On the class of univalent functions starlike with respect to \(N\)-symmetric points
Let \(f(z)=z+a_2z^2+a_3z^3+\cdots\) be an univalent function in the unit disk \(U\) and let the class of univalent functions starlike with respect to \(N\)-symmetric points, \(SSP_N\) be described with \(\text{Re}\{zf'(z)/f_N(z)\}>0\), \(z\in U\), where \(f_N(z)=z+\sum^\infty_{m=1}a_{mN+1} z^{mN+1}\). In this paper authors give a structural formulae for functions in \(SSP_N\) and use it to obtain a sharp upper bound of \(|f'(z)|\) when \(N\) is even. This completes the distortion theorem for \(SSP_N\) since the rest is given by \textit{P. Singh} and \textit{R. Chand} [Indian. J. Pure Appl. Math. 10, 1167--1174 (1979; Zbl 0416.30010)]. The authors also give a characterization of the class \(SSP_N\) in terms of Hadamard convolution and prove that for any \(N\geq 2\) the class \(SSP_N\) contains non-starlike functions motivated by the fact that \(SSP_1\equiv S^*\).
scientific article; zbMATH DE number 1751372
scientific article