On some classes of holomorphic functions whose derivatives have positive real part

Abstract

In this paper we discuss about normalized holomorphic functions whose derivatives have positive real part. For this class of functions, denoted $R$, we present a general distortion result (some upper bounds for the modulus of the $k$-th derivative of a function). We present also some remarks on the functions whose derivatives have positive real part of order $\alpha$. More details about these classes of functions can be found in \cite{macgregor}, \cite{thomas}, \cite[Chapter 4]{mocanu} and \cite{krishna}. In the last part of this paper we present two new subclasses of normalized holomorphic functions whose derivatives have positive real part which generalize the classes $R$ and $R(\alpha)$. For these classes we present some general results and examples.