Mapping properties of nonlinear integral operators and pre-Schwarzian derivatives
Let \(A\) be the class of functions analytic in the unit disk \({\mathbf D}= \{{\mathbf z}:|{\mathbf z}|< 1\}\) and normalized by \(f(0)= 0\), \(f'(0)= 1\). In 1969 Hornich investigated the operators \[ f\oplus g(z)= \int^z_0 f'(w) g'(w)\,dw;\quad \alpha* f(z)= \int^z_0 (f'(w))^\alpha\, dw \] acting for \(f,g\in LU\) (locally univalent functions from \(U\)) and complex parameter \(\alpha\). For \(f\in LU\) we define the pre-Schwarzian derivative \[ T_f(z)= f''(z)/f'(z) \] and the norm \[ \| f\|= \sup_{z\in D}\,|(1- |z|^2)\,T_f(z)|. \] These operators and norm give the structure of vector space to the family \(LU\). In this paper some properties of the following integral operators \[ I_\alpha[f]= \alpha* f\quad\text{and}\quad J_\alpha[f]= I_\alpha[J[f]], \] where \[ J[f](z)= \int^z_0 {f(w)\over w}\,dw, \] are investigated. In particular, the following results are presented: (i) For every univalent function \(f\in A\) and every complex number \(\alpha\) the following inequality \[ \| J_\alpha[f]\|\leq 4|\alpha| \] holds. (ii) The class \(S^*\) of starlike functions is not convex concerning Hornich operation. (iii) Suppose that \(f\in A\) and the function \(zf''(z)/f'(z)\) takes no values in the set \(E_0= \{z:\text{Re\,}z\leq -1,|\text{Im\,}z|\geq -\sqrt{3} \text{Re\,}z\}\). Then \(f_\alpha= I_\alpha[f]\in S^*\) for all \(\alpha\in [0,1]\).
scientific article; zbMATH DE number 2126899
scientific article