On a conjecture for trigonometric sums and starlike functions

Abstract

We pose and discuss the following conjecture: let snμ (z) {colon equals} ∑k = 0n frac((μ)k, k !) zk, and for ρ ∈ (0, 1] let μ* (ρ) be the unique solution μ ∈ (0, 1] of ∫0(ρ + 1) π sin fenced(t - ρ π) tμ - 1 dt = 0 .Then for 0 < μ ≤ μ* (ρ) and n ∈ N we have | arg [(1 - z)ρ snμ (z)] | ≤ ρ π / 2, | z | < 1. We prove this for ρ = frac(1, 2), and in a somewhat weaker form, for ρ = frac(3, 4). Far reaching extensions of our conjectures and results to starlike functions of order 1 - μ / 2 are also discussed. Our work is closely related to recent investigations concerning the understanding and generalization of the celebrated Vietoris' inequalities. © 2007 Elsevier Inc. All rights reserved.