bibkey: karpsitnik2010logconvexity
authors: D. Karp and S. M. Sitnik
year: 2010
title: Log-convexity and log-concavity of hypergeometric-like functions
doi: 10.1016/j.jmaa.2009.10.057
url: https://arxiv.org/abs/0902.3073
claim: "The unnumbered Conjecture after Theorem 4 asks for monotonicity and exact ranges of the shifted upper-parameter Kummer ratio in inequalities (10) and (11)."
strata_touched:
  - D5/S3/Analytic/KummerRatio/Result
license: citation-only
triage: anchor

Shifted upper-parameter Kummer ratios

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DOI: 10.1016/j.jmaa.2009.10.057

Source: https://arxiv.org/abs/0902.3073

Pinned version: https://arxiv.org/abs/0902.3073v1, the unnumbered Conjecture after Theorem 4, inequalities (10) and (11). Journal publication: Journal of Mathematical Analysis and Applications 364(2) (2010), 384–394.

Mathematical correspondence

The source function is

$$M(s;c;x)={}_1{F}_1(s;c;x)=\sum _{n\ge 0}\frac{(s{)}_n}{(c{)}_{n}n!}{x}^n,$$

with ascending Pochhammer symbols. The ratio is

$$R(x)=\frac{M(b+\delta ;c;x)M(a;c;x)}{M(a+\delta ;c;x)M(b;c;x)}.$$

For $c,\delta >0$ and $0<a<b$, the formal image on the positive ray is $(A,1)$, where $A=\Gamma (a+\delta )\Gamma (b)/(\Gamma (b+\delta )\Gamma (a))$. The printed phrase “onto (1, A)” lists the limiting values in the order $x\to 0^{+}$ and $x\to +\infty$; the increasing order of the interval endpoints is $(A,1)$. For $a<b<c-\delta$, the negative-ray image is $(B,1)$, where $B=\Gamma(c-a-\delta)\Gamma(c-b)/ (\Gamma(c-b-\delta)\Gamma(c-a))$.

Defs.kummerCoeff, Defs.kummer, Defs.ratio and Defs.claim in D5/S3/Analytic/KummerRatio/Defs encode these source objects. D5/S3/Analytic/KummerRatio/Result.result proves both strict monotonicity statements and both exact images with all the displayed parameter ranges.

The bounded literature scope and its limitations are stated in Problems/karp-sitnik-2010-kummer-ratio-monotonicity.md. Exhaustive literature completeness and priority are ASSUMED-UNVERIFIED.