A new inclusion for Bavrin's families of holomorphic functions in \(n\)-circular domains (Q2836163)

The paper refers a relation \(R_{G}\backsim P_{G}\) between two Bavrin's families of holomorphic functions \(f\:G\rightarrow \mathbb {C}\), \(f(0)=1\), satisfying, on some planar intersections of \(n\)-circular domains \(G\subset \mathbb {C}^{n}\), conditions of close-to-convexity and close-to-starlikeness, respectively. The main theorem says that there holds the proper inclusion \(R_{G}\subsetneqq P_{G}\), similar as well known inclusions \(M_{G}\subsetneqq N_{G},V_{G}\subsetneqq C_{G}\) for Bavrin's families \(M_{G},N_{G},V_{G},C_{G}\), defined as in above by starlikenes, convexity and so on. Let us note that a function \(f\in P_{G}\setminus R_{G}\) has been given universally for all domains \(G\), using a number characteristic \(\Delta =\Delta (G)\) of domains \(G\). It is very important that the authors consider functions in a wide class of domains \(G\) \(\subset \mathbb {C}^{n}\), because in \(\mathbb {C}^{n}\) the Riemann mapping theorem is false. Let us note also that a well-known analytic characterization of the above families use the collection \(C_{G}\) of holomorphic functions \(p\:G\rightarrow \mathbb {C}\), \(p(0)=1\), \(\operatorname {Re} p(z)>0\), \(z\in G\), and the Temljakov's differential operator \(Lf\) defined as follows NEWLINE\[NEWLINE Lf(z)=Df(z)(z), \qquad z\in G, NEWLINE\]NEWLINE where \(Df(z)\) means the Frechet derivative of \(f\) at \(z\). The mentioned characterization has the form: a holomorphic function \(f\:G\rightarrow \mathbb {C}\), \(f(0)=1\), belongs to \(V_{G},M_{G},N_{G},R_{G},P_{G}\), if it satisfies the factorization NEWLINE\[NEWLINE \begin{aligned} Lf&=p\cdot 1, \quad p\in C_{G}, \\ Lf&=p\cdot f, \quad p\in C_{G}, \\ Lf&=p\cdot (LLf), \quad p\in C_{G}, \\ Lf&=p\cdot (Lf) \cdot (L\varphi), \quad \varphi \in N_{G}, \, p\in C_{G}, \\ f&=p\cdot \psi, \quad \psi \in M_{G}, \, p\in C_{G}, \end{aligned} NEWLINE\]NEWLINE respectively.