Univalent functions with missing Taylor coefficients
Let \(S^*(\alpha)\), \(K(\alpha)\) and \(S_\gamma\) be the usual classes of starlike functions of order \(\alpha\), convex functions of order \(\alpha\) and strongly starlike functions of order \(\gamma\) respectively. Let \(S^*_\alpha\), \(K_\alpha\) and \(R_\alpha\) be the classes consisting of normalized analytic functions defined by the conditions \(| zf'(z)/f(z)-1| <1-\alpha\), \(| zf''(z)/f'(z)| <1-\alpha\) and \(| \arg f'(z)| <\pi \gamma/2\) respectively. For \(n\geq 2\), \({\mathcal U}(\lambda)\) be the class of all analytic functions of the form \(f(z)=z+a_{n+1}z^{n+1}+\cdots\) satisfying \(| f'(z)(z/f(z))^2 -1| <\lambda\). In the present paper, the authors have found conditions on \(\lambda\) so that each of the functions in \({\mathcal U}(\lambda)\) is in \(S^*(\alpha)\), \(S_\gamma\) or \(R_\gamma\). Also they have found conditions on \(\lambda\) such that for every function \(f\in {\mathcal U}(\lambda)\), the integral operator \(cz^{1-c}\int_0^z\zeta^c/f(\zeta) d\zeta\) (\(c>0\)) is in \(S^*_\alpha\), \(K(\alpha)\) or \(K_\alpha\).
scientific article; zbMATH DE number 2158040