Simple efficient bounds for arcsine and arctangent functions

Abstract

The aim of this paper is to present new, simple and sufficiently sharp bounds for arcsine and arctangent functions. Some of the bounds are computationally efficient while others are efficient to approximate the integrals Int_{a}^{b} (arcsin x)/x dx and Int_{a}^{b} (arctan x)/x dx. As a matter of interest, several other sharp and generalized inequalities for (arcsin x)/x and (arctan x)/x are also established which are efficient to give some known and other trigonometric inequalities

The Natural Approaches of Shafer-Fink Inequality for Inverse Sine Function

“…Some new proof and various improvements of Shafer-Fink inequality can be found in [3][4][5][6][7][8][9][10][11][12][13]. In [14], Bercu obtained the generalizations and refinements of Shafer-Fink inequality as follows.…”