An application of hypergeometric functions to a problem in function theory

• Main Author: Daniel S. Moak
• Format: Article
• Language: English
• Published: Hindawi Limited 1984-01-01
• Series: International Journal of Mathematics and Mathematical Sciences
• Subjects: hypergeometric functions; Jacobi polynomials; maximum property; and positive maximum property.
• Online Access: http://dx.doi.org/10.1155/S0161171284000545
• ISSN: 0161-1712; 1687-0425

In some recent work in univalent function theory, Aharonov, Friedland, and Brannan studied the series (1+xt)α(1−t)β=∑n=0∞An(α,β)(x)tn. Brannan posed the problem of determining S={(α,β):|An(α,β)(eiθ)|<|An(α,β)(1)|,   0<θ<2π,   α>0,   β>0,   n=1,2,3,…}. Brannan showed that if β≥α≥0, and α+β≥2, then (α,β)∈S. He also proved that (α,1)∈S for α≥1. Brannan showed that for 0<α<1 and β=1, there exists a θ such that |A2k(α,1)e(iθ)|>|A2k(α,1)(1)| for k any integer. In this paper, we show that (α,β)∈S for α≥1 and β≥1.