International Journal of Mathematics and Mathematical Sciences
Volume 7 (1984), Issue 3, Pages 503-506

An application of hypergeometric functions to a problem in function theory

Daniel S. Moak

Department of Mathematics, Texas Tech University, Lubbock 79409, Texas, USA

Received 20 November 1980; Revised 9 March 1981

Copyright © 1984 Daniel S. Moak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


In some recent work in univalent function theory, Aharonov, Friedland, and Brannan studied the series (1+xt)α(1t)β=n=0An(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 βα0, and α+β2, then (α,β)S. He also proved that (α,1)S for α1. 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 α1 and β1.