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

Abstract

In this paper we propose a new method for sharpening and refinements of some trigonometric inequalities. We apply these ideas to some inequalities of Wilker–Cusa–Huygens type.

Double–Sided Taylor’s Approximations and Their Applications in Theory of Analytic Inequalities

“…which converges for x ∈ 0, . Based on this, the following result was proposed and proved in [23] (Theorem 5):…”

“…In [23] the polynomials T f , 0+ m (x) and Ì f ; 0+, π/2− m (x) are calculated and the concrete inequalities…”

“…The same approach, based on Theorem 1 and Theorem 4, provides generalizations of the inequalities from [23] related to the function f (x) = 3…”

“…Let us name this theorem the Theorem on double-sided TAYLOR's approximations. In papers [23], [24], [25], [26] and [27] Theorem 1 was denoted by Theorem WD. Let us note that the proof of Theorem 1 (Theorem 2 in [7]) was based on the L'HOSPITAL's rule for monotonicity.…”

Sharpening and generalizations of Shafer-Fink and Wilker type inequalities: a new approach

“…It is important to say that inequalities of this type have applications in various fields of engineering [1,5,16]; see also [4,10,12].…”