Some Remarks on Close-to-Convex and Strongly Convex Functions

Abstract

We consider questions of the following kind: When does boundedness of $|{\arg \{1+zp'(z)/p(z)\}}|$, for a given analytic function $p$, imply boundedness of $|{\arg\{p(z)\}}|$? The paper determines the order of strong close-to-convexity in the class of strongly convex functions. Also, we consider conditions that are sufficient for a function to be a Bazilevič function.