On a conjecture for trigonometric sums and starlike functions, II Original Research Article

We prove the case View the MathML sourceρ=14 of the following conjecture of Koumandos and Ruscheweyh: let View the MathML sourcesnμ(z)≔∑k=0n(μ)kk!zk, and for ρ∈(0,1]ρ∈(0,1] let μ∗(ρ)μ∗(ρ) be the unique solution of View the MathML source∫0(ρ+1)πsin(t−ρπ)tμ−1dt=0 Turn MathJax on in (0,1](0,1]. Then we have View the MathML source|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 View the MathML sourcex−Γ(x+μ)Γ(x+1)x2−μ is completely monotonic on (0,∞)(0,∞) for μ0≤μ<1μ0≤μ<1.