Refinements and generalizations of some inequalities of Shafer-Fink’s type for the inverse sine function

In this paper, we give some sharper refinements and generalizations of inequalities related to Shafer-Fink’s inequality for the inverse sine function stated in Theorems 1, 2, and 3 of Bercu (Math. Probl. Eng. 2017: Article ID 9237932, 2017).


Introduction
Inverse trigonometric functions, particularly the inverse sine function, have many applications in computer science and engineering. They are widely used in many fields, such as telecommunications, especially optical fiber telecommunications, signal processing, machine learning, and so on.
The main objective of the research presented in this paper is a refinement of Shafer-Fink's inequality for x ∈ [, ]; see [, ]. Various improvements of Shafer-Fink's inequality have been considered so far in [] and [-]. Also, let us mention that one refinement of Shafer-Fink's inequality was given in [], and it had applications in [, ] (see also []).
In this paper, we focus on the results of Bercu [] related to Shafer-Fink's inequality and give generalizations and refinements of the inequalities stated in Theorems , , and  in that paper. For e convenience of the reader, we further cite them.

Main results
The main results of this paper are generalizations and improvements of the inequalities related to Shafer-Fink's inequalities given in Theorems , , and  by Bercu [], here Statements , , and . First, let us recall some well-known power series expansions.
For |x| ≤ , where for m ∈ N  . Also, for |x| ≤ , where for m ∈ N  .

Refinements of the inequalities in Statements 1 and 2
Let us consider the function for x ∈ [, ] and k =  or k = π .
Then, for x ∈ [, ], we have: for m ∈ N and C k () = k  . Equality () is obtained by applying Cauchy's product to the corresponding series.
It is easy to verify that the following recurrence relations hold: and for m ∈ N  and k =  or k = π . Next, let us consider the function for x ∈ [, ] and k =  or k = π . Then, for x ∈ [, ], we have: Let us prove that D k (n) >  for all n ∈ N , n ≥ . First, we note that for k =  or k = π , we have: Now, let us assume that the statement holds for n = m, that is, D k (m) > . We will prove that the statement holds for n = m + , that is, D k (m + ) > . Using the recurrence relations (), (), and (), we get: Observing the above expression and using the induction hypothesis (D k (m) > ), we conclude that D k (m + ) > . Hence, by the principle of mathematical induction it follows that D k (n) >  for all n ∈ N , n ≥ , that is, Thus, we have proved the following theorem.
Theorem  For x ∈ [, ], n ∈ N , and k =  or k = π , we have the inequality Remark  For n =  and n = , we get the left-hand sides of the inequalities stated in Statements  and , respectively (Theorems  and  from Bercu []).
Example  For k = , the following statements are true for every x ∈ [, ].
• If n = , then • If n = , then • If n = , then etc. Also, for k = π , the following statements are true for every x ∈ [, ].

Refinements of the inequality in Statement 3
In [, Theorem ], Bercu proved the following inequalities for every x ∈ [, ]: where a(x) = (/)x  + (/)x  . We propose the following improvement and generalization of ().
Theorem  If n ∈ N and n ≥ , then for every x ∈ [, ], where Remark  Note that inequality () is a particular case of () for n = .
Example  For n > , inequality () refines inequality (), and we have the following new results.

etc.
Proof of Theorem  Based on Cauchy's product of power series () and (), the real analytical function has the power series where for m = , , . . . . First, we prove relation (). Consider the sequence (S(m)) m∈N,m≥ where and Finally, as that is, () Statement () is true for m = , that is, T() =  > . Observing that and using the induction hypothesis (i.e., T(k) = (k-)!  k k- (k-)!  >  for some positive integer k ≥ ), we conclude, by the principle of mathematical induction, that T(k + ) > . Therefore, inequalities () and () are true, and consequently E(m) >  for m ∈ N , m ≥ .

Conclusion
In this paper, we proposed and proved new inequalities, which present refinements and generalizations of inequalities stated in [], related to Shafer-Fink's inequality for the inverse sine function.
Also, our approach provides inequalities that allow new approximations of the functions