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.
The Gaussian hypergeometric function is given by
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.
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.
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:
2. Main results
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. .
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 ,
The author would like to thank the anonymous referees for their thoughtful comments and suggestions, which improved the paper.