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 $0<\mu\leq\mu^{\ast}(\rho)$, the partial sum $s_n^{\mu}(z)=\displaystyle\sum_{k=0}^n \frac{(\mu)_k}{k!}z^k$, $0<\mu\leq1$, $|z|<1$, 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.