Sharp Refinements for the Inverse Sine Function Related to Shafer-Fink’s Inequality

Abstract

The aim of our work is to give new sharp refinements of Shafer-Fink’s inequality, using suitable changes of variables.

1. Introduction and Motivation

The inverse sine function is an elementary function that appears in many fields of engineering. In the communication theory and signal processing, it is used to describe the phase of a complex-valued signal. The inverse sine function also appears in the field of control theory, where a nonlinear network unit is modeled by a nonlinear function.

There are many applications in which the inverse sine must be replaced by an approximated function, for example, by a rational function. But we have to mention that finding a replacement of simple form for the inverse sine function is in fact difficult. That is why in our work we focus only on special analytic inequalities which have some interesting properties.

The starting point of this paper is Shafer-Fink’s double inequality for the arc sine function:

Furthermore, 3 and are best constants in (1). Many refinements and extensions of this inequality have been provided (see, e.g., [1–9] and closely related references therein).

2. Results and Discussion

Since , it results that (1) gives a good results near zero, as .

Related to the above approximation, we present the following inequality, which provides improvement of Shafer-Fink’s inequality (1).

Theorem 1. For every real number , the following two-sided inequality holds:

Proof. The function has the derivative In order to prove that , for every , we have to establish that or equivalentlySince both sides of the above inequality are positive for all , we can rise to the second power and obtain the following true result: We thus find that , which imply that the function is strictly increasing on .
For proving the right-hand side inequality from Theorem 1, we introduce the function Its derivative is positive for all . Thus the function is strictly increasing on with , so on .
The proof is completed.

Another way to extend the left-hand side inequality from (1) is to consider the approximation near the origin of the form , where as .

The result is stated as Theorem 2.

Theorem 2. For every , one haswhere .

Proof. The function has the derivative For proving that for all , we have to show or equivalently Since both sides of the above inequality are positive for all , we can rise to the second power and deduce that which is true for every .
The proof of Theorem 2 is completed.

In the following, we will discuss the right-hand side inequality from (1). More precisely, we state and prove the following results.

Theorem 3. For every in the left-hand side and for every in the right-hand side, the following inequalities hold true:

Remark 4. Using MATLAB software, we found that the equation has the real roots , .

We consider the function and its derivative

We have to find the real number so that for all or equivalently

Both sides of the above inequality are positive on , and hence we can rise to the second power and we find

The polynomial function from the left-hand side has the real roots and ±3 ≈ .

We choose .

Therefore, for all and for all .

Since and , we obtain that on .

For proving the left-hand side inequality from Theorem 3, we introduce the function

Its derivative is

The inequality on is equivalent to or

Both sides of the above inequality are positive on ; therefore we rise to the second power and we get the following true inequality on :

In Theorem 5, we will state the following inequalities which give good results near the number for the approximation .

Theorem 5. For every , one has

Remark 6. In order to improve the right-hand side of Shafer-Fink’s inequality (1), we impose that , which is true for , where .

Proof. We consider the function and its derivative We note that . In order to obtain on , we have to prove that for all or equivalentlyThe real roots of the function from the left-hand side of (29) are Case  1. For all , the function from the left-hand side of (29) is nonpositive and the function from the right-hand side of (29) is positive; hence inequality (29) holds true.
Case  2. For , both sides of inequality (29) are positive, and thus we can rise to the second power and obtainThe polynomial function has the real roots , , or ; hence for all .
The polynomial function has the real roots , , , , or ; hence for all .
Therefore, both sides of inequality (31) are nonpositive, so we multiply it by and then we can rise to the second power and obtain where Using MATLAB software, we find that the polynomial function has only one real root ; hence on .
For proving the right-hand side of the inequality from Theorem 5 we introduce the function We note that ; therefore in order to demonstrate the inequality on , we have to establish that on or equivalently Since both sides of the above inequality are positive on , we can rise to the second power and getwhereThe polynomial function has the real roots , , and ; hence on .
The polynomial function has the real roots , , , , and ; hence on .
Therefore, inequality (36) is true on .
This completes the proof.

Remark 7. Combining the results of Theorems 3 and 5, respectively, we note that we improve the right-hand side of Shafer-Fink’s inequality on as follows:

3. Conclusion

In the present work we investigated the approximation of the inverse sine function and obtained new bounds. We have deduced lower and upper bounds which are sharp and very accurate and also improve Shafer-Fink’s inequality.

Conflicts of Interest

The author declares that he has no conflicts of interest.