In this paper we, respectively, give simple proofs of some remarkable trigonometric inequalities, based on the Padé approximation method. We also obtain rational refinements of these inequalities. We are convinced that the Padé approximation method offers a general framework for solving many other similar inequalities.MSC: 41A21; 26D05; 26D15; 33B10

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

Abstract. The aim of this work is to provide simple proofs of some remarkable trigonometric inequalities: Jordan inequality, Kober inequality, Becker-Stark inequality, Wu-Srivastava inequality. The proofs are based on Padé approximant method. We also obtain rational refinements for these inequalities.Mathematics subject classification (2010): 41A21, 26D05, 26D15, 33B10.

The aim of this paper is to deal with the refinements of certain inequalities for hyperbolic functions using Padé approximation method. We provide a useful way of improving the inequalities for trigonometric functions and hyperbolic functions.

In this paper we provide approximations for the error function using the Padé approximation method and the Fourier series method. These approximations have simple forms and acceptable bounds for the absolute error. Then we use them in diffusion theory.