Integral transforms of functions with the derivative in a halfplane
Let \({\mathcal A}\) denote the class of functions \(f\) holomorphic in the unit disc \(\Delta\) with normalization \(f(0)= 0\), \(f'(0)= 1\). Denote by \({\mathcal K}\), \({\mathcal S}_\gamma^*\) and \({\mathcal S}\) the subclasses of \({\mathcal A}\) containing functions that are convex, starlike of order \(\gamma\) and univalent, respectively. For a function \(f\in{\mathcal A}\) we define the known integral transform \(F_c(z)= (c+1) \int_0^1 t^{c-1} f(tz) dt\), \(c>-1\). The authors define the classes \[ {\mathcal P}_\beta= \{f\in {\mathcal A}\mid \exists \alpha\in \mathbb{R}\mid \operatorname {Re} \{e^{i\alpha} (f'(z)- \beta)\}> 0,\;z\in \Delta\}. \] The object of this paper is to establish a sharp relation between \(\beta\) and \((c,\gamma)\) such that \(f\in {\mathcal P}_\beta\) implies \(F_c\in {\mathcal S}_\gamma^*\), \(0\leq \gamma\leq \frac 12\). The cases \(c=0\) (Alexander transform) and \(c=1\) (Libera transform) are considered in detail.
scientific article; zbMATH DE number 1415563