2019
Том 71
№ 7

# Sheremeta M. M.

Articles: 17
Article (Ukrainian)

### Estimations of the Laplace – Stieltjes integrals

Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1467-1482

We study the Laplace – Stieltjes integrals with an arbitrary abscissa of convergence. The lower and upper estimates for these integrals are established. The accumulated results are used to deduce the relationships between the growth of the integral and the maximum of the integrand.

Article (Ukrainian)

### On the regular growth of Dirichlet series absolutely convergent in a half-plane

Ukr. Mat. Zh. - 2011. - 63, № 5. - pp. 686-698

For the Dirichlet series $F(s) = \sum^{\infty}_{n=1}a_n \exp \{s \lambda_n\}$ with the abscissa of absolute convergence $\sigma a = 0$, conditions on $(λ_n)$ and $(a_n)$ (λn) are established under which $\ln M(\sigma, F) = T_R(1 + o(1)) \exp\{\varrho R/|\sigma|\}$ as $\sigma \uparrow 0$, where$M(σ, F) = \sup\{|F(\sigma + it)| : t \in R\}$ and $T_R$ and $\varrho_R$ are positive constants.

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.

Article (Ukrainian)

### Properties of entire solutions of differential equations

Ukr. Mat. Zh. - 2006. - 58, № 12. - pp. 1693–1703

We investigate the close-to-convexity and l-index boundedness of entire solutions of the differential equations $z^2w'' + \beta zw' + (\gamma z^2 — \beta)w = 0$ і$zw'' + \beta w' + \gamma zw = 0$.

Article (Ukrainian)

### On the Mean Values of the Dirichlet Series

Ukr. Mat. Zh. - 2004. - 56, № 11. - pp. 1501-1502

For Dirichlet series with arbitrary abscissa of absolute convergence, we investigate the relationhip between the increase in the maximum term and $\left( {\mathop \sum \nolimits_{n = 1}^\infty \left| {a_n } \right|^q \exp \{ q\sigma \lambda _n \} } \right)^{1/q}$ , q ∈ (0,+∞).

Article (Ukrainian)

### Boundedness of the l-Index of the Naftalevich–Tsuji Product

Ukr. Mat. Zh. - 2004. - 56, № 2. - pp. 247-256

We investigate conditions for zeros under which the Naftalevich–Tsuji product is a function of a bounded l-index analytic in the unit disk.

Article (Ukrainian)

### On the Regular Variation of Main Characteristics of an Entire Function

Ukr. Mat. Zh. - 2003. - 55, № 6. - pp. 840-849

We establish a necessary and sufficient condition for the coefficients a n of an entire function $f(z) = \sum {_{n = 0}^\infty } {\text{ }}a_n z^n$ under which its central index and the logarithms of the maximum of the modulus and the maximum term are regularly varying functions. We construct an entire function the logarithm of the maximum of whose modulus is a regularly varying function, whereas the Nevanlinna characteristic function is not a regularly varying function.

Article (Ukrainian)

### On the Asymptotic Behavior of the Remainder of a Dirichlet Series Absolutely Convergent in a Half-Plane

Ukr. Mat. Zh. - 2003. - 55, № 3. - pp. 379-388

For a Dirichlet series $\sum\nolimits_{n = 1}^\infty {a_n \exp \{ s{\lambda}_n \} }$ with nonnegative exponents and zero abscissa of absolute convergence, we study the asymptotic behavior of the remainder $\sum\nolimits_{k = n}^\infty {\left| {a_k } \right|\exp \{ {\delta \lambda}_k \} }$ , δ < 0, as n → ∞.

Article (Ukrainian)

### Boundedness of the l-Index of Laguerre–Pólya Entire Functions

Ukr. Mat. Zh. - 2003. - 55, № 1. - pp. 91-99

We investigate conditions on zeros of an entire function f of the Laguerre–Pólya class under which f is a function of bounded l-index.

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 the Binomial Asymptotics of an Entire Dirichlet Series

Ukr. Mat. Zh. - 2001. - 53, № 4. - pp. 542-549

Let M(σ) be the maximum modulus and let μ(σ) be the maximum term of an entire Dirichlet series with nonnegative exponents λ n increasing to ∞. We establish a condition for λ n under which the relations $$\ln {\mu }\left( {{\sigma ,}F} \right) \leqslant \Phi _1 \left( {\sigma } \right) + \left( {1 + o\left( 1 \right)} \right){\tau }\Phi _{2} \left( {\sigma } \right)\quad \left( {{\sigma } \to + \infty } \right)$$ and $$\ln M\left( {{\sigma ,}F} \right) \leqslant \Phi _1 \left( {\sigma } \right) + \left( {1 + \left( 1 \right)} \right){\tau }\Phi _{2} \left( {\sigma } \right)\quad \left( {{\sigma } \to + \infty } \right)$$ are equivalent under certain conditions on the functions Φ1 and Φ2.

Brief Communications (Ukrainian)

### On the growth of an entire dirichlet series

Ukr. Mat. Zh. - 1999. - 51, № 8. - pp. 1149–1153

We establish the relation between the increase of the quantityM(σ,F) = ∣a 0∣ + ∑ n=1 a n ∣ exp (σλ n ) and the behavior of sequences (|a n |) and (λ n ), where (λ n ) is a sequence of nonnegative numbers increasing to + ∞, andF(s) =a 0 + ∑ n=1 a n e sλn ,s=σ+it, is the Dirichlet entire series.

Article (Ukrainian)

### A generalization of the Lindelöf theorem

Ukr. Mat. Zh. - 1998. - 50, № 9. - pp. 1177–1192

We present a generalization of the Lindelöf theorem on conditions that should be imposed on the coefficients of the Taylor series of an entire transcendental function ƒ in order that the relation $ln M_f (r) - \tau r^\rho , r \to \infty , M_f (r) = \max \left\{ {\left| {f(r)} \right|:|z| = r} \right\}$ , be satisfied.

Article (Russian)

### On the existence of entire functions of bounded l-index and l-regular growth

Ukr. Mat. Zh. - 1996. - 48, № 9. - pp. 1166–1182

We prove that, under certain conditions on a positive functionl continuous on [0, +∞], there exists an entire transcendental functionf of boundedl-index such that lnlnM f(r)lnL(r),r→∞, whereM f (r)=max {|f(z)|: |z|=r} andL(r)=∫ 0 r l(t)dt. Ifl(r)=r p-1 forr≥1, 0<ρ<∞, then there exists an entire functionf of boundedl-index such thatM f (r)≈r p .

Article (Ukrainian)

### Generalization of the fricke theorem on entire functions of finite index

Ukr. Mat. Zh. - 1996. - 48, № 3. - pp. 412-417

We prove that, for every sequence (a k) of complex numbers satisfying the conditions Σ(1/|a k |) < ∞ and |a k+1| − |a k | ↗ ∞ (k → ∞), there exists a continuous functionl decreasing to 0 on [0, + ∞] and such that f(z) = Π(1 −z/|a k |) is an entire function of finite l-index.

Article (Ukrainian)

### On the eigenvalues of the fredholm operator

Ukr. Mat. Zh. - 1996. - 48, № 1. - pp. 116-123

We prove that if ω(t, x, K 2 (m) )⩽c(x)ω(t) for allxε[a, b] andx ε [0,b-a] wherecL 1(a, b) and ω is a modulus of continuity, then λ n =O(n m-1/2ω(1/n)) and this estimate is unimprovable.

Article (Ukrainian)

### On the Radii of univalence of Gel'fond-Leont'ev derivatives

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 390–399

Let $0 < R < +\infty,$ let $A(R)$ bethe class of functions $$f(z) = \sum_{k=0}^{\infty}f_kz^k,$$ analytic in $\{ z: |z| < R \}$, and let $$l(z) = \sum_{k=0}^{\infty}l_kz^k,\; l_k > 0$$ be a formal power series. We prove that if $l^2_k/l_{k+1}l_{k-1}$ is a nonincreasing sequence, $f \in A(R)$, and $|f_k/f_{k+1} \nearrow R,\; k \rightarrow \infty,\; 0 < R < +\infty$, then the sequence $(\rho_n)$ of radii of univalence of the Gel'fondLeont'ev derivatives satisfies the relation $$D^n_lf(z) = \sum_{k=0}^{\infty}\frac{l_kf_{k+n}}{l_{k+n}}z_k$$ The case where the condition $|f_k/f_{k+1}|\nearrow R,\quad k \rightarrow \infty$, is not satisfied is also considered.