and the analogue for corresponding functions. J. Inequal. Appl. 2019, 93 (2019)

### Acknowledgements

The authors would like to thank the editor and the anonymous referees for their valuable suggestions and comments which helped us to improve this paper greatly.

Not applicable.

## Funding

This research work was partially supported by Zhejiang Key Research and Development Project of China (LY19F020041, 2018C01030), the National Science Foundation of China (61972120,61672009).

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All authors contributed equally to the manuscript and read and approved the final manuscript.

### Corresponding author

Correspondence to Xiao-Diao Chen.

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Chen, XD., Nie, L. & Huang, W. New inequalities between the inverse hyperbolic tangent and the analogue for corresponding functions. J Inequal Appl 2020, 131 (2020). https://doi.org/10.1186/s13660-020-02396-8