Abstract
This paper introduces a new Carathéodory function \(\varphi_{0}(z)\) that maps the unit disk onto the domain bounded by a catenary of equal resistance. We establish its fundamental properties and examine the class \(\mathcal{P}_{n}(a)\) of functions \(\varphi(z)=1+c_{n}z^{n}+c_{n+1}z^{n+1}+\cdots\) ( \(n\geq 1\) ) subordinate to \(\varphi_{0}(z)\) . For this class, we derive sharp estimates of \(\left|z\frac{\varphi^{\prime}(z)}{\varphi(z)}\right|\) , \(|\varphi(z)|\) , and \(\textrm{Re}\varphi(z)\) , which generalize known results in special cases. We also introduce a new Ma-Minda type class of starlike functions and solve distortion, growth, covering, and convexity radius problems for this class. Particular cases yield known properties of starlike functions with gap series. Furthermore, we apply the \(\mathcal{P}_{n}(a)\) estimates to study a general class of doubly close-to-starlike functions associated with \(\varphi_{0}(z)\) , obtaining sharp growth theorems and radii of starlikeness. These results extend known theorems for close-to-starlike functions and provide precise starlikeness radii for several classical function classes.
