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Convexity of the incomplete beta functions. (English) Zbl 1200.30025
Summary: The author finds the condition on b and c such that the incomplete beta function φ(b,c;z) defined by φ(b,c;z):=z 2 F 1 (1,b;c;z) and the normalized confluent hypergeometric function z 1 F 1 (1;c;z)= k=1 (b,k-1)/(c,k-1)z k are convex in Δ. Conditions on a,b, and c for the convexity of z 2 F 1 (a,b;c;z) and z 1 F 1 (a;c;z) for a1 remains an open problem.
30C45Special classes of univalent and multivalent functions
30C80Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable)
33B20Incomplete beta and gamma functions
33C45Orthogonal polynomials and functions of hypergeometric type