Subordination by convex functions (Q5926160)

Let \(K(\alpha)\), \(0\leq\alpha< 1\) be the class of functions \(g(z)= z+\sum^\infty_{n= 2}a_n z^n\), which are convex of order \(\alpha\) in the unit disk \(U\) and suppose that for the function \(f\) holomorphic in \(U\), \(f(z)+ zf'(z)\) is subordinate to \(g(z)+ zg'(z)\) in \(U(\prec)\). Then:NEWLINENEWLINENEWLINE1. \(g\in K(0)\Rightarrow f\prec g\) in \(|z|< 0.745\dots\),NEWLINENEWLINENEWLINE2. \(g\in K({1\over 2})\Rightarrow f\prec g\) in \(|z|< 0,8612\dots\),NEWLINENEWLINENEWLINE3. \(g(z)= {z\over 1-z}\rightarrow f\prec g\) in \(U\),NEWLINENEWLINENEWLINE4. \(g(z)=-\log(1- z)\Rightarrow f\prec g\) in \(|z|< 0,98\dots\),NEWLINENEWLINENEWLINE5. \(g(z)= z+\lambda z^2\), \(|\lambda|\leq{1\over 5}\Rightarrow f\prec p\) in \(U\),NEWLINENEWLINENEWLINE6. \(g(z)= e^z- 1\Rightarrow f\prec g\) in \(|z|< 0,8138\dots\)\ .