Series of univalent functions

We are considering a class S of functions F(z), F(0) = 0, F′(0) = 1 that are univalent and regular in the circle ¦z¦ < 1, and its subclasses s _h ^* and K of starlike functions of order h and of convex functions respectively. Among others, we establish the following results: If F(z)εs and 0 < α < 1, then IfF (z) ε s (0 < a < 1) and $$begin{gathered} 1 + operatorname{Re} {{z_1 F^n left( {z_1 } right)} mathord{left/ {vphantom {{z_1 F^n left( {z_1 } right)} {F'left( {z_1 } right)}}} right. kern-nulldelimiterspace} {F'left( {z_1 } right)}} = operatorname{Re} {{alpha z_1 F''left( {alpha z_1 } right)} mathord{left/ {vphantom {{alpha z_1 F''left( {alpha z_1 } right)} {F'left( {alpha z_1 } right)}}} right. kern-nulldelimiterspace} {F'left( {alpha z_1 } right)}} hfill left( {2 - sqrt 3< left| {z_1 } right| = r< 1} right) hfill end{gathered} $$ then we obtain the domain of values of the point αz₁.