On the maximum value of a confluent hypergeometric function
type: Item
instance of: scholarly article
title: On the maximum value of a confluent hypergeometric function
zbMATH Open document ID: 1505.33004
author: Bujar Fejzullahu
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2022-02-03T00:00:00Z
review text: The author studies the maximum value of the confluent hypergeometric function under oscillatory parameter conditions. As a result, the author discovers a new inequality for the Gauss hypergeometric function. This work's quick literature review can be seen. From a mathematical perspective, the confluent hypergeometric function has been thoroughly studied (see, for instance, [\textit{Y. L. Luke}, The special functions and their approximations. Vol. I, II. New York-London: Academic Press (1969; Zbl 0193.01701); \textit{W. Magnus} et al., Formulas and theorems for the special functions of mathematical physics. 3rd enlarged ed. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0143.08502); \textit{F. W. Olver} et al., NIST Digital Library of Mathematical Functions, \url{https://dlmf.nist.gov/13}]). Particularly, when $x>0$ and $b>a>0$, the estimate of the confluent hypergeometric function $_1F_1(a;b;x)$ has received extensive study (see [\textit{B. C. Carlson}, Proc. Am. Math. Soc. 17, 32--39 (1966; Zbl 0137.26803); \textit{Y. L. Luke}, J. Approx. Theory 5, 41--65 (1972; Zbl 0225.33004)], and references therein). For instance, Luke [loc. cit.] and others demonstrated the following inequalities \[ \exp(ax/b)<_1F_1(a;b;x)<1-a/b+\exp(x)*a/b, x>0, b>a>0. \] \[ _1F_1(a;b;x)<(b-1)\exp(x)/(b-a-1)(1+x) , x>0, b-1>a>0. \] We point out that \textit{E. R. Love} [J. Inequal. Appl. 1, No. 3, 293--299 (1997; Zbl 0908.33002), Corollary 2] demonstrated that (i) $\dots$ $|_1F_1(a;b;x)|\leq\Gamma(b-a)*\exp(x)|/|\Gamma(1-a)|\Gamma(b)$ exists when $x>0$, $b>0$, and $b>1/2$. Unfortunately, it does not appear that the conditions provided for the above equation (i) are accurate. For example, equation (i) yields $_1F_1(a; b; x)=0$ for any $x>0$ for $a\in N$ and $b>a$. The estimate of $_1F_1(a; b; x)$ is much more complicated when the parameters confirm the so-called oscillatory conditions $a<0$ and $b-a>1$, and, to the author's knowledge, it has not been studied in the literature, with the…
reviewed by: Showkat Ahmad Dar
Mathematics Subject Classification ID: 33C15
Mathematics Subject Classification ID: 33C20
zbMATH DE Number: 7468755
zbMATH Keywords: confluent hypergeometric functions
zbMATH Keywords: generalized hypergeometric series
MaRDI profile type: Publication
full work available at URL: crmath.256
OpenAlex ID: W4205247495
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DOI: 10.5802/CRMATH.256
describes a project that uses: Digital Library of Mathematical Functions
description: scientific article; zbMATH DE number 7468755