On a conjecture for trigonometric sums by S. Koumandos and S. Ruscheweyh

Abstract

S. Koumandos and S. Ruscheweyh posed the following conjecture: For $ρ\in(0,1]$ and $0<μ\leqμ^{\ast}(ρ)$, the partial sum $s_n^μ(z)=\displaystyle\sum_{k=0}^n \frac{(μ)_k}{k!}z^k$, $0<μ\leq1$, $|z|<1$, satisfies % \begin{align*} (1-z)^ρs_n^μ(z) \prec \left(\frac{1+z}{1-z}\right)^ρ, \qquad n\in \mathbb{N}, \end{align*} where $μ^{\ast}(ρ)$ is the unique solution of \begin{align*} \int_0^{(ρ+1)π} \sin(t-ρπ)t^{μ-1}dt=0. \end{align*} This conjecture is already settled for $ρ=\frac{1}{2}$, $\frac{1}{4}$, $\frac{3}{4}$ and $ρ=1$. In this work, we validate this conjecture for an open neighbourhood of $ρ=\frac{1}{3}$ and in a weaker form for $ρ=\frac{2}{3}$. The particular value of the conjecture leads to several consequences related to starlike functions.