Central functions for classes of concave univalent functions (Q2811004)

For a class \(\mathcal F\) of functions holomorphic in a neighborhood of the origin and with a Taylor expansion of the form \(f_c(z) = z + a_{2}z^{2}+a_{3}z^{3}+\cdots \), the authors introduce the following definition of a central function \(f_c\). A function \(f_c(z) = z + b_{2}z^{2}+b_{3}z^{3}+\cdots \) is called a central function for the class \(\mathcal F\) with respect to a sequence of constants \(\{k_n\}\), \(n\geq 2\), if and only if \(f_c\in \mathcal F\) and the regions of variability for the functionals \(a_n\), \(n\geq 2\), on the class \(\mathcal F\) are given by the inequalities \(| a_n - b_n| \leq k_n\). NEWLINENEWLINENEWLINENEWLINE In the paper, a central function \(f_c\) for the class \({\operatorname {Co}}(p)\) with respect some nontrivial sequence \(\{k_n\}\), \(n\geq 2\), is described, where \({\operatorname {Co}}(p)\), \(0<p<1\), is the class of univalent meromorphic functions \(f\) in \(\{| z|<1\}\) which have a Taylor series expansion of the form \(f(z)=z+a_2z^2+a_3z^3+\cdots \) at the origin, are holomorphic in \(\{| z|<p\}\), have a simple pole at \(p\), and are such that \(\mathbb {C} \setminus f(| z| <1)\) is a bounded convex set.NEWLINENEWLINESome other considerations on central functions \(f_c\) in the classes of starlike or convex functions are mentioned in the paper. Also the numbers \(\delta \) are established for which the \(T_{\delta}\)-neighborhood of \(f_c\) contains the above mentioned classes. The introduced notion of a central function \(f_c\) is inspired by the paper of \textit{U. Bednarz} and \textit{J. Sokół} [J. Math. Appl. 32, 25--32, (2010; Zbl 1382.30016)].