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

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