Properties of certain transforms defined by convolution of analytic functions

摘要

Let A be the class of all analytic functions f in the open unit disk U of the form f(z)=z+∑k=2∞akzk. For λ>0,Rec>0 and α<1, two subclasses P(λ) and Sα∗ of A are introduced. In this paper, we find suitable conditions on λ, c and α such that for each f∈P(λ) satisfying (z/f(z))*F(1,c;c+1;z)≠0 for all z∈U, the functionG(z)=z(z/f(z))∗F(1,c;c+1;z)(z∈U)belongs to P(λ′),Sα∗ or S∗(α). Here S∗(α) denotes the usual class of starlike of order α(0⩽α<1) in U. We also determine necessary conditions so that f∈P(λ) implies thatzG′(z)G(z)-12β<12β,|z|