Subordination by univalent functions

Abstract:

Let $K$ be the class of functions $f(z) = z + {a_2}{z^2} + \cdots$, which are regular and univalently convex in $\left | z \right | < 1$. In this paper we establish certain subordination relations between an arbitrary member $f$ of $K$, its partial sums and the functions $(\lambda /z)\int _0^z {f(t)dt}$ and $\mu \int _0^z {{t^{ - 1}}f(t)dt}$. The well-known result that $z/2$ is subordinate to $f(z)$ in $\left | z \right | < 1$ for every $f$ belonging to $K$ follows as a particular case from our results. We also improve certain results of Robinson regarding subordination by univalent functions. A sufficient condition for a univalent function to be convex of order $\alpha$ is also given.