Mathematical Problems in Engineering
Volume 2017 (2017), Article ID 9237932, 5 pages
https://doi.org/10.1155/2017/9237932

Research Article

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

Gabriel Bercu

Department of Mathematics and Computer Sciences, “Dunărea de Jos” University of Galaţi, 111 Domnească Street, 800201 Galaţi, Romania

Correspondence should be addressed to Gabriel Bercu; gbercu@ugal.ro

Received 9 April 2017; Accepted 2 July 2017; Published 2 August 2017

Academic Editor: Kishin Sadarangani

Copyright © 2017 Gabriel Bercu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 equivalently

Since 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 has

where .

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 equivalently

The 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 obtain

The 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 get

where

The 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.

References

B.-N. Guo and F. Qi, “Sharpening and generalizations of Carlson's inequality for the arc cosine function,” Hacettepe Journal of Mathematics and Statistics, vol. 39, no. 3, pp. 403–409, 2010.

B. J. Malešević, “An application of -method on Shafer-Fink’s inequality,” Univerzitet u Beogradu. Publikacije Elektrotehni\v ckog Fakulteta. Serija Matematika, vol. 8, pp. 90–92, 1997.

B. J. Malešević, “An application of -method on inequalities of Shafer-Fink-type,” Mathematical Inequalities & Applications, vol. 10, no. 3, pp. 529–534, 2007.

D. S. Mitrinović, Analytic Inequalities, Springer, Berlin, Germany, 1970.

C. Mortici and H. M. Srivastava, “Estimates for the arctangent function related to Shafer's inequality,” Colloquium Mathematicum, vol. 136, no. 2, pp. 263–270, 2014.

Y. Hu and C. Mortici, “A lower bound on the sinc function and its application,” The Scientific World Journal, vol. 2014, Article ID 571218, 4 pages, 2014.

F. Qi and B.-N. Guo, “Sharpening and generalizations of Shafer's inequality for the arc sine function,” Integral Transforms and Special Functions. An International Journal, vol. 23, no. 2, pp. 129–134, 2012.

L. Zhu, “On Shafer-Fink inequalities,” Mathematical Inequalities & Applications, vol. 8, no. 4, pp. 571–574, 2005.

L. Zhu, “On Shafer-Fink-type inequality,” Journal of Inequalities and Applications, vol. 2007, Article ID 67430, 4 pages, 2007.