On a conjecture for trigonometric sums and starlike functions, II

We prove the case ρ=14 of the following conjecture of Koumandos and Ruscheweyh: let snμ(z)≔∑k=0n(μ)kk!zk, and for ρ∈(0,1]ρ∈(0,1] let μ∗(ρ)μ∗(ρ) be the unique solution of ∫0(ρ+1)πsin(t−ρπ)tμ−1dt=0 in (0,1](0,1]. Then we have |arg[(1−z)ρsnμ(z)]|≤ρπ/2 for 0<μ≤μ∗(ρ)0<μ≤μ∗(ρ), n∈Nn∈N and zz in the unit disk of CC and μ∗(ρ)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 μ0μ0 such that the derivative of x−Γ(x+μ)Γ(x+1)x2−μ is completely monotonic on (0,∞)(0,∞) for μ0≤μ<1μ0≤μ<1.