PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 148, Number 2, February 2020, Pages 777–786 https://doi.org/10.1090/proc/14751 Article electronically published on August 28, 2019 ON SHARP BOUNDS FOR RATIOS OF k-BALANCED HYPERGEOMETRIC FUNCTIONS ROGER W. BARNARD, KENDALL C. RICHARDS, AND ELYSSA N. SLIHEET (Communicated by Mourad Ismail) Abstract. We extend recently obtained sharp bounds for ratios of zero- balanced hypergeometric functions to the general k-balanced case, k ∈ N. We also discuss the absolute monotonicity of generalizations of previously studied functions involving generalized complete elliptic integrals. 1. Introduction The hypergeometric function studied by Gauss is given by 2 F 1 (a, b; c; r) := ∞  n=0 (a) n (b) n (c) n n! r n , where (a) n := Γ(a + n) Γ(a) is referred to as the Pochhammer symbol. In the case that the denominator pa- rameter satisfies c = a + b + k (with k ∈ N 0 := N ∪{0}), the resulting 2 F 1 (a, b; a + b + k; r) is said to be k-balanced. In the important case that k = 1, the series is also called Saalsch¨ utzian or simply balanced (see [7, p. 140]). The complete elliptic integral of the first kind is defined by (1.1) K(r)=  π/2 0 1  1 − r 2 sin 2 (t) dt, and the complete elliptic integral of the second kind is defined by (1.2) E (r)=  π/2 0  1 − r 2 sin 2 (t)dt, which satisfy the well-known Legendre relation given by K(r ′ )E (r)+ K(r)E (r ′ ) −K(r)K(r ′ )= π 2 , Received by the editors June 9, 2019, and, in revised form, June 10, 2019, and June 20, 2019. 2010 Mathematics Subject Classification. Primary 33C05, 33C75; Secondary 26D15. Key words and phrases. Gaussian hypergeometric function, complete elliptic integrals, gener- alized complete elliptic integrals. c 2019 American Mathematical Society 777 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use