Distance between a maximum modulus point and zero set of an analytic function

S. I. Fedynyak; P. V. Filevych

doi:10.30970/ms.52.1.10-23

Let $f$ be an analytic function in the disk $\mathbb{D}_R=\{z\in \mathbb{C}\colon |z|\le R\}$, $R\in (0,+\infty]$. A point $w\in\mathbb{D}_R$ is called a maximum modulus point of $f$ if $|f(w)|=M(|w|,f)$, where $M(r,f)=\max\{|f(z)|\colon |z|=r\}$. Denote by $d(w, f)$ the distance between a maximum modulus point $w$ and the zero set of $f$, i.e., $d(w,f)=\inf\{|w-z|\colon f(z)=0\}$. Let $\Phi$ be a continuous function on $[a,\ln R)$ such that $x\sigma-\Phi(\sigma)\to-\infty$, $\sigma\uparrow \ln R$, for every $x\in\mathbb{R}$. Let also $\widetilde{\Phi}$ be the Young-conjugate function of $\Phi$ and $\overline{\Phi}(x)=\widetilde{\Phi}(x)/x$ for all sufficiently large $x$. We prove that if $$ \ln M(r,f)\le (1+o(1))\Phi(\ln r),\quad r\uparrow R, $$ then $$ \varliminf_{|w|\uparrow R}d(w,f)\frac{\overline{\Phi}\,^{-1}(\ln |w|)}{|w|}\ge C_0, $$ where $C_0=0,5416\dots$. When the Taylor coefficients of $f$ are nonnegative, the constant $C_0$ can be replaced by $\pi$, and the inequality obtained in this case is sharp.

1. I.V. Ostrovskii, Distance between a maximum modulus point of an entire function and its zero set, Operator theory, subharmonic functions, Kyiv: Naukova Dumka, 1991, 67-75. (in Russian)

2. I. Ostrovskii, A.E. Ureyen, Distance between a maximum modulus point and zero set of an entire function, Complex Variables, Theory Appl., 48 (2003), №7, 583-598.

3. I. Ostrovskii, A.E. Ureyen, Maximum modulus points and zero sets of entire functions of regular growth, C. R. Acad. Sci. Paris. Ser. I., 341 (2005), №8, 481-484.

4. I. Ostrovskii, A.E. Ureyen, On maximum modulus points and zero sets of entire functions of regular growth, Rocky Mt. J. Math., 38 (2008), №2, 583-618.

5. A.E. Ureyen, On maximum modulus points and the zero set for an entire function of either zero or infinite order, Comput. Methods Funct. Theory, 4 (2005), №2, 341-354.

6. S.I. Fedynyak, On maximum modulus points and zero set for an entire function, Mat. Stud., 30 (2008), №2, 169-172.

7. M.A. Evgrafov, Asymptotic estimates and entire functions, Moscow: Nauka, 1979. (in Russian)

8. T.Ya. Hlova, P.V. Filevych, Generalized types of the growth of Dirichlet series, Carpathian Math. Publ., 7 (2015), №2, 172–187.

9. P.V. Filevych, On the slow growth of power series convergent in the unit disk, Mat. Stud., 16 (2001), №2, 217–221.

	Distance between a maximum modulus point and the zero set of an analytic function
Author	S. I. Fedynyak¹, P. V. Filevych² napets.fed@gmail.com¹, p.v.filevych@gmail.com² 1) Department of Applied Mathematics and Statistics Ukrainian Catholic University, Lviv, Ukraine; 2) Department of Computational Mathematics and Programming Lviv Polytechnic National University, Lviv, Ukraine
Abstract	Let $f$ be an analytic function in the disk $\mathbb{D}_R=\{z\in \mathbb{C}\colon \|z\|\le R\}$, $R\in (0,+\infty]$. A point $w\in\mathbb{D}_R$ is called a maximum modulus point of $f$ if $\|f(w)\|=M(\|w\|,f)$, where $M(r,f)=\max\{\|f(z)\|\colon \|z\|=r\}$. Denote by $d(w, f)$ the distance between a maximum modulus point $w$ and the zero set of $f$, i.e., $d(w,f)=\inf\{\|w-z\|\colon f(z)=0\}$. Let $\Phi$ be a continuous function on $[a,\ln R)$ such that $x\sigma-\Phi(\sigma)\to-\infty$, $\sigma\uparrow \ln R$, for every $x\in\mathbb{R}$. Let also $\widetilde{\Phi}$ be the Young-conjugate function of $\Phi$ and $\overline{\Phi}(x)=\widetilde{\Phi}(x)/x$ for all sufficiently large $x$. We prove that if $$ \ln M(r,f)\le (1+o(1))\Phi(\ln r),\quad r\uparrow R, $$ then $$ \varliminf_{\|w\|\uparrow R}d(w,f)\frac{\overline{\Phi}\,^{-1}(\ln \|w\|)}{\|w\|}\ge C_0, $$ where $C_0=0,5416\dots$. When the Taylor coefficients of $f$ are nonnegative, the constant $C_0$ can be replaced by $\pi$, and the inequality obtained in this case is sharp.
Keywords	analytic function; maximum modulus; maximum modulus point; zero set
DOI	doi:10.30970/ms.52.1.10-23
Reference	1. I.V. Ostrovskii, Distance between a maximum modulus point of an entire function and its zero set, Operator theory, subharmonic functions, Kyiv: Naukova Dumka, 1991, 67-75. (in Russian) 2. I. Ostrovskii, A.E. Ureyen, Distance between a maximum modulus point and zero set of an entire function, Complex Variables, Theory Appl., 48 (2003), №7, 583-598. 3. I. Ostrovskii, A.E. Ureyen, Maximum modulus points and zero sets of entire functions of regular growth, C. R. Acad. Sci. Paris. Ser. I., 341 (2005), №8, 481-484. 4. I. Ostrovskii, A.E. Ureyen, On maximum modulus points and zero sets of entire functions of regular growth, Rocky Mt. J. Math., 38 (2008), №2, 583-618. 5. A.E. Ureyen, On maximum modulus points and the zero set for an entire function of either zero or infinite order, Comput. Methods Funct. Theory, 4 (2005), №2, 341-354. 6. S.I. Fedynyak, On maximum modulus points and zero set for an entire function, Mat. Stud., 30 (2008), №2, 169-172. 7. M.A. Evgrafov, Asymptotic estimates and entire functions, Moscow: Nauka, 1979. (in Russian) 8. T.Ya. Hlova, P.V. Filevych, Generalized types of the growth of Dirichlet series, Carpathian Math. Publ., 7 (2015), №2, 172–187. 9. P.V. Filevych, On the slow growth of power series convergent in the unit disk, Mat. Stud., 16 (2001), №2, 217–221.
Pages	10-23
Volume	52
Issue	1
Year	2019
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Distance between a maximum modulus point and the zero set of an analytic function