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Zbl 1200.30025
Swaminathan, A.
Convexity of the incomplete beta functions.
(English)
[J] Integral Transforms Spec. Funct. 18, No. 7, 521-528 (2007). ISSN 1065-2469; ISSN 1476-8291/e

Summary: The author finds the condition on $b$ and $c$ such that the incomplete beta function $\varphi(b,c;z)$ defined by $\varphi(b,c;z):= z_2F_1(1,b;c;z)$ and the normalized confluent hypergeometric function $z_1F_1(1;c;z)= \sum_{k=1}^\infty (b,k-1)/(c,k-1)z^k$ are convex in $\Delta$. Conditions on $a, b$, and $c$ for the convexity of $z_2F_1(a,b;c;z)$ and $z_1F_1(a;c;z)$ for $a\ne 1$ remains an open problem.
MSC 2000:
*30C45 Special classes of univalent and multivalent functions
30C80 Maximum principle, etc. (one complex variable)
33B20 Incomplete beta and gamma functions
33C45 Orthogonal polynomials and functions of hypergeometric type

Keywords: Gaussian hypergeometric functions; incomplete beta functions

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