Some Rational Approximations and Bounds for Bateman’s G-Function

Abstract

Symmetrical patterns exist in the nature of inequalities, which play a basic role in theoretical and applied mathematics. In several studies, inequalities present accurate approximations of functions based on their symmetry properties. In this paper, we present the following rational approximations for Bateman’s G-function G(w)=1w+2w2+∑j=1n4αjw2−2j−1+O1w2n+2, where α1=14, and αj=(1−22j+2)B2j+2j+1+∑ν=1j−1(1−22j−2ν+2)B2j−2ν+2ανj−ν+1,j>1. As a consequence, we introduced some new bounds of G(w) and a completely monotonic function involving it.

Two Approximation Formulas for Bateman’s G-Function with Bounded Monotonic Errors

“…. and they proved the following inequality: Recently, Ahfaf, Mahmoud, and Talat [8] introduced the following rational approximations…”