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

Abstract

In this paper the double-sided TAYLOR's approximations are studied. A short proof of a well-known theorem on the double-sided TAYLOR's approximations is introduced. Also, two new theorems are proved regarding the monotonicity of such approximations. Then we present some new applications of the double-sided TAYLOR's approximations in the theory of analytic inequalities.Similarly, the first TAYLOR's approximation in the left neighborhood of b is defined by:

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…”

“…We shall call T f, a+ n (x) the rst Taylor approximation in the right neighbourhood of a [8]. For n ∈ N 0 , we dene the remainder of rst Taylor approximation in the right neighbourhood of a by R f, a+ n (x) = f (x) − T f, a+ n (x).…”

“…For n ∈ N 0 , we dene the remainder of rst Taylor approximation in the right neighbourhood of a by R f, a+ n (x) = f (x) − T f, a+ n (x). In the paper [8] are considered the polynomials:…”

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

“…Particularly, their application in proving various analytic inequalities is of great importance [6], [12], [16], [17]. Recently, numerous inequalities have been generalized and improved by the use of the so-called doublesided TAYLOR's approximations [16], [20], [21]- [25] and [29]. Many topics regarding these approximations are presented in [29].…”

“…Recently, numerous inequalities have been generalized and improved by the use of the so-called doublesided TAYLOR's approximations [16], [20], [21]- [25] and [29]. Many topics regarding these approximations are presented in [29]. Some of the basic concepts and results about the double-sided TAYLOR's approximations presented in [29], which will be used in this paper, are given in the next section.…”