Pfluger-type theorem for functions of refined regular growth

I. Chyzhykov

doi:10.15330/ms.47.2.169-178

Without a priori assumptions on zero distribution we prove that if an entire function $f$ of noninteger order $\rho$ has an asymptotic of the form $\log|f(re^{i\theta})|=r^\rho h_f(\theta)+ O(\frac{r^{\rho}}{\delta(r)})$, $ E\not \ni re^{i\theta}\to \infty$, where $h$ is the indicator of $f$, $\delta$ is an unbounded regularly growing function, and $E$ is an appropriate exceptional set, then the counting function of zeros and the integrated counting function of zeros in the angle $\{z: \alpha\le\arg z\le\beta\}$ have similar asymptotic for almost all $\alpha\le\beta$. It complements results on functions of completely regular growth due to P. Agranovich and V. Logvinenko, B. Vynnyts'kyi and R. Khats'.

1. Agranovich P.Z., Approximation of subharmonic functions and problems of polynomial asymptotics connected with it, Math. Physics, Analysis, Geometry, 7 (2000), №3, 255–265. (in Russian)

2. Agranovich P.Z., Logvinenko V.N., An analog of the Titchmarsh-Valiron theorem on the two-term asymptotics of a subharmonic function with masses on a finite system of rays, Sib. Math. Zh., 26 (1985), №5, 3–19. (in Russian)

3. Agranovich P.Z., V.N. Logvinenko V.N., On massivity of exceptional set of multi-term asymptotic representation of a subharmonic function, Kharkov, FTINT AN USSR, Preprint 45–87, 18 p. (in Russian)

4. Agranovich P.Z., Logvinenko V.N., Polynomial asymptotic representation of subharmonic functions in the plane, Siberain Math. J., 32 (1991), №1, 1–16.

5. Azarin V.S., Asymptotic behaviour of subharmonic functions of finite order, Mat. Sb., N. Ser., 36 (1980), №2, 135–154.

6. Azarin V.S., On the polynomial asymptotics of subharmonic functions of finite order and their mass distributions, J. Math. Phys. Analysis, Geom., 3 (2007), №1, 5–12.

7. Goldberg A.A., Ostrovskii I.V., Value distribution of meromorphic functions. Transl. Math. Monographs 236, American Math. Soc., Providence, R. I., 2008.

8. Seneta Eu., Regularly varying functions. Lecture Notes in Mathematics, 508, Berlin-Heidelberg-New York: Springer-Verlag. 1976, 112 p.

9. Khabibullin B.N., Asymptotic behavior of the difference of subharmonic functions, Mat. Stud., 21 (2004), №1, 47–63.

10. Kondratyuk A.A., Fourier series and meromorphic functions. – Lvov, Vyshcha shkola, 1988. – 196 p. (in Russian)

11. Levin B.Ya., Distribution of zeros of entire functions. – Amer. Math. Soc. Providence, R.I., 1980.

12. Pfluger A., Die Wertvertailung und das Verhalten von Betrag und Argument einer speziellen Klasse analytisher Funktionen, II, Comm. Math. Helv., 12 (1939), 25–69.

13. Vynnyts’kyi B.V., Khats’ R.V., On growth regularity of an entire function of nonentire order with zeros on a finite system of rays, Mat. Stud., 22 (2005), №1, 31–38. (in Ukrainian)

14. Yulmukhametov R.S., Asymptotics of the difference of subharmonic functions, Mathematical notes, 41 (1987), №3, 199–204. (in Russian)

	Pfluger-type theorem for functions of refined regular growth
Author	I. Chyzhykov chyzhykov@yahoo.com Ivan Franko National University of Lviv, Lviv, Ukraine
Abstract	Without a priori assumptions on zero distribution we prove that if an entire function $f$ of noninteger order $\rho$ has an asymptotic of the form $\log\|f(re^{i\theta})\|=r^\rho h_f(\theta)+ O(\frac{r^{\rho}}{\delta(r)})$, $ E\not \ni re^{i\theta}\to \infty$, where $h$ is the indicator of $f$, $\delta$ is an unbounded regularly growing function, and $E$ is an appropriate exceptional set, then the counting function of zeros and the integrated counting function of zeros in the angle $\{z: \alpha\le\arg z\le\beta\}$ have similar asymptotic for almost all $\alpha\le\beta$. It complements results on functions of completely regular growth due to P. Agranovich and V. Logvinenko, B. Vynnyts'kyi and R. Khats'.
Keywords	entire function; completely regular growth; indicator; refined regular growth; zero distribution
DOI	doi:10.15330/ms.47.2.169-178
Reference	1. Agranovich P.Z., Approximation of subharmonic functions and problems of polynomial asymptotics connected with it, Math. Physics, Analysis, Geometry, 7 (2000), №3, 255–265. (in Russian) 2. Agranovich P.Z., Logvinenko V.N., An analog of the Titchmarsh-Valiron theorem on the two-term asymptotics of a subharmonic function with masses on a finite system of rays, Sib. Math. Zh., 26 (1985), №5, 3–19. (in Russian) 3. Agranovich P.Z., V.N. Logvinenko V.N., On massivity of exceptional set of multi-term asymptotic representation of a subharmonic function, Kharkov, FTINT AN USSR, Preprint 45–87, 18 p. (in Russian) 4. Agranovich P.Z., Logvinenko V.N., Polynomial asymptotic representation of subharmonic functions in the plane, Siberain Math. J., 32 (1991), №1, 1–16. 5. Azarin V.S., Asymptotic behaviour of subharmonic functions of finite order, Mat. Sb., N. Ser., 36 (1980), №2, 135–154. 6. Azarin V.S., On the polynomial asymptotics of subharmonic functions of finite order and their mass distributions, J. Math. Phys. Analysis, Geom., 3 (2007), №1, 5–12. 7. Goldberg A.A., Ostrovskii I.V., Value distribution of meromorphic functions. Transl. Math. Monographs 236, American Math. Soc., Providence, R. I., 2008. 8. Seneta Eu., Regularly varying functions. Lecture Notes in Mathematics, 508, Berlin-Heidelberg-New York: Springer-Verlag. 1976, 112 p. 9. Khabibullin B.N., Asymptotic behavior of the difference of subharmonic functions, Mat. Stud., 21 (2004), №1, 47–63. 10. Kondratyuk A.A., Fourier series and meromorphic functions. – Lvov, Vyshcha shkola, 1988. – 196 p. (in Russian) 11. Levin B.Ya., Distribution of zeros of entire functions. – Amer. Math. Soc. Providence, R.I., 1980. 12. Pfluger A., Die Wertvertailung und das Verhalten von Betrag und Argument einer speziellen Klasse analytisher Funktionen, II, Comm. Math. Helv., 12 (1939), 25–69. 13. Vynnyts’kyi B.V., Khats’ R.V., On growth regularity of an entire function of nonentire order with zeros on a finite system of rays, Mat. Stud., 22 (2005), №1, 31–38. (in Ukrainian) 14. Yulmukhametov R.S., Asymptotics of the difference of subharmonic functions, Mathematical notes, 41 (1987), №3, 199–204. (in Russian)
Pages	169-178
Volume	47
Issue	2
Year	2017
Journal	Matematychni Studii
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