Positive Gegenbauer polynomial sums and applications to starlike functions

Abstract

Let sn(f,z):=∑k=0n akz k be the nth partial sum of f(z)= ∑k=0 ∞ akzk. We show that Re s n(f/z,z)>0 holds for all z ∈ D, n ∈ N, and all starlike functions f of order λ iff λ0 ≤ λ 0 for all n ∈ N, x ∈ [-1,1], and 0 < μ ≤ μ0 := 1 - λ0 = 0.345778.... This result complements an inequality of Brown, Wang, and Wilson (1993) and extends a result of Ruscheweyh and Salinas (2000). © 2004 Springer Science+Business Media, Inc.