# A simple geometric criterion for non-Bazilevicness

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## Abstract

In 1955 I.E. Bazilevie introduced a class of functions analytic and univalent in the unit disc, which have since come to be known as Bazilevie functions. Let S be the 9 n class of functions defined by f(z) = z + a^z which are regular and univalent in D = {z:|z| < 1}. It is known that the usual subclasses of S eonsiting of functions which are elose-to-eonvex, starlike (w.r.t. the origin), or convex in D, are also contained in the class of normalized Bazilevie functions.

Unfortunately, the original characterization of Bazilevie functions was not very geometric. Then in 1971, T. Sheil-Small gave a very nice geometric characterization, similar to that given by Kaplan for elose-to-eonvex functions. Still not too much has been done to capture a geometric feeling for them.

The main purpose of the dissertation is to give a very simple visual criterion, sufficient to show non-Bazilevicness. Using this technique we exhibit a class of non-Bazilevie polynomials, answering the problem posed by D. Campbell as to the smallest degree of a non-Bazilevie polynomial. We also use a similar line of reasoning to significantly lower the best known upper bound for radius of Bazilevicness to <0.888.