On the partial sums of convex functions of order $1/2$

Abstract:

Let $f\left ( z \right ) = z + {a_2}{z^2} + \ldots$ be regular and univalently convex of order $1/2$ in the unit disc $U$ and let ${s_n}\left ( {z,f} \right )$ denote its $n$th partial sum. In the present note we determine the radius of convexity of ${s_n}\left ( {z,f} \right )$, depending on $n$, and generalize and sharpen a result of Ruscheweyh concerning the partial sums of convex functions. We also prove that for every $n \geq 1,{\text {Re}}\left ( {{s_n}\left ( {z,f} \right )/z} \right ) > 1/2$ in $U$.