2019
Том 71
№ 9

All Issues

Boundedness of $L$-index for the composition of entire functions of several variables

Bandura A. І., Skaskiv O. B.


Abstract

We consider the following compositions of entire functions $F(z) = f \bigl( \Phi (z)\bigr) $ and $H(z,w) = G(\Phi 1(z),\Phi 2(w))$, where f$f : C \rightarrow C, \Phi : C^n \rightarrow C,\; \Phi_1 : C^n \rightarrow C, \Phi_2 : C^m \rightarrow C$, and establish conditions guaranteeing the equivalence of boundedness of the $l$-index of the function $f$ to the boundedness of the $L$-index of the function $F$ in joint variables, where $l$ : $C \rightarrow R_{+}$ is a continuous function and $$L(z) = \Bigl( l\bigl( \Phi (z)\bigr) \bigm| \frac{\partial \Phi (z)}{\partial z_1}\bigm| ,..., l \bigl( \Phi (z) \bigr) \bigm|\frac{\partial \Phi (z)}{\partial z_n} \bigm| \Bigr).$$ Under certain additional restrictions imposed on the function $H$, we construct a function $\widetilde{L} $ such that $H$ has a bounded $\widetilde{ L}$ -index in joint variables provided that the function $G$ has a bounded $L$-index in joint variables. This solves a problem posed by Sheremeta.

Citation Example: Bandura A. І., Skaskiv O. B. Boundedness of $L$-index for the composition of entire functions of several variables // Ukr. Mat. Zh. - 2018. - 70, № 10. - pp. 1334-1344.