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α-logarithmically convex functions. (English) Zbl 0924.30012

Let A denote the class of normalised analytic functions f defined by f(z)=z+a 2 z 2 + for zD={z:|z|<1}.

For α0 the authors introduce the class M α of normalised analytic α-logarithmically convex functions defined in the open unit disc D by

Re1+zf '' (z) f ' (z) α zf ' (z) f(z) 1-α >0·

For fM α , a best possible subordination theorem is obtained which implies that M α forms a subset of the starlike functions S * . Some extreme coefficient problems are also solved.

This definition and some properties are inspired by the well known α-convex functions, introduced in 1969 by P. T. Mocanu. The principal proofs of this paper are based on the so called “admissible functions method” defined and developed by P. T. Mocanu and S. S. Miller, method which used differential subordinations.

MSC:
30C45Special classes of univalent and multivalent functions