2019
Том 71
№ 7

# Mulyava O. M.

Articles: 4
Brief Communications (Ukrainian)

### On conditions for Dirichlet series absolutely convergent in a half-plane to belong to the class of convergence

Ukr. Mat. Zh. - 2008. - 60, № 6. - pp. 851–856

For a Dirichlet series $F(s) = \sum^{\infty}_{n=0}a_n \exp \{s\lambda_n\}$ with the abscissa of absolute convergence $\sigma_a = 0$, let $M(\sigma) = \sup\{|F(\sigma+it)|:\;t \in {\mathbb R}\}$ and $\mu(\sigma) = \max\{|a_n| \exp(\sigma \lambda_n):\;n \geq 0\},\quad \sigma < 0.$ It is proved that the condition $\ln \ln n = o(\ln \lambda_n),\;n\rightarrow\infty$, is necessary and sufficient for equivalence of relations $\int^0_{-1}|\sigma|^{\rho-1}\ln M(\sigma)d\sigma < +\infty$ and $\int^0_{-1}|\sigma|^{\rho-1}\ln \mu(\sigma)d\sigma < +\infty,\quad \rho > 0,$ for each such series.

Brief Communications (Ukrainian)

### Integral analog of one generalization of the Hardy inequality and its applications

Ukr. Mat. Zh. - 2006. - 58, № 9. - pp. 1271–1275

Under certain conditions on continuous functions $μ, λ, a$, and $f$, we prove the inequality $$\int\limits_0^y {\mu (x)\lambda (x)f\left( {\frac{{\int_0^x {\lambda (t)a(t)dt} }}{{\int_0^x {\lambda (t)dt} }}} \right)dx \leqslant K\int\limits_0^y {\mu (x)\lambda (x)f(a(x))} dx,} y \leqslant \infty ,$$ and describe its application to the investigation of the problem of finding conditions under which Laplace integrals belong to a class of convergence.

Article (Ukrainian)

### On Entire Functions Belonging to a Generalized Class of Convergence

Ukr. Mat. Zh. - 2002. - 54, № 4. - pp. 439-446

In terms of Taylor coefficients and distribution of zeros, we describe the class of entire functions f defined by the convergence of the integral $\int\limits_{r_0 }^\infty {\frac{{\gamma (\ln M_{f} (r))}}{{r^{\rho + 1} }}} dr$ , where γ is a slowly increasing function.

Article (Ukrainian)

### On convergence classes of Dirichlet series

Ukr. Mat. Zh. - 1999. - 51, № 11. - pp. 1485–1494

We establish conditions for the coefficients of a Dirichlet series under which this series belongs to a certain class of convergence.