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Sufficient conditions for starlikeness. (English) Zbl 0998.30010

Let A be the class of functions f of the form f(z)=z+a 2 z 2 + which are holomorphic in the unit disc U={z:|z|<1}. A function fA is said to be starlike of order α if it satisfies Re{zf ' (z)/f(z)}>α for zU. We denote by S * (α), 0α<1, the class of such functions and S * =S * (0).

In this paper the authors obtain some sufficient conditions for starlikeness and for starlikeness of order α for the functions fA. For example: Theorem 2: If fA satisfies

Rezf ' (z) f(z)αzf '' (z) f ' (z)+1>-α 2 4(1-α),zU

for some α0,2), then fS * (α 2).

30C45Special classes of univalent and multivalent functions