Geometric properties of Cesaro averaging operators

In this paper, using positivity of trigonometric cosine and sine sums whose coefficients are generalization of Vietoris numbers, we find the conditions on the coefficient $\{a_k\}$ to characterize the geometric properties of the corresponding analytic function $f(z)=z+\displaystyle\sum_{k=2}^{\infty} a_kz^k$ in the unit disc $\mathbb{D}$. As an application we also find geometric properties of a generalized Ces\`aro type polynomials.