On a conjecture for trigonometric sums and starlike functions, II

We prove the case ρ = 4 of the following conjecture of Koumandos and Ruscheweyh: let s μ n (z) := ∑n k=0 (μ)k k! z k , and for ρ ∈ (0, 1] let μ(ρ) be the unique solution of ∫ (ρ+1)π 0 sin(t − ρπ)tμ−1dt = 0 in (0, 1]. Then we have | arg[(1− z)ρs n (z)]| ≤ ρπ/2 for 0 < μ ≤ μ(ρ), n ∈ N and z in the unit disk of C and μ(ρ) is the largest number with this property. For the proof of this other new results are required that are of independent interest. For instance, we find the best possible lower bound μ0 such that the derivative of x − Γ (x+μ) Γ (x+1) x 2−μ is completely monotonic on (0,∞) for μ0 ≤ μ < 1. c © 2009 Elsevier Inc. All rights reserved.