Odd starlike functions

Abstract:

In the present paper, among other things, we prove that if $f(f(0) = 0,f’(0) = 1)$ is regular and odd starlike in $\left | z \right | < 1$, then ${\text {Re }}f(z)/{s_n}(z,f) > 1/2,\;\left | z \right | < 1$, where ${s_n}(z,f)$ denotes the $n$th partial sum of $f,\;n = 1,2,3, \ldots ,$, thus generalising the known result: $\operatorname {Re} {\text { }}f(z)/z > 1/2,\;\left | z \right | < 1$. As an application, we show that each partial sum of an odd convex function is close-to-convex in $\left | z \right | < 1$.