Asymptotic Relations between Maximums of Absolute Values and Maximums of Real Parts of Entire Functions

Filevich, P. V.

doi:10.1023/b:matn.0000023320.27440.57

Cited by 5 publications

(6 citation statements)

References 2 publications

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“…In the past several decades, the problem on the growth and value distribution of analytic functions has been an important and interesting subject in the fields of complex analysis. Moreover, considerable attention has been paid to the growth and the value distribution of analytic functions defined by Dirichlet series and Laplace-Stieltjes transforms, and a great deal of interesting results focusing on the growth and value distribution of such functions can be found in (see [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]). For example, Yu [18] in 1963 first proved a series of theorems about the Valiron-Knopp-Bohr formula of the associated abscissas of bounded convergence, absolute convergence and uniform convergence of Laplace-Stieltjes transforms, the maximal molecule M u ðσ, GÞ, the maximal term μðσ, GÞ, the Borel line and the order of entire functions represented by Laplace-Stieltjes transforms convergent in the complex plane.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The Approximation of Laplace-Stieltjes Transforms Concerning Sun’s Type Function

Shen

2021

Journal of Function Spaces

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The main aim of this paper is to establish some theorems concerning the error E n F , β , the Sun’s type function U r , and M u σ , F of entire functions defined by Laplace-Stieltjes transforms with infinite order converge in the whole complex plane. Our results exhibit the growth of Laplace-Stieltjes transforms from the point of view of approximation.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

The Approximation of Laplace-Stieltjes Transforms Concerning Sun’s Type Function

Shen

2021

Journal of Function Spaces

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“…Let Φ ∈ Ω, and let ψ ∈ L be an arbitrary function satisfying (7). We prove that there exist a sequence ζ ∈ Z such that N ζ (r) ≥ Φ(log r) for all sufficiently large r and sequences (s k ) and (r k ) satisfying (3) such that N ζ (r k ) = Φ(log r k ) for all integers k ≥ 1 and for any function f ∈ E ζ we have (12). We may suppose without loss of generality that there exists a number σ 0 > 1 such that Φ(σ) = 0 for all σ ≤ σ 0 .…”

Section: Proof Of Theoremsmentioning

confidence: 99%

Comparative growth of an entire function and the integrated counting function of its zeros

Andrusyak,

Filevych

2024

Carpathian Math. Publ.

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Let $(\zeta_n)$ be a sequence of complex numbers such that $\zeta_n\to\infty$ as $n\to\infty$, $N(r)$ be the integrated counting function of this sequence, and let $\alpha$ be a positive continuous and increasing to $+\infty$ function on $\mathbb{R}$ for which $\alpha(r)=o(\log (N(r)/\log r))$ as $r\to+\infty$. It is proved that for any set $E\subset(1,+\infty)$ satisfying $\int_{E}r^{\alpha(r)}dr=+\infty$, there exists an entire function $f$ whose zeros are precisely the $\zeta_n$, with multiplicities taken into account, such that the relation $$ \liminf_{r\in E,\ r\to+\infty}\frac{\log\log M(r)}{\log r\log (N(r)/\log r)}=0 $$ holds, where $M(r)$ is the maximum modulus of the function $f$. It is also shown that this relation is best possible in a certain sense.

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“…At the end of the introductory part, we note that some other problems concerning comparisons of the growth of an entire function f to the distribution of its zeros were considered, in particular, in [3,4,5,6,7,8,9,10]. We also note that questions regarding the sizes of exceptional sets in various asymptotic relations between characteristics of entire functions were investigated, for example, in [12,13,14,15,16,17,18,19].…”

Section: Theorem C ([2]mentioning

confidence: 99%

Minimal growth of entire functions with prescribed zeros outside exceptional sets

2022

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Let $h$ be a positive continuous increasing to $+\infty$ function on $\mathbb{R}$. It is proved that for an arbitrary complex sequence $(\zeta_n)$ such that $0<|\zeta_1|\le|\zeta_2|\le\dots$ and $\zeta_n\to\infty$ as $n\to\infty$, there exists an entire function $f$ whose zeros are the $\zeta_n$, with multiplicities taken into account, for which$$\ln m_2(r,f)=o(N(r)),\quad r\notin E,\ r\to+\infty.$$with a set $E$ satisfying $\int_{E\cap(1,+\infty)}h(r)dr<+\infty$, if and only if $\ln h(r)=O(\ln r)$ as $r\to+\infty$.Here $N(r)$ is the integrated counting function of the sequence $(\zeta_n)$ and$$m_2(r,f)=\left(\frac{1}{2\pi}\int_0^{2\pi}|\ln|f(re^{i\theta})||^2d\theta\right)^{1/2}.$$

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Asymptotic Relations between Maximums of Absolute Values and Maximums of Real Parts of Entire Functions

Cited by 5 publications

References 2 publications

The Approximation of Laplace-Stieltjes Transforms Concerning Sun’s Type Function

The Approximation of Laplace-Stieltjes Transforms Concerning Sun’s Type Function

Comparative growth of an entire function and the integrated counting function of its zeros

Minimal growth of entire functions with prescribed zeros outside exceptional sets

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