Libera transform of functions with bounded turning (Q1856966)

Let \({\mathcal A}_n\), \(n=1,2,\dots\) be the class of functions \(f\), which are holomorphic in the unit disc \(U=\{z:|z|<1\}\), of the form \(f(z)=z+ a_{n+1}z^{n+1} +a_{n+2}z^{n+2} +\cdots+\), \(z\in U\); \({\mathcal A}={\mathcal A}_1\). Let \(S\subset {\mathcal A}\) denote the class of univalent functions, \(S^* \subset S\)-starlike functions, \(S^*(\beta)\)-strongly starlike functions of order \(\beta\in (0,1)\), \({\mathcal R}= \{f\in A:\text{Re} f'(z)>0\}\). In the article the authors discuss a relation between these classes which involve the Libera operator \(L\) defined by formula \(L(f)(z)= {2\over z}\int^z_0 f(t)dt\). In particular they proved that \(L({\mathcal R}\cap {\mathcal A}_2)\subset S^*(2/3)\).