On a conjecture for trigonometric sums and starlike functions

We pose and discuss the following conjecture: 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 μ∈(0,1]μ∈(0,1] of ∫0(ρ+1)πsint-ρπtμ-1dt=0.Then for 0<μ⩽μ*(ρ)0<μ⩽μ*(ρ) and n∈Nn∈N we have |arg[(1-z)ρsnμ(z)]|⩽ρπ/2,|z|<1. We prove this for ρ=12, and in a somewhat weaker form, for ρ=34. Far reaching extensions of our conjectures and results to starlike functions of order 1-μ/21-μ/2 are also discussed. Our work is closely related to recent investigations concerning the understanding and generalization of the celebrated Vietoris’ inequalities.