Properties of certain transforms defined by convolution of analytic functions

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