2019
Том 71
№ 9

All Issues

Skaskiv O. B.

Articles: 11
Article (Ukrainian)

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

Bandura A. І., Skaskiv O. B.

↓ Abstract

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.

Article (Ukrainian)

Wiman-type inequality for functions analytic in a polydisc

Kurylyak A. O., Shapovalovs’ka L. O., Skaskiv O. B.

↓ Abstract   |   Full text (.pdf)

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

Skaskiv O. B., Trakalo O. M.

↓ Abstract   |   Full text (.pdf)

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

Skaskiv O. B.

↓ Abstract   |   Full text (.pdf)

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

Skaskiv O. B., Trusevich О. M.

↓ Abstract   |   Full text (.pdf)

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

Salo T. M., Skaskiv O. B.

↓ Abstract   |   Full text (.pdf)

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

Skaskiv O. B.

↓ Abstract   |   Full text (.pdf)

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

Lutsyshyn M. R., Skaskiv O. B.

↓ Abstract   |   Full text (.pdf)

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

Skaskiv O. B., Sorokivskii V. M.

Full text (.pdf)

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

Skaskiv O. B.

Full text (.pdf)

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