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

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:


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).
Theorem 1.For every real number 0 ≤  ≤ 1, the following two-sided inequality holds: Proof.The function has the derivative In order to prove that   () > 0, for every 0 ≤  ≤ 1, we have to establish that Since both sides of the above inequality are positive for all 0 ≤  ≤ 1, we can rise to the second power and obtain the following true result: We thus find that   () ≥ 0, which imply that the function  is strictly increasing on [0, 1].
For proving the right-hand side inequality from Theorem 1, we introduce the function The proof is completed.
The result is stated as Theorem 2.
Proof.The function has the derivative For proving that ℎ  () ≥ 0 for all 0 ≤  ≤ 1, we have to show or equivalently Since both sides of the above inequality are positive for all 0 ≤  ≤ 1, we can rise to the second power and deduce that which is true for every 0 ≤  ≤ 1.
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  ∈ [0, 1] in the left-hand side and for every  ∈ [0, 0.871433] in the right-hand side, the following inequalities hold true: Remark 4. Using MATLAB software, we found that the equation has the real roots  1 = 0,  2,3 ≈ ±0.871433.

Mathematical Problems in Engineering 3
We consider the function and its derivative ) . ( We have to find the real number  ∈ [0, 0.871433] so that   () ≤ 0 for all 0 ≤  ≤  or equivalently Both sides of the above inequality are positive on [0, 0.871433], and hence we can rise to the second power and we find We choose  = 0.735975.Therefore,   () ≤ 0 for all 0 ≤  ≤  and   () > 0 for all  <  ≤ 0.871433.
For proving the left-hand side inequality from Theorem 3, we introduce the function Its derivative is The inequality   () ≥ 0 on [0, 1] is equivalent to or Both sides of the above inequality are positive on [0, 1]; therefore we rise to the second power and we get the following true inequality on [0, 1]: In Theorem 5, we will state the following inequalities which give good results near the number 1 for the approximation arcsin  ≈ /(2 + √ 1 −  2 ) + ().

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.