The natural approach of trigonometric inequalities - Padé approximant

Abstract

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.

Extensions of the natural approach to refinements and generalizations of some trigonometric inequalities

“…Similarly, Theorems 1 , 2 , 3 , 4 , 5 , and 6 can be applied to other results and inequalities from [ 13 , 16 ], as well as to a broad category of analytical inequalities.…”

“…The results of the present research can be used to verify as well as to refine a broad category of inequalities. For example, Bercu ([ 13 ], Theorem 2.4) proved the following inequalities: for every , where .…”

The natural algorithmic approach of mixed trigonometric-polynomial problems

“…Using Padé approximations, Bercu [ 7 , 13 ] recently refined certain trigonometric inequalities over various intervals . All such inequalities can be proved in a similar way and using the Natural Approach algorithm as in the proof of Theorem 9 .…”

New Polynomial Bounds for Jordan’s and Kober’s Inequalities Based on the Interpolation and Approximation Method

“…It is very obvious that the right sides of Inequality (16), Inequality (17), and Inequality (18) are the same. Recently, Bercu [20] provided a Padé-approximant-based method and obtained the following inequalities:…”