2019
Том 71
№ 11

# Skaskiv O. B.

Articles: 11
Article (Ukrainian)

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

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

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.

Article (Ukrainian)

### Wiman-type inequality for functions analytic in a polydisc

Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 78-86

We prove an analog of Wiman-type inequality for analytic functions in a polydisc $\mathbb{D}^p = \{z \in \mathbb{C}^p : |z_j| < 1,\; j \in \{ 1, . . . ,p\} \} , p \in N.$ The obtained inequality is sharp.

Brief Communications (Ukrainian)

### On the Stability of the Maximum Term of the Entire Dirichlet Series

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 571–576

We establish necessary and sufficient conditions for logarithms of the maximal terms of the entire Dirichlet series $F(z) = \sum^{+\infty}_{n=0}a_n e^{z\lambda_n}$ and $A(z) = \sum^{+\infty}_{n=0}a_n b_n e^{z\lambda_n}$ to be asymptotically equivalent as ${\rm Re}\;z \rightarrow +\infty$ outside some set of finite measure.

Article (Ukrainian)

### Rate of Convergence of Positive Series

Ukr. Mat. Zh. - 2004. - 56, № 12. - pp. 1665-1674

We investigate the rate of convergence of series of the form $$F(x) = \mathop \sum \limits_{n = 0}^{ + \infty } \;a_n e^{x\lambda _n + \tau (x)\beta _n } ,\quad a_n \geqslant 0,\quad n \geqslant 1,\quad a_0 = 1$$ where λ = (λn), 0 = λ0 < λn ↑ + ∞, n → + ∞, β = {βn: n ≥ 0} ⊂ ℝ+, and τ(x) is a nonnegative function nondecreasing on [0; +∞), and $$F(x) = \mathop \sum \limits_{n = 0}^{ + \infty } \;a_n f(x\lambda _n ),\quad a_n \geqslant 0,\quad n \geqslant 1,\quad a_0 = 1,$$ where the sequence λ = (λn) is the same as above and f (x) is a function decreasing on [0; +∞) and such that f (0) = 1 and the function ln f(x) is convex on [0; +∞).

Brief Communications (Ukrainian)

### Relations of Borel Type for Generalizations of Exponential Series

Ukr. Mat. Zh. - 2001. - 53, № 11. - pp. 1580-1584

We prove that the condition $\sum\nolimits_{n = 1}^{ + \infty } {\left( {n{\lambda }_n } \right)^{ - 1} < + \infty }$ is necessary and sufficient for the validity of the relation ln F(σ) ∼ ln μ(σ, F), σ → +∞, outside a certain set for every function from the class $H_ + \left( {\lambda } \right)\mathop = \limits^{{df}} \cup _f H\left( {{\lambda,}f} \right)$ . Here, H(λ, f) is the class of series that converge for all σ ≥ 0 and have a form $$F\left( {\sigma} \right) = \sum\limits_{n = 0}^{ + \infty } {a_n f\left( {{\sigma \lambda}_n } \right),\quad a_n \geqslant 0,\;n \geqslant 0,}$$ and f(σ) is a positive differentiable function increasing on [0, +∞) and such that f(0) = 1 and ln f(σ) is convex on [0, +∞).

Article (Ukrainian)

### Entire Dirichlet Series of Rapid Growth and New Estimates for the Measure of Exceptional Sets in Theorems of the Wiman–Valiron Type

Ukr. Mat. Zh. - 2001. - 53, № 6. - pp. 830-839

For entire Dirichlet series of the form $F\left( z \right) = \sum\nolimits_{n = 0}^{ + \infty } {a_n e^{z{\lambda }_n } ,0 \leqslant {\lambda }_n \uparrow + \infty ,\;n \to + \infty }$ , we establish conditions under which the relation $$F\left( {{\sigma } + iy} \right) = \left( {1 + o\left( 1 \right)} \right)a_{{\nu }\left( {\sigma } \right)} e^{\left( {{\sigma + }iy} \right){\lambda }_{{\nu }\left( {\sigma } \right)} }$$ holds uniformly in $y \in \mathbb{R}\;{as}\;{\sigma } \to + \infty$ outside a certain set E for which $$DE = \mathop {\lim \sup }\limits_{{\sigma } \to + \infty } h\left( {\sigma } \right)\;{meas}\;\left( {E \cap \left[ {{\sigma ,} + \infty } \right)} \right) = 0$$ where h(σ) is a positive continuous function increasing to +∞ on [0, +∞).

Article (Russian)

### On the growth of analytic functions represented by the Dirichlet series on semistrips

Ukr. Mat. Zh. - 1993. - 45, № 5. - pp. 681–693

The behavior of the Dirichlet series with null abscissa of absolute convergence is studied on semistrips.

Brief Communications (Ukrainian)

### Minimum of modulus of dirichlet multisequence

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1295–1297

Conditions are established under which the following relation is satisfied: $$M(x) = (1 + o(1))m(x) = (1 + o(1))\mu (x)$$ as $|x |→ + ∞$ outside a sufficiently small set, for an entire function $F(z)$ of several complex variables $z ∈ ℂ_p,p ≥ 2$, represented by a Dirichlet series. Here $M(x) = \sup \{|F(x+iy) |: y ∈ ℝ^p\}$ and $m(x) = \inf \{ |F(x+iy) |:y ∈ ℝ^p,$ with $μ(x)$ the maximal term of the Dirichlet series, $x ∈ ℝ^p$.

Article (Ukrainian)

### Growth of horizontal rays of analytic functions represented by Dirichlet series

Ukr. Mat. Zh. - 1990. - 42, № 3. - pp. 363-371

Article (Ukrainian)

### A theorem of Borel type for a dirichlet series having abscissa of absolute convergence zero

Ukr. Mat. Zh. - 1989. - 41, № 11. - pp. 1532–1541