On certain subclass of Dirichlet series absolutely convergent in half-plane

M. M. Sheremeta

doi:10.30970/ms.57.1.32-44

M. M. Sheremeta Ivan Franko National University of Lviv, Lviv

DOI: https://doi.org/10.30970/ms.57.1.32-44

Keywords: Dirichlet series; Hadamard composition; neighborhood of the function; differential equation.

Abstract

Denote by $\mathfrak{D}_0$ a class of absolutely convergent in half-plane $\Pi_0=\{s\colon \text{Re}\,s<0\}$ Dirichlet series
$F(s)=e^{sh}-\sum_{k=1}^{\infty}f_k\exp\{s(\lambda_k+h)\},\, s=\sigma+it$, where $h> 0$, $h<\lambda_k\uparrow+\infty$ and $f_k>0$.
For $0\le\alpha<h$ and $l\ge 0$ we say that $F$ belongs to the class $\mathfrak{DF}_h(l,\alpha)$ if and only if
$\text{Re}\{e^{-hs}((1-l)F(s)+\frac{l}{h}F'(s))\}>\frac{\alpha}{h}$,
and belongs to the class $\mathfrak{DG}_h(l,\alpha)$ if and only if
$\text{Re}\{e^{-hs}((1-l)F'(s)+\frac{l}{h}F''(s))\}>\alpha$ for all $s\in \Pi_0$. It is proved
that $F\in \mathfrak{DF}_h(l,\alpha)$ if and only if $ \sum_{k=1}^{\infty}(h+l\lambda_k)f_k\le h-\alpha$, and
$F\in \mathfrak{DG}_h(l,\alpha)$ if and only if $\sum_{k=1}^{\infty}(h+l\lambda_k)(\lambda_k+h)f_k\le h(h-\alpha)$.

If $F_j\in \mathfrak{DF}_h(l_j,\alpha_j)$, $j=1, 2$, where $l_j\ge0$ and $0\le \alpha_j<h$, then Hadamard composition
$(F_1*F_2)\in \mathfrak{D}F_h(l,\alpha)$, where $l=\min\{l_1,l_2\}$ and
$\alpha=h-\frac{(h-\alpha_1)(h-\alpha_2)}{h+l\lambda_1}$. Similar statement is correct for the class $F_j\in \mathfrak{DG}_h(l,\alpha)$.

For $j>0$ and $\delta>0$ the neighborhood of the function $F\in \mathfrak{D}_0$ is defined as follows
$O_{j,\delta}(F)=\{G(s)=e^{s}-\sum_{k=1}^{\infty}g_k\exp\{s\lambda_k\}\in \mathfrak{D}_0\colon
\sum_{k=1}^{\infty}\lambda^j_k|g_k-f_k|\le\delta\}$. It is described the neighborhoods of functions from classes $\mathfrak{DF}_h(l,\alpha)$ and $\mathfrak{DG}_h(l,\alpha)$.

Conditions on real parameters $\gamma_0,\,\gamma_1,\,\gamma_2,\,a_1$ and $a_2$ of the differential equation
$w''+(\gamma_0e^{2hs}+\gamma_1e^{hs}+\gamma_2) w=a_1e^{hs}+a_2e^{2hs}$ are found, under which this equation has a solution
either in $\mathfrak{DF}_h(l,\alpha)$ or in $\mathfrak{DG}_h(l,\alpha)$.

Author Biography

M. M. Sheremeta, Ivan Franko National University of Lviv, Lviv

Department of Mechanics and Mathematics, Professor

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