On  Dirichlet series like to compositions of Hadamard

Mulyava, O. M.; Sheremeta, М. М.

doi:10.15330/ms.51.1.25-34

Cited by 3 publications

(3 citation statements)

References 5 publications

(10 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Theorem C ( [1]). Let α ∈ L and β ∈ L be positive continuously differentiable functions such that d ln α −1 (ϱβ(σ)) dσ = O(1) as σ → ∞ for each ϱ ∈ (0, +∞).…”

Section: Theorem B ([6]mentioning

confidence: 99%

On entire Dirichlet series similar to Hadamard compositions

Mulyava

Sheremeta

2023

Mat. Stud.

Self Cite

View full text Add to dashboard Cite

A function $F(s)=\sum_{n=1}^{\infty}a_n\exp\{s\lambda_n\}$ with $0\le\lambda_n\uparrow+\infty$ is called the Hadamard composition of the genus $m\ge 1$ of functions $F_j(s)=\sum_{n=1}^{\infty}a_{n,j}\exp\{s\lambda_n\}$ if $a_n=P(a_{n,1},...,a_{n,p})$, where$P(x_1,...,x_p)=\sum\limits_{k_1+\dots+k_p=m}c_{k_1...k_p}x_1^{k_1}\cdot...\cdot x_p^{k_p}$ is a homogeneous polynomial of degree $m\ge 1$. Let $M(\sigma,F)=\sup\{|F(\sigma+it)|:\,t\in{\Bbb R}\}$ and functions $\alpha,\,\beta$ be positive continuous and increasing to $+\infty$ on $[x_0, +\infty)$. To characterize the growth of the function $M(\sigma,F)$, we use generalized order $\varrho_{\alpha,\beta}[F]=\varlimsup\limits_{\sigma\to+\infty}\dfrac{\alpha(\ln\,M(\sigma,F))}{\beta(\sigma)}$, generalized type$T_{\alpha,\beta}[F]=\varlimsup\limits_{\sigma\to+\infty}\dfrac{\ln\,M(\sigma,F)}{\alpha^{-1}(\varrho_{\alpha,\beta}[F]\beta(\sigma))}$and membership in the convergence class defined by the condition$\displaystyle \int_{\sigma_0}^{\infty}\frac{\ln\,M(\sigma,F)}{\sigma\alpha^{-1}(\varrho_{\alpha,\beta}[F]\beta(\sigma))}d\sigma<+\infty.$Assuming the functions $\alpha, \beta$ and $\alpha^{-1}(c\beta(\ln\,x))$ are slowly increasing for each $c\in (0,+\infty)$ and $\ln\,n=O(\lambda_n)$ as $n\to \infty$, it is proved, for example, that if the functions $F_j$ have the same generalized order $\varrho_{\alpha,\beta}[F_j]=\varrho\in (0,+\infty)$ and the types $T_{\alpha,\beta}[F_j]=T_j\in [0,+\infty)$, $c_{m0...0}=c\not=0$, $|a_{n,1}|>0$ and $|a_{n,j}|= o(|a_{n,1}|)$ as $n\to\infty$ for $2\le j\le p$, and $F$ is the Hadamard composition of genus$m\ge 1$ of the functions $F_j$ then $\varrho_{\alpha,\beta}[F]=\varrho$ and $\displaystyle T_{\alpha,\beta}[F]\le \sum_{k_1+\dots+k_p=m}(k_1T_1+...+k_pT_p).$It is proved also that $F$ belongs to the generalized convergence class if and only ifall functions $F_j$ belong to the same convergence class.

show abstract

“…Theorem C ( [1]). Let α ∈ L and β ∈ L be positive continuously differentiable functions such that d ln α −1 (ϱβ(σ)) dσ = O(1) as σ → ∞ for each ϱ ∈ (0, +∞).…”

Section: Theorem B ([6]mentioning

confidence: 99%

On entire Dirichlet series similar to Hadamard compositions

Mulyava

Sheremeta

2023

Mat. Stud.

Self Cite

View full text Add to dashboard Cite

show abstract

“…be analytic functions. As in [1], I say that the function is similar to the Hadamard composition of the functions f j if a n = w(a n,1 , . .…”

Section: Introductionmentioning

confidence: 99%

“…If R[ f ] > 0, then, for 0 ≤ r ≤ R[ f ], let M(r, f ) = max{| f (z)| : |z| = r} and µ(r, f ) = max{|a n |r n : n ≥ 0} be the maximal term of series (1). M. K. Sen [12,13] researched a connection between the growth of the maximal term of the derivative ( f 1 * f 2 ) (k) of the usual Hadamard composition f 1 * f 2 of entire functions f and g and the growth of the maximal term of derivative f…”

Section: Introductionmentioning

confidence: 99%

Hadamard Compositions of Gelfond–Leont’ev Derivatives

Sheremeta

2022

Axioms

View full text Add to dashboard Cite

For analytic functions fj(z)=∑n=0∞an,jzn, 1≤j≤p, the notion of a Hadamard composition (f1*…*fp)m=∑n=0∞∑k1+⋯+kp=mck1…kpan,1k1·…·an,pkpzn of genus m is introduced. The relationship between the growth of the Gelfond–Leont’ev derivative of the Hadamard composition of functions fj and the growth Hadamard composition of Gelfond–Leont’ev derivatives of these functions is studied. We found conditions under which these derivatives and the composition have the same order and a lower order. For the maximal terms of the power expansion of these derivatives, I describe behavior of their ratios.

show abstract

Hadamard Composition of Series in Systems of Functions

Sheremeta

2023

BMJ

View full text Add to dashboard Cite

For regularly converging in ${\Bbb C}$ series $A_j(z)=\sum\limits_{n=1}^{\infty}a_{n,j}f(\lambda_nz)$, $1\le j\le p$, where $f$ is an entire transcendental function, the asymptotic behavior of a Hadamard composition $A(z)=\break=(A_1*...*A_p)_m(z)=\sum\limits_{n=1}^{\infty} \left(\sum\limits_{k_1+\dots+k_p=m}c_{k_1...k_p}a_{n,1}^{k_1}\cdot...\cdot a_{n,p}^{k_p}\right)f(\lambda_nz)$ of genus m is investigated. The function $A_1$ is called dominant, if $|c_{m0...0}||a_{n,1}|^m \not=0$ and $|a_{n,j}|=o(|a_{n,1}|)$ as $n\to\infty$ for $2\le j\le p$. The generalized order of a function $A_j$ is called the quantity $\varrho_{\alpha,\beta}[A_j]=\break=\varlimsup\limits_{r\to+\infty}\dfrac{\alpha(\ln\,\mathfrak{M}(r,A_j))}{\beta(\ln\,r)}$, where $\mathfrak{M}(r,A_j)=\sum\limits_{n=1}^{\infty} |a_{n,j}|M_f(r\lambda_n)$, $ M_f(r)=\max\{|f(z)|:\,|z|=r\}$ and the functions $\alpha$ and $\beta$ are positive, continuous and increasing to $+\infty$. Under certain conditions on $\alpha$, $\beta$, $M_f(r)$ and $(\lambda_n)$, it is proved that if among the functions $A_j$ there exists a dominant one, then $\varrho_{\alpha,\beta}[A]=\max\{\varrho_{\alpha,\beta}[A_j]:\,1\le j\le p\}$. In terms of generalized orders, a connection is established between the growth of the maximal terms of power expansions of the functions $(A^{(k)}_1*...*A^{(k)}_p)_m$ and $((A_1*...*A_p)_m)^{(k)}$. Unresolved problems are formulated

show abstract

On Dirichlet series like to compositions of Hadamard

Cited by 3 publications

References 5 publications

On entire Dirichlet series similar to Hadamard compositions

On entire Dirichlet series similar to Hadamard compositions

Hadamard Compositions of Gelfond–Leont’ev Derivatives

Hadamard Composition of Series in Systems of Functions

Contact Info

Product

Resources

About