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

Ahfaf, Omelsaad; Mahmoud, Mansour; Talat, Ahmed

doi:10.3390/sym14050929

Open AccessArticle

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

by

Omelsaad Ahfaf

¹,

Mansour Mahmoud

^2,*

and

Ahmed Talat

³

¹

Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

²

Mathematics Department, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

³

Mathematics and Computer Sciences Department, Faculty of Science, Port Said University, Port Said 42526, Egypt

^*

Author to whom correspondence should be addressed.

Symmetry 2022, 14(5), 929; https://doi.org/10.3390/sym14050929

Submission received: 9 April 2022 / Revised: 25 April 2022 / Accepted: 27 April 2022 / Published: 2 May 2022

(This article belongs to the Special Issue Mathematical Inequalities, Special Functions and Symmetry)

Download Versions Notes

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) = \frac{1}{w} + {[2 w^{2} + \sum_{j = 1}^{n} 4 α_{j} w^{2 - 2 j}]}^{- 1} + O (\frac{1}{w^{2 n + 2}}),

where

α_{1} = \frac{1}{4},

and

α_{j} = \frac{(1 - 2^{2 j + 2}) B_{2 j + 2}}{j + 1} + \sum_{ν = 1}^{j - 1} \frac{(1 - 2^{2 j - 2 ν + 2}) B_{2 j - 2 ν + 2} α_{ν}}{j - ν + 1}, j > 1 .

As a consequence, we introduced some new bounds of

G (w)

and a completely monotonic function involving it.

Keywords: psi function; Padé approximant; Bateman’s G-function; bounds; completely monotonic; applications

Share and Cite

MDPI and ACS Style

Ahfaf, O.; Mahmoud, M.; Talat, A. Some Rational Approximations and Bounds for Bateman’s G-Function. Symmetry 2022, 14, 929. https://doi.org/10.3390/sym14050929

AMA Style

Ahfaf O, Mahmoud M, Talat A. Some Rational Approximations and Bounds for Bateman’s G-Function. Symmetry. 2022; 14(5):929. https://doi.org/10.3390/sym14050929

Chicago/Turabian Style

Ahfaf, Omelsaad, Mansour Mahmoud, and Ahmed Talat. 2022. "Some Rational Approximations and Bounds for Bateman’s G-Function" Symmetry 14, no. 5: 929. https://doi.org/10.3390/sym14050929

APA Style

Ahfaf, O., Mahmoud, M., & Talat, A. (2022). Some Rational Approximations and Bounds for Bateman’s G-Function. Symmetry, 14(5), 929. https://doi.org/10.3390/sym14050929

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Menu

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

Abstract

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI