On a conjecture for trigonometric sums and starlike functions, II

Abstract

We prove the case ρ=1/4 of the following conjecture of Koumandos and Ruscheweyh: let snμ(z)=Σk=0n(μ)k/ k!zk, and for ρε(0,1] let μ≤(ρ) be the unique solution of 0(ρ+1)πsin(t-ρπ)tμ-1dt =0 in (0,1]. Then we have pipearg[(1-z)ρsnμ(z)]pipe≤ ρπ/2 for 0<μ≤μ*(ρ), nΣN{double-struck} and z in the unit disk of C{double-struck} 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-g{cyrillic}(x+μ)/g{cyrillic}(x+1)x2-μ is completely monotonic on (0,∞) for μ0≤μ<1. © 2009 Elsevier Inc.