Sufficient conditions for the improved regular growth of entire functions in terms of their averaging
Let $f$ be an entire function of order $\rho\in (0,+\infty)$ with zeros on a finite system of rays $\{z: \arg z=\psi_{j}\}$, $j\in\{1,\ldots,m\}$, $0\le\psi_1<\psi_2<\ldots<\psi_m<2\pi$ and $h(\varphi)$ be its indicator. In 2011, the author of the article has been proved that if $f$ is o...
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| Format: | Article |
| Language: | English |
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Vasyl Stefanyk Precarpathian National University
2020-06-01
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| Series: | Karpatsʹkì Matematičnì Publìkacìï |
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| Online Access: | https://journals.pnu.edu.ua/index.php/cmp/article/view/3877 |
| _version_ | 1797205847270490112 |
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| author | R.V. Khats' |
| author_facet | R.V. Khats' |
| author_sort | R.V. Khats' |
| collection | DOAJ |
| description | Let $f$ be an entire function of order $\rho\in (0,+\infty)$ with zeros on a finite system of rays $\{z: \arg z=\psi_{j}\}$, $j\in\{1,\ldots,m\}$, $0\le\psi_1<\psi_2<\ldots<\psi_m<2\pi$ and $h(\varphi)$ be its indicator. In 2011, the author of the article has been proved that if $f$ is of improved regular growth (an entire function $f$ is called a function of improved regular growth if for some $\rho\in (0,+\infty)$ and $\rho_1\in (0,\rho)$, and a $2\pi$-periodic $\rho$-trigonometrically convex function $h(\varphi)\not\equiv -\infty$ there exists a set $U\subset\mathbb C$ contained in the union of disks with finite sum of radii and such that $\log |{f(z)}|=|z|^\rho h(\varphi)+o(|z|^{\rho_1})$, $U\not\ni z=re^{i\varphi}\to\infty$), then for some $\rho_3\in (0,\rho)$ the relation \begin{equation*} \int_1^r {\frac{\log |{f(te^{i\varphi})}|}{t}}\, dt=\frac{r^\rho}{\rho}h(\varphi)+o(r^{\rho_3}),\quad r\to +\infty, \end{equation*} holds uniformly in $\varphi\in [0,2\pi]$. In the present paper, using the Fourier coefficients method, we establish the converse statement, that is, if for some $\rho_3\in (0,\rho)$ the last asymptotic relation holds uniformly in $\varphi\in [0,2\pi]$, then $f$ is a function of improved regular growth. It complements similar results on functions of completely regular growth due to B. Levin, A. Grishin, A. Kondratyuk, Ya. Vasyl'kiv and Yu. Lapenko. |
| first_indexed | 2024-04-24T08:57:37Z |
| format | Article |
| id | doaj.art-7ef3e939018b4ead8eefe42633225d9d |
| institution | Directory Open Access Journal |
| issn | 2075-9827 2313-0210 |
| language | English |
| last_indexed | 2024-04-24T08:57:37Z |
| publishDate | 2020-06-01 |
| publisher | Vasyl Stefanyk Precarpathian National University |
| record_format | Article |
| series | Karpatsʹkì Matematičnì Publìkacìï |
| spelling | doaj.art-7ef3e939018b4ead8eefe42633225d9d2024-04-16T07:00:37ZengVasyl Stefanyk Precarpathian National UniversityKarpatsʹkì Matematičnì Publìkacìï2075-98272313-02102020-06-01121465410.15330/cmp.12.1.46-543370Sufficient conditions for the improved regular growth of entire functions in terms of their averagingR.V. Khats'0Drohobych Ivan Franko State Pedagogical University, 24 Franko Str., 82100, Drohobych, UkraineLet $f$ be an entire function of order $\rho\in (0,+\infty)$ with zeros on a finite system of rays $\{z: \arg z=\psi_{j}\}$, $j\in\{1,\ldots,m\}$, $0\le\psi_1<\psi_2<\ldots<\psi_m<2\pi$ and $h(\varphi)$ be its indicator. In 2011, the author of the article has been proved that if $f$ is of improved regular growth (an entire function $f$ is called a function of improved regular growth if for some $\rho\in (0,+\infty)$ and $\rho_1\in (0,\rho)$, and a $2\pi$-periodic $\rho$-trigonometrically convex function $h(\varphi)\not\equiv -\infty$ there exists a set $U\subset\mathbb C$ contained in the union of disks with finite sum of radii and such that $\log |{f(z)}|=|z|^\rho h(\varphi)+o(|z|^{\rho_1})$, $U\not\ni z=re^{i\varphi}\to\infty$), then for some $\rho_3\in (0,\rho)$ the relation \begin{equation*} \int_1^r {\frac{\log |{f(te^{i\varphi})}|}{t}}\, dt=\frac{r^\rho}{\rho}h(\varphi)+o(r^{\rho_3}),\quad r\to +\infty, \end{equation*} holds uniformly in $\varphi\in [0,2\pi]$. In the present paper, using the Fourier coefficients method, we establish the converse statement, that is, if for some $\rho_3\in (0,\rho)$ the last asymptotic relation holds uniformly in $\varphi\in [0,2\pi]$, then $f$ is a function of improved regular growth. It complements similar results on functions of completely regular growth due to B. Levin, A. Grishin, A. Kondratyuk, Ya. Vasyl'kiv and Yu. Lapenko.https://journals.pnu.edu.ua/index.php/cmp/article/view/3877entire function of completely regular growthentire function of improved regular growthindicatorfourier coefficientsaveragingfinite system of rays |
| spellingShingle | R.V. Khats' Sufficient conditions for the improved regular growth of entire functions in terms of their averaging Karpatsʹkì Matematičnì Publìkacìï entire function of completely regular growth entire function of improved regular growth indicator fourier coefficients averaging finite system of rays |
| title | Sufficient conditions for the improved regular growth of entire functions in terms of their averaging |
| title_full | Sufficient conditions for the improved regular growth of entire functions in terms of their averaging |
| title_fullStr | Sufficient conditions for the improved regular growth of entire functions in terms of their averaging |
| title_full_unstemmed | Sufficient conditions for the improved regular growth of entire functions in terms of their averaging |
| title_short | Sufficient conditions for the improved regular growth of entire functions in terms of their averaging |
| title_sort | sufficient conditions for the improved regular growth of entire functions in terms of their averaging |
| topic | entire function of completely regular growth entire function of improved regular growth indicator fourier coefficients averaging finite system of rays |
| url | https://journals.pnu.edu.ua/index.php/cmp/article/view/3877 |
| work_keys_str_mv | AT rvkhats sufficientconditionsfortheimprovedregulargrowthofentirefunctionsintermsoftheiraveraging |