On a sufficient condition for function to be $p$-valent close-to-convex

Mamoru Nunokawa, Janusz Sokol, Lucyna Trojnar-Spelina


The property of close-to-convexity
of analytic function, that generalizes that of starlikeness, was
introduced by Kaplan in 1952. He also gave the geometric
interpretation of this property and proved that all close-to-convex
functions in the unit disc $\mathbb D$, are univalent. Another
geometric interpretation was given later by Lewandowski. In this
paper we established some sufficient conditions for functions
analytic in the unit disc $\mathbb D$ to be  $p$-valently
close-to-convex  in $\mathbb D$.


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