On the maximum value of a confluent hypergeometric function
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 exception for the case where $a$ is a negative integer and $b>0$ (see, for instance, [\textit{I. Krasikov} and \textit{A. Zarkh}, J. Approx. Theory 162, No. 11, 2021--2047 (2010; Zbl 1210.33017); \textit{E. Levin} and \textit{D. Lubinsky}, J. Approx. Theory 139, No. 1--2, 107--143 (2006; Zbl 1127.42023); \textit{H. N. Mhaskar} and \textit{E. B. Saff}, Constr. Approx. 1, 71--91 (1985; Zbl 0582.41009); \textit{G. Szegö}, Orthogonal polynomials. 4th ed. Providence, RI: American Mathematical Society (AMS) (1975; Zbl 0305.42011); \textit{G. N. Watson}, J. Lond. Math. Soc. 14, 19--22 (1939; Zbl 0020.21805)], and references therein). It is widely known that $_1F_1(a; b; x)$ has a finite number of real zeros for $a<0$ and $b>0$ (see [\textit{F. W. Olver} et al., NIST Digital Library of Mathematical Functions, Section 13.9]). Using the Kummer transformation, \[ _1F_1(a; b; x)=\exp(x)* _1F_1(b-a; b; -x) \] The real zeros of the function $_1F_1(a;b;x)$ are therefore positive when $a<0$ and $b>0$. The maximum value of $_1F_1(a,b,x)$ with oscillatory conditions of parameters is examined in this paper using the Watson-Glaeske product formula and the Sonin-Pólya theorems for confluent hypergeometric functions. This paper has produced excellent results. The author focused their research on a single hypergeometric function. This method can be used to enhance any hypergeometric function.
scientific article; zbMATH DE number 7468755
scientific article