On certain subclasses of bounded univalent functions

TL;DRAbstract

Let D = {z C; |z| < 1}, T = {z C; |z| = 1}. Denote by S the class of functions f of the form f (z) = z + a 2 z 2 + . . . holomorphic and univalent in D, and by S(M ), M > 1, the subclass of functions f of the family S such that |f (z)| < M in D. We introduce (and investigate the basic properties of) the class S(M, m; ), 0 < m M < , 0 1, of bounded functions f of the family S for which there exists an open arc I = I(f ) T of length 2 such that lim zz 0 ,zD |f (z)| M for every z 0 I and lim zz 0 ,zD |f (z)| m for every z 0 T \ I.

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