2019
Том 71
№ 11

All Issues

Bandura A. І.

Articles: 2
Article (Ukrainian)

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

Bandura A. І., Skaskiv O. B.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2018. - 70, № 10. - pp. 1334-1344

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.

Brief Communications (Ukrainian)

Directional logarithmic derivative and the distribution of zeros of an entire function of bounded $L$-index in the direction

Bandura A. І., Skaskiv O. B.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2017. - 69, № 3. - pp. 426-432

We establish new criteria of boundedness of the $L$-index in the direction for entire functions in $C^n$. These criteria are formulated as estimate of the maximum modulus via the minimum modulus on a circle and describe the distribution of their zeros and the behavior of the directional logarithmic derivative. In this way, we prove Hypotheses 1 and 2 from the article [Bandura A. I., Skaskiv O. B. Open problems for entire functions of bounded index in direction // Mat. Stud. – 2015. – 43, № 1. – P. 103 – 109]. The obtained results are also new for the entire functions of bounded index in $C$. They improve the known results by M. N. Sheremeta, A. D. Kuzyk, and G. H. Fricke.