An application of hypergeometric functions to a problem in function theory

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 determiningS={(α,β):|An(α,β)(eiθ)|<|An(α,β)(1)|, 0<θ<2π, α>0, β>0, n=1,2,3,…}. Brannan showed that ifβ≥α≥0, andα+β≥2, then(α,β)∈S. He also proved that(α,1)∈Sforα≥1. Brannan showed that for0<α<1andβ=1, there exists aθsuch that|A2k(α,1)e(iθ)|>|A2k(α,1)(1)|forkany integer. In this paper, we show that(α,β)∈Sforα≥1andβ≥1.