# A note on inequalities for the ratio of zero-balanced hypergeometric functions

## Abstract

Motivated by a question suggested by M. E. H. Ismail in 2017, we present sharp inequalities for the ratio of zero-balanced Gaussian hypergeometric functions. The main theorems generalize known results for complete elliptic integrals of the first kind.

## 1. Introduction

The Gaussian hypergeometric function is given by

where

is referred to as the Pochhammer symbol, or rising factorial, and simplifies to In the case that the denominator parameter satisfies . the resulting , is said to be “zero-balanced”. To provide context, we include a brief history of related estimates involving important special cases of zero-balanced hypergeometric functions. In particular, the complete elliptic integral of the first kind is defined by

It is well known that

Similarly, the generalized complete elliptic integrals of the first kind are defined for

For more information on these functions, we refer the reader to Reference 6Reference 9 and, for recently obtained related results, to Reference 7Reference 10Reference 13Reference 14 and the references contained therein.

The initial thread for this investigation begins with the following elegant inequality obtained by G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen Reference 4 in 1990:

In light of this result, it is natural to ask the following questions:

• What is the best value

• Can this be extended to the ratio of generalized complete elliptic integrals?

Motivated by these questions, H. Alzer and the author of this paper obtained the following result.

M. E. H. Ismail Reference 1, p. 1669, Reference 11 asked whether Theorem 1.1 can be extended to the zero-balanced hypergeometric function. It is this question that serves as the catalyst and focal point of this paper, which answers the question in the affirmative.

Before presenting our main results, we provide some additional context and note one important refinement of (Equation 1.2) due to Anderson et al. First note that Theorem 1.1, with

with the best possible constant factors

As noted in Reference 1, (Equation 1.4) improves the lower bound in (Equation 1.3) with

## 2. Main results

In order to prove Theorem 2.1, we will make use of a result that is an immediate corollary to (the proof of) the following result which was proved in Reference 8:

The proof of Lemma 2.2 uses a generating function argument applied to the function given by

The following corollary generalizes the result in (Equation 1.4) by Anderson et al. in Reference 5.

## 3. A further simplification

Before concluding, we will make a simplifying observation that allows us to relax the condition that

## Acknowledgments

The author would like to thank the anonymous referees for their thoughtful comments and suggestions, which improved the paper.