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.
is referred to as the Pochhammer symbol, or rising factorial, and simplifies to $(a)_n=a(a+1)\cdots (a+n-1)$. In the case that the denominator parameter satisfies $c=a+b$, the resulting ${}_2F_1(a,b;a+b;r)$ 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
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:
$$\begin{equation} \frac{1}{1+r}<\frac{\mathcal{K}(r)}{\mathcal{K}(\sqrt {r})} \quad \text{for all $r\in (0,1)$.} \tag{1.2}\cssId{eq1}{} \end{equation}$$
In light of this result, it is natural to ask the following questions:
• What is the best value $\lambda$ such that
$$\begin{equation*} \frac{1}{1+\lambda r}<\frac{\mathcal{K}(r)}{\mathcal{K}(\sqrt {r})}\quad \text{for all $r\in (0,1)$?} \end{equation*}$$
• 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.
with the best possible factors $\lambda _a=a(1-a)$ and $\mu _a=0$.
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 $a=1/2$, implies that
with the best possible constant factors $\lambda =1/4$ and $\mu =0$. 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 $\lambda = 1/4$. 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 $a, b >0$ with $a+b>ab$. For all $r\in (0,1)$ we have
is not necessarily of constant sign when $\lambda >1$.
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 $r\mapsto (1-r)^\lambda \,{}_2F_1(a,b;c;r)$ whose series coefficients can be expressed in terms of ${}_3F_2$. Placing our attention on the generating function rather than on its series coefficients, we arrive at the following basic result.
Lemma 2.3.
Suppose $a, b>0$ and $1>\lambda \geq ab/(a+b).$ Define
Then $-f^\prime$ is absolutely monotonic on $(0,1)$.
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 $(1-r)^{1/4}\,{}_2F_1(a,b;a+b;r)$ for $4ab/\leq a+b$ was verified in Reference 2, Theorem 1.7. Also, the monotonicity and concavity properties of ${r^\prime }^c{\mathcal{K}}(r)$ for $c\geq 1/2$ were presented in Reference 6, Theorem 3.21 and generalized in 2000 to ${r^\prime }^c{\mathcal{K}}_a(r)$ for $a\in (0,1/2]$ and $c\geq 2a(1-a)$ in Reference 3, Lemma 5.4 (1), where $r^\prime = \sqrt {1-r^2}$.
with the best possible exponents $\lambda _a=a(1-a)$ and $\mu _a=0$.
3. A further simplification
Before concluding, we will make a simplifying observation that allows us to relax the condition that $a+b>ab$ 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 $F(r):= {}_2F_1(a,b;c;r)$, it follows that
for $r\in (0,1)$. As a natural extension of Reference 2, Theorem 1.7 and its proof, one can easily show that $f$ is decreasing. In particular, an application of (Equation 3.1) followed by (Equation 3.2) reveals that
Since $\frac{(a)_n(b)_n}{(a+b+1)_n n!}< \frac{(a)_n(b)_n}{(a+b)_n n!}$, it follows that $f$ is strictly decreasing on $(0,1)$, and the conclusion follows as in the proof of Theorem 2.1.
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Acknowledgments
The author would like to thank the anonymous referees for their thoughtful comments and suggestions, which improved the paper.