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

By Kendall C. Richards

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 can be expressed in terms of in the following form:

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

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 such that

• 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.

Theorem 1.1 (Reference 1, Theorem 4.1).

Let . For all we have

with the best possible factors and .

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 , implies that

with the best possible constant factors and . While it follows that (Equation 1.3) refines (Equation 1.2), it is important to note that Anderson, Vamanamurthy, and Vuorinen Reference 5 proved the following inequality in 1992:

As noted in Reference 1, (Equation 1.4) improves the lower bound in (Equation 1.3) with . Moreover, it is this result by Anderson et al. that suggests a path toward answering the question by Ismail.

2. Main results

Theorem 2.1.

Suppose with . For all we have

with the best possible exponents and .

Remarks.

The zero-balanced extension of Theorem 1.1 follows from Theorem 2.1 by noting that for ,

and It is also worth noting that the value of

is not necessarily of constant sign when .

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:

Lemma 2.2 (Reference 8, Lemma 1).

Suppose and . Then

where is the generalized hypergeometric function given by

The proof of Lemma 2.2 uses a generating function argument applied to the function given by whose series coefficients can be expressed in terms of . Placing our attention on the generating function rather than on its series coefficients, we arrive at the following basic result.

Lemma 2.3.

Suppose and Define

Then is absolutely monotonic on .

Remarks.

As noted above, the basic result in Lemma 2.3 is a direct corollary to Reference 8, Lemma 1, which appeared in 2001. The monotonicity of the special case for was verified in Reference 2, Theorem 1.7. Also, the monotonicity and concavity properties of for were presented in Reference 6, Theorem 3.21 and generalized in 2000 to for and in Reference 3, Lemma 5.4 (1), where .

Sketch of proof of Lemma 2.3.

Suppose and An argument similar to that used in the proof of Lemma 2.2 as given in Reference 8 yields that

Since and

it follows that is absolutely monotonic since it is the product of two absolutely monotonic functions.

Proof of Theorem 2.1.

Suppose and We will prove the result that

In this direction, let be defined as in Lemma 2.3 and, without loss of generality, suppose . For , it follows that

which directly implies that

The sharpness of follows from the fact that

The upper bound follows from the fact that the function is increasing. The sharpness of is obtained by using

which follows from the asymptotic relation (see Reference 9, eq. (2), p. 74)

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

Corollary 2.4.

Let . For all we have

with the best possible exponents and .

3. A further simplification

Before concluding, we will make a simplifying observation that allows us to relax the condition that in Theorem 2.1. The proof will incorporate the following classical results (see Reference 12, 15.5.1 and Reference 12, 15.8.1, respectively). With , it follows that

Theorem 3.1.

Suppose . For all we have

with the best possible exponents and .

Proof:.

Let and define

for . As a natural extension of Reference 2, Theorem 1.7 and its proof, one can easily show that is decreasing. In particular, an application of (Equation 3.1) followed by (Equation 3.2) reveals that

Since , it follows that is strictly decreasing on , and the conclusion follows as in the proof of Theorem 2.1.

Acknowledgments

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

Mathematical Fragments

Equation (1.2)
Theorem 1.1 (Reference 1, Theorem 4.1).

Let . For all we have

with the best possible factors and .

Equation (1.3)
Equation (1.4)
Theorem 2.1.

Suppose with . For all we have

with the best possible exponents and .

Lemma 2.2 (Reference 8, Lemma 1).

Suppose and . Then

where is the generalized hypergeometric function given by