Bounds for Quotients of Inverse Trigonometric and Inverse Hyperbolic Functions

We establish new simple bounds for the quotients of inverse trigonometric and inverse hyperbolic functions such as sin −1 x sinh−1 x and tanh −1 x tan−1 x . The main results provide polynomial bounds using even quadratic functions and exponential bounds under the form eax 2 . Graph validation is also performed.

We contribute to the subject by establishing polynomial and exponential bounds for the functions sin −1 x sinh −1 x and tanh −1 x tan −1 x , which are motivated by these works. In the whole paper, it is to be noted that the superscript "−" for trigonometric and hyperbolic functions is used for their inverses.

Statements
Our main results are the following theorems.
− 1 ≈ 0.78221397 are the best possible constants such that the inequalities hold.

Graphical Illustrations
In this part, we compare the obtained bounds by the means of graphics, with a discussion. Figure 1 presents the bounds obtained in Theorems 1 and 3 for the "ratio sin" function defined by sin It can be observed that the exponential bounds are sharper.  Again, it can be observed that the exponential bounds are sharper. Thus, the graphical illustrations reveal that the upper bounds of (6) and (7) are sharper than those of (4) and (5), respectively.
We end by illustrating the ratio comparison states in Corollary 1 in Figure 3.

Auxiliary Results
In order to prove our main results, we need the following lemmas from the existing literature.
Then, (i) r 1 (x) and r 2 (x) are increasing on (a, b) if f g is increasing on (a, b); and (ii) r 1 (x) and r 2 (x) are decreasing on (a, b) if f g is decreasing on (a, b). The strictness of the monotonicity of r 1 (x) and r 2 (x) depends on the strictness of the monotonicity of f g .
The series for (sin −1 x) 2 can also be found in [24]. For series expansions of powers of sin −1 x we refer to [25] and references therein.
We also prove some other lemmas that are required to prove our main results.

Lemma 3.
The following inequality is true.
Differentiation gives (9) is a refinement of the inequality See, for instance, [5].

Remark 1.
It is worth noting that an upper bound of tanh −1 x tan −1 x in (12) is sharper than those in (5) and (7) as r → 1.

Proofs of Theorems
Proof of Theorem 1. Let us set By differentiating with respect to x, we obtain 1+x 2 with f 3 (0) = 0 and f 4 (0) = 0. By differentiating again with respect to x, we get . Now we need to show that f 8 (x) is strictly increasing on (0, 1). To demonstrate the required monotonicity of f 8 (x), we must prove that f 8 (x) > 0. First, we show that the numerator in f 8 (x), say N 1 , is positive on (0, 1). We have Simplifying the above expression we get the following So N 1 > 0 and hence f 8 (x) is positive. As a result, f 8 (x) is strictly increasing on (0, 1). By successive application of Lemma 1, we conclude that f (x) is strictly increasing on (0, 1).
This completes the proof of Theorem 1.
Proof of Corollary 1. It is an immediate consequence of Theorems 1 and 2, and Remark 2.

Remark 3.
A better upper bound for tanh −1 x tan −1 x in (0, 1) can be found in Lemma 6, as stated in Remark 1.

Conclusions and Direction for Further Research
Polynomial and exponential bounds for bell-shaped functions involving only trigonometric or only hyperbolic functions or their inverses are present in the literature. Recently, these types of bounds have been obtained for the quotients of trigonometric and hyperbolic functions. We contributed to the field by establishing similar bounds for the quotients of inverse trigonometric and inverse hyperbolic functions, which can be useful in the theory of analytical inequalities. The exponential bounds were sharper than the polynomial bounds.
Wilker-type and Huygens-type inequalities for inverse trigonometric and inverse hyperbolic function quotients may also be obtained.