On minimum modulus of an entire function of zero
order

I. E. Chyzhykov

For a non-negative function $\psi\in C^2[1,+\infty)$ we define $\psi_j(r)=\frac{d^j\psi(r)}{(d\ln r)^j}$, $j\in \{1, 2\}$. Suppose that $\psi_2(r)\to+\infty$ and $\psi_2(2r)\sim\psi_2(r)$ $(r\to+\infty)$. Then there exist an entire function $g(z)$ of zero order with the following properties: a) $n(r,0,g)\sim\psi_1(r)$, $\ln M(r,g)\sim \psi(r)$ $(r\to+\infty)$; b) $\ln M(r,g)-\ln \mu(r,g) \ge (1+o(1)) \frac{\pi^2}2 \psi_2(r)$ $(r\to+\infty)$, where $M(r,g)=\max\{|g(z)|:|z|=r\}$, $\mu(r,g)=\min\{|g(z)|:|z|=r\}$, $n(r,0,g)$ is the zero counting functions of $g$.

1. Hayman W.K. Subharmonic functions, Vol 2. London e. a.: Academic Press, 1989. XXI+591 pp.

2. Chyzhykov I.E. An addition to $\cos\pi\rho$-theorem for subharmonic and entire functions of zero lower order, Proc.\ Amer.\ Math.\ Soc. 130 (2002), no. 2, 517--528.

3. Barry P.D. The minimum modulus of small integral and subharmonic functions, Proc. London Math. Soc. (3) 12 (1962), no. 47, 445–495.

4. Fenton P.C. The infimum of small subharmonic functions, Proc. Amer. Math. Soc. 78 (1980), no. 1, 43–47.

5. Гольдберг А. А. О минимуме модуля мероморфной функции медленного роста // Мат. заметки. 25 (1979), no. 6, 835–844. Engl. trans. in Math. Notes (1979), 432–437.

6. Fenton P.C. The minimum of small entire functions, Proc. Amer. Math. Soc. 81 (1981), no. 4, 557–561.

7. Fenton P.C. The minimum modulus of certain small entire functions, Proc. Amer. Math. Soc. 271 (1982), no. 1, 183–195 .

8. Barry P.D. On integral functions which grow little more rapidly then do polynomials, Proc. R. Ir. Acad. 82A (1982) no. 1, 55–95.

9. Lyubarskii Yu., Malinnikova Eu. On approximation of subharmonic functions, J. d'Analyse Math. 83 (2001), 121–149.

10. Arsov M. G. Functions representable as differences of subharmonic functions // Trans. Amer. Math. Soc. – 1953. – V.75. P.327–365.

11. Hayman W.K., Kennedy P.B. Subharmonic functions, Vol 1, London e. a.: Academic Press, 1976.

	On minimum modulus of an entire function of zero order
Author	I. E. Chyzhykov Faculty of Mechanics and Mathematics, Lviv National University
Abstract	For a non-negative function $\psi\in C^2[1,+\infty)$ we define $\psi_j(r)=\frac{d^j\psi(r)}{(d\ln r)^j}$, $j\in \{1, 2\}$. Suppose that $\psi_2(r)\to+\infty$ and $\psi_2(2r)\sim\psi_2(r)$ $(r\to+\infty)$. Then there exist an entire function $g(z)$ of zero order with the following properties: a) $n(r,0,g)\sim\psi_1(r)$, $\ln M(r,g)\sim \psi(r)$ $(r\to+\infty)$; b) $\ln M(r,g)-\ln \mu(r,g) \ge (1+o(1)) \frac{\pi^2}2 \psi_2(r)$ $(r\to+\infty)$, where $M(r,g)=\max\{\|g(z)\|:\|z\|=r\}$, $\mu(r,g)=\min\{\|g(z)\|:\|z\|=r\}$, $n(r,0,g)$ is the zero counting functions of $g$.
Keywords	non-negative function, entire function, zero counting functions
DOI	doi:10.30970/ms.17.1.41-46
Reference	1. Hayman W.K. Subharmonic functions, Vol 2. London e. a.: Academic Press, 1989. XXI+591 pp. 2. Chyzhykov I.E. An addition to $\cos\pi\rho$-theorem for subharmonic and entire functions of zero lower order, Proc.\ Amer.\ Math.\ Soc. 130 (2002), no. 2, 517--528. 3. Barry P.D. The minimum modulus of small integral and subharmonic functions, Proc. London Math. Soc. (3) 12 (1962), no. 47, 445–495. 4. Fenton P.C. The infimum of small subharmonic functions, Proc. Amer. Math. Soc. 78 (1980), no. 1, 43–47. 5. Гольдберг А. А. О минимуме модуля мероморфной функции медленного роста // Мат. заметки. 25 (1979), no. 6, 835–844. Engl. trans. in Math. Notes (1979), 432–437. 6. Fenton P.C. The minimum of small entire functions, Proc. Amer. Math. Soc. 81 (1981), no. 4, 557–561. 7. Fenton P.C. The minimum modulus of certain small entire functions, Proc. Amer. Math. Soc. 271 (1982), no. 1, 183–195 . 8. Barry P.D. On integral functions which grow little more rapidly then do polynomials, Proc. R. Ir. Acad. 82A (1982) no. 1, 55–95. 9. Lyubarskii Yu., Malinnikova Eu. On approximation of subharmonic functions, J. d'Analyse Math. 83 (2001), 121–149. 10. Arsov M. G. Functions representable as differences of subharmonic functions // Trans. Amer. Math. Soc. – 1953. – V.75. P.327–365. 11. Hayman W.K., Kennedy P.B. Subharmonic functions, Vol 1, London e. a.: Academic Press, 1976.
Pages	41-46
Volume	17
Issue	1
Year	2002
Journal	Matematychni Studii
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Table of content of issue	html

On minimum modulus of an entire function of zero order