Quasi-convex mappings of order \(\alpha\) on the unit polydisk in \(\mathbb{C}^n\)
Let \(X\) be a complex Banach space with the norm \(\left\|.\right\|\), let \(X^{*}\) denotes the dual space of \(X\), \(E\) be the open unit ball in \(X\). Also, let \(U^{n}\) be the open unit polydisc in \(\mathbb{C}^{n}\). For each \(x\in X\setminus\left\{0\right\}\) consider \(T(x)=\left\{T_{*}\in X^{*}:\left\|T_{x}\right\|=1, T_{x}(x)=\left\|x\right\|\right\}\). Let \(D^{n}f\) denotes the nth Fr\(\acute{e}\)chet derivative of a holomorphic mapping \(f:E\rightarrow X\). Suppose \(\alpha\in[0,1)\). A normalized locally biholomorphic mapping \(f:E\rightarrow X\) is said to be quasi-convex of type \(B\) and order \(\alpha\) if \[ Re\left\{T_{x}\left[(Df(x))^{-1}(D^{2}f(x)(x^{2})+Df(x)x)\right]\right\}\geq\alpha\left\|x\right\|,\quad x\in E. \] In the paper the authors obtain a sufficient condition for the quasi-convex mappings of type \(B\) and order \(\alpha\) on \(E\). Sharp estimations of all homogeneous expansions are established for quasi-convex mappings of type \(B\) and order \(\alpha\) defined on \(U^{n}\), which satisfy some conditions.
quasiconvex mapping of type B and order \(\alpha\)
scientific article; zbMATH DE number 5828485