Double-Sided Taylor’s Approximations and Their Applications in Theory of Trigonometric Inequalities

Abstract

In this paper the double-sided TAYLOR's approximations are used to obtain generalisations and improvements of some trigonometric inequalities.

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

“…The left-hand side inequality of (22) holds for all x ∈ [0, 1] due to b n ≥ 0, and the light-hand side inequality of (22) holds just due to 0 ≤ x ≤ 1.…”

“…Remark 5. In the process of proving Theorems 1 and 2, we prove that a n , b n > 0, which just meet a condition in a theorem called "Theorem on double-sided TAYLOR's approximations" (see [21] (Theorem 4), [22] (Theorem 2), [23] (Theorem 22)). Therefore, the proofs of Theorems 1 and 2 can be completed by "Theorem on double-sided TAYLOR's approximations".…”

Some new estimates of precision of Cusa-Huygens and Huygens approximations

“…Let us introduce some notation and the basic claims that shall be used according to the papers [8] and [9]. Let us begin from real function f : (a, b) −→ R for which there are the nite values f (k) (a+) = lim…”

“…The previous statement we call the Theorem on double-sided Taylor's approximations in [8] and [9], i.e. Theorem WD in [26]- [30].…”

Convexity and double-sided Taylor's approximations

“…Concerning the already established results about the polynomial, rational and irrational bounds of functions sin x/x and cos x, we would like to recommend reading the survey article [10] by F. Qi, D.-W. Niu and B.-N. Guo (see also Example 3.4, where we will use some particular results quoted in this article). On the other hand, the double-sided Taylor's approximations play important role in the theory of analytic inequalities; see, e.g., [5,6,8,13] and references cited therein for further information in this direction. As mentioned in the abstract, the main aim of this paper is to establish some general results about the double-sided Taylor's approximations of functions.…”