Cesáro partial sums of certain analytic functions (Q395806)

The authors study geometric properties such as starlikeness and convexity of the Cesáro partial sums of certain analytic functions in the open unit disk. For \[ f_3(z)=z+\frac{k-1}{k}z^2+\frac{k-2}{k}z^3, \] where \(k\geq 2\), they show that \(f_3(z)\) belongs to the set of univalent starlike functions of order \(\frac{1}{2}\) denoted by \(S^*(\frac{1}{2})\), and the Cesáro sum of \(\frac{z}{(1-z)}\), \(\sigma_3(z)=z+\frac{2}{3}z^2+\frac{1}{3}z^3\), is starlike of order \(\alpha=\frac{1}{5}\). Results are proved using convexity of similar functions.