Norm estimates for the Bernardi integral transform of functions defined by subordination
Let \(\mathcal{A}\) denote the class of functions \(f\) analytic in the unit disc \(\Delta=\left\{z\in\mathbb{C}:|z|<1\right\}\) with the normalization \(f(0)=0=f'(0)-1\). Let \(\mathcal{LU}\) be the subclass of \(\mathcal{A}\) consisting of all locally univalent functions. The norm of a function \(f\in\mathcal{LU}\) is given by \[ \left\|f\right\|=\sup_{z\in\Delta}(1-|z|^{2})\left|\frac{zf''(z)}{f'(z)}\right|. \] Consider \(S^{*}(A,B)\) and \(K(A,B)\) the subclasses of \(\mathcal{A}\) defined by \[ S^{*}(A,B)=\left\{f\in\mathcal{A}:\frac{zf'(z)}{f(z)}\prec\frac{1+Az}{1+Bz}\right\} \] and \[ K(A,B)=\left\{f\in\mathcal{A}:1+\frac{zf''(z)}{f'(z)}\prec\frac{1+Az}{1+Bz}\right\} \] \[ (-1\leq B<A\leq1). \] For \(\gamma>-1\), the Bernardi integral transform \(J_{\gamma}[f]\) of \(f\in\mathcal{A}\) is defined by \[ J_{\gamma}[f](z)=\frac{\gamma+1}{z^{\gamma}}\int_{0}^{z}t^{\gamma-1}f(t)dt. \] Let \(f\in K(A,B)\). In the paper, the authors obtain sharp estimates for \(\left\|J_{\gamma}[f]\right\|\). A number of open questions concerning \(S^{*}(A,B)\) and other subclasses of \(\mathcal{A}\) are considered.
univalent, starlike and convex functions
scientific article; zbMATH DE number 5289872