一类指数系数微分方程的拟星解和拟凸解

Q3 Mathematics Matematychni Studii Pub Date : 2021-10-27 DOI:10.30970/ms.56.1.39-47
M. Sheremeta
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引用次数: 1

摘要

狄利克雷级数 $F(s)=e^{s}+\sum_{k=1}^{\infty}f_ke^{s\lambda_k}$ 用指数表示 $1<\lambda_k\uparrow+\infty$ 以及绝对收敛的横坐标 $\sigma_a[F]\ge 0$ 据说是有序的伪星形 $\alpha\in [0,\,1)$ 输入 $\beta \in (0,\,1]$ 如果$\left|\dfrac{F'(s)}{F(s)}-1\right|<\beta\left|\dfrac{F'(s)}{F(s)}-(2\alpha-1)\right|$ 对所有人 $s\in \Pi_0=\{s\colon \,\text{Re}\,s<0\}$. 类似地,函数 $F$ 说是阶伪凸吗 $\alpha\in [0,\,1)$ 输入 $\beta \in (0,\,1]$ 如果$\left|\dfrac{F''(s)}{F'(s)}-1\right|<\beta\left|\dfrac{F''(s)}{F'(s)}-(2\alpha-1)\right|$ 对所有人 $s\in \Pi_0$. 在参数上发现了一些条件 $b_0,\,b_1,\,c_0,\,c_1,\,\,c_2$ 系数 $a_n$下的微分方程$\dfrac{d^2w}{ds^2}+(b_0e^{s}+b_1)\dfrac{dw}{ds}+(c_0e^{2s}+c_1e^{s}+c_2)w=\sum\limits_{n=1}^{\infty}a_ne^{ns}$有一个完整的解是伪星形的还是有序的伪凸的 $\alpha\in [0,\,1)$ 输入 $\beta \in (0,\,1]$。在此解的某些条件下,证明了渐近等式成立$\ln\,\max\{|F(\sigma+it)|\colon t\in {\mathbb R}\}=\dfrac{1+o(1)}{2}\left(|b_0|+\sqrt{|b_0|^2+4|c_0|}\right)$ as $\sigma \to+\infty$.
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Pseudostarlike and pseudoconvex solutions of a differential equation with exponential coefficients
Dirichlet series $F(s)=e^{s}+\sum_{k=1}^{\infty}f_ke^{s\lambda_k}$ with the exponents $1<\lambda_k\uparrow+\infty$ and the abscissa of absolute convergence $\sigma_a[F]\ge 0$ is said to be pseudostarlike of order $\alpha\in [0,\,1)$ and type $\beta \in (0,\,1]$ if$\left|\dfrac{F'(s)}{F(s)}-1\right|<\beta\left|\dfrac{F'(s)}{F(s)}-(2\alpha-1)\right|$\ for all\ $s\in \Pi_0=\{s\colon \,\text{Re}\,s<0\}$. Similarly, the function $F$ is said to be pseudoconvex of order $\alpha\in [0,\,1)$ and type $\beta \in (0,\,1]$ if$\left|\dfrac{F''(s)}{F'(s)}-1\right|<\beta\left|\dfrac{F''(s)}{F'(s)}-(2\alpha-1)\right|$\ for all\ $s\in \Pi_0$. Some conditions are found on the parameters $b_0,\,b_1,\,c_0,\,c_1,\,\,c_2$ and the coefficients $a_n$, under which the differential equation $\dfrac{d^2w}{ds^2}+(b_0e^{s}+b_1)\dfrac{dw}{ds}+(c_0e^{2s}+c_1e^{s}+c_2)w=\sum\limits_{n=1}^{\infty}a_ne^{ns}$has an entire solution which is pseudostarlike or pseudoconvex of order $\alpha\in [0,\,1)$ and type $\beta \in (0,\,1]$. It is proved that by some conditions for such solution the asymptotic equality holds  $\ln\,\max\{|F(\sigma+it)|\colon t\in {\mathbb R}\}=\dfrac{1+o(1)}{2}\left(|b_0|+\sqrt{|b_0|^2+4|c_0|}\right)$ as $\sigma \to+\infty$.
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来源期刊
Matematychni Studii
Matematychni Studii Mathematics-Mathematics (all)
CiteScore
1.00
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0.00%
发文量
38
期刊介绍: Journal is devoted to research in all fields of mathematics.
期刊最新文献
On the h-measure of an exceptional set in Fenton-type theorem for Taylor-Dirichlet series Almost periodic distributions and crystalline measures Reflectionless Schrodinger operators and Marchenko parametrization Existence of basic solutions of first order linear homogeneous set-valued differential equations Real univariate polynomials with given signs of coefficients and simple real roots
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