On Ramanujan asymptotic expansions and inequalities for hypergeometric functions
The complete elliptic integral \(K(r)\) can be expressed in terms of the Gaussian hypergeometric function \(F(a,b;c;r)\) as \(K(r)={\pi \over 2} F({1 \over 2} , {1 \over 2} ;1;r^2) .\) In the 1990's, several results were proved for \(K(r)\) and, more generally, for the zero-balanced hypergeometric function (defined by the requirement \(c=a+b\)) when \(a,b 0.\) The following facts serve, in part, as motivation for such investigations: (1) \(K(r)\) is closely connected with conformal invariants, see [\textit{G. D. Anderson, M. K. Vamanamurthy, M. Vuorinen,} Conformal invariants, inequalities, and quasiconformal maps (1997; Zbl 0885.30012)]), (2) \(K(r)\) can be expressed in terms of the arithmetic-geometric mean, and hence it is natural to ask what connections exist between \(K(r)\) and mean values (see \textit{M. K. Vamanamurthy, M. Vuorinen} [J. Math. Anal. Appl. 183, No. 1, 155-166 (1994; 802.26009)] and \textit{J. M. Borwein} and \textit{P. B. Borwein} [J. Math. Anal. Appl. 177, No. 2, 572-582 (1993; 783.33001)]) (3) many of the properties of \(K(r)\) can be extended to the zero-balanced case. Some results inspired, in part, by these ideas can be found for instance in the following papers: [\textit{G. D. Anderson, R. W. Barnard, K. C. Richards, M. K. Vamanamurthy} and \textit{M. Vuorinen}, Trans. Am. Math. Soc. 347, No. 5, 1713-1723 (1995; Zbl 0826.33003)], [\textit{S.-L. Qiu}, \textit{M. Vuorinen} Nagoya Math. J. 154, 31-56 (1999; Zbl 0944.33004)], and [\textit{R. Balasubramanian, S. Ponnusmay, M. Vuorinen}, J. Math. Anal. Appl. 218, 256-268 (1998; Zbl 0783.33001)], \textit{H. Alzer} [Math. Proc. Cambridge Philos. Soc. 124, No. 2, 309-314 (1998; Zbl 0906.33012)] \textit{B. C. Carlson} and \textit{J. L. Gustafson} [SIAM J. Math. Anal. 25, No.~25, No.2, 288-303 (1994; Zbl 0794.41021)], \textit{R. Kühnau} [Z. Angew. Math. Mech. 74, 140-142 (1994; Zbl 0844.65009)], \textit{A. Laforgia} and \textit{S. Sismondi} [Rend. Circ. Mat. Palermo, II. Ser. 41, No. 2, 302-308 (1992; Zbl 0759.33009)], \textit{R. W. Barnard, K. Pearce}, and \textit{K. C. Richards}, Monotonicity in \({}_3 F_2\) and comparisions of classical approximations of elliptical arc length [SIAM J. Math. Anal. Appl. (to appear), ] The present paper belongs also to this company. It refines Ramanujan's approximation of \(F(a,b;a+b;r)\) in the neighborhood of the singularity \(r=1.\) The authors point out that \(F(a,b;c;r), 0 < r< 1,\) is real for \(c> 0\) and \(b=\overline{a}.\) This fact enables the authors to apply power series techniques in the proofs. The specialist reader will observe some new ideas and novel features in the proofs.
scientific article; zbMATH DE number 1334249