Starlike functions in an elliptical domain (Q2711262)

An analytic function \(f: E\to\mathbb{C}\), \(f(0)= 0\) is called starlike in a domain \(E\) if \(f\) is univalent in \(E\) and \(f(E)\) is a starlike domain w.r.t. the origin.NEWLINENEWLINENEWLINELet \(E= h(U)\), \(h(z)= {a+ b\over 2} z+ {a- b\over 2}\overline z\), \(U= \{z:|z|< 1\}\).NEWLINENEWLINENEWLINETheorems. Suppose that \(f\) is analytic in \(E\), \(f(0)= 0\), \(f(z)\neq 0\) in \(E\setminus\{0\}\), \(f'(z)\neq 0\) in \(E\). IfNEWLINENEWLINENEWLINE(i) \((a^2+ b^2)\text{ Re}{zf'(z)\over f(z)}- (a^2- b^2)\text{ Re}{\overline zf'(z)\over f(z)}> 0\) \((z\in E)\),NEWLINENEWLINENEWLINEor ifNEWLINENEWLINENEWLINE(ii) \(|\text{arg}{zf'(z)\over f(z)}|< \arccos {a^2- b^2\over a^2+ b^2}\) \((z\in E)\),NEWLINENEWLINENEWLINEthen \(f\) is starlike in \(E\).