Criterion for the family of rounded \(\varepsilon\) starlike mappings (Q1856881)

The authors introduce the family of \(\varepsilon\) starlike mappings and study the criterion for this family. The results obtained for this family cover the known results about the family of convex mappings and the family of starlike mappings, and describe how one family can transit to another one. They give the criterion for the family of \(\varepsilon\) starlike mappings on the unit ball in the complex Banach space and also in a bounded convex circular domain in \(\mathbb{C}^n\). One of the results obtained is: Let \(X,Y\) be two complex Banach spaces, and let \(B\subset X\) be the open unit ball. If \(f:B\to Y\) is a locally holomorphic mapping on \(B\) with \(f(0)=0\), then \(f\) is \(\varepsilon\) starlike if and only if \[ \text{Re} \Bigl\{T_x\biggl[ \bigl(Df(x) \bigr)^{-1}\bigl( f(x)-\varepsilon f(y)\bigr) \biggr]\Bigr\}\geq 0 \] holds for any \(x\in B\) and \(y\in B\) whenever \(\|y\|\leq \|x\|<1\), where \(T_x\in T(x)\), \(Df(x)\) is the Fréchet derivative of \(f\) at \(x\).