The angular value distribution of random analytic functions

Author M. P. Mahola, P. V. Filevych
marichka_stanko@ukr.net, filevych@mail.ru
Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of NASU, S. Z. Gzhytskyy Lviv National University of Veterinary Medicine and Biotechnologies

Abstract Let $\mathcal{R}\in(0,+\infty]$, $f(z)=\sum c_nz^n$ be an analytic function in the disk $\{z\colon |z|<\mathcal{R}\}$, $T_f(r)$ be the Nevanlinna characteristic, $N_f(r,\alpha,\beta,a)$ be the integrated counting function of $a$-points of $f$ in the sector $0<|z|\le r$, $\alpha\le\arg_{\alpha} z<\beta$, and $(\omega_n(\omega))$ be a~sequence of independent equidistributed on $[0,1]$ random variables. Under some conditions on the growth of $f$ it is proved that for random analytic function $f_\omega(z)=\sum e^{2\pi i\omega_n(\omega)}a_nz^n$ almost surely for every $a\in\mathbb{C}$ and all $\alpha<\beta\le\alpha+2\pi$ the relation $N_{f_\omega}(r,\alpha,\beta,a)\sim\frac{\beta-\alpha}{2\pi}T_{f_\omega}(r)$, $r\to\mathcal{R}$, holds outside some exceptional set $E\subset(0,\mathcal{R})$.
Keywords meromorphic function, value distribution; random analytic function; Nevanlinna characteristic; counting function; integrated counting function
Reference 1. Hayman W.K. Meromorphic functions. – Clarendon Press (Oxford), 1964.

 2. Goldberg A.A., Ostrovskii I.V. Value distribution of meromorphic functions. – Translations of Mathematical Monographs, V.236. American Mathematical Society, Providence, R.I., 2008.

 3. Kahane J.-P. Some Random Series of Functions. – Cambridge University Press, 1994.

 4. Offord A.C. The distribution of the values of a random function in the unit disk// Studia Math. – 1972. – V.41. – P. 71–106.

 5. Mahola M.P., Filevych P.V. The value distribution of a random entire function// Mat. Stud. – 2010. – V.34, Ή2. – P. 120–128.

 6. Hayman W.K. Angular value distribution of power series with gaps// Proc. London Math. Soc. (3). – 1972. – V.24. – P. 590–624.

 7. Murai T. The deficiency of entire functions with Fej.er gaps// Ann. Inst. Fourier (Grenoble). – 1983. – V.33. – P. 39–58.

 8. Hayman W.K., Rossi J.F. Characteristic, maximum modulus and value distribution// Trans. Amer. Math. Soc. – 1984. – V.284, Ή2. – P. 651–664.

 9. Offord A.C. The distribution of the values of an entire function whose coefficients are independent random variables (I)// Proc. London Math. Soc. – 1965. – V.14A. – P. 199–238.

 10. Offord A.C. The distribution of the values of an entire function whose coefficients are independent random variables (II)// Math. Proc. Cambridge Phil. Soc. – 1995. – V.118, Ή3. – P. 527–542.

 11. Filevych P.V. Some classes of entire functions in which the Wiman-Valiron inequality can be almost certainly improved// Mat. Stud. – 1996. – V.6. – P. 59–66. (in Ukrainian)

 12. Filevych P.V. Correlations between the maximum modulus and maximum term of random entire functions// Mat. Stud. – 1997. – V.7, Ή2. – P. 157–166. (in Ukrainian)

 13. Filevych P.V. Wiman–Valiron type inequalities for entire and random entire functions of finite logarithmic order// Sib. Mat. Zhurn. – 2003. – V.42, Ή3. – P. 683–694. (in Russian). English translation in: Siberian Math. J. – 2003. – V.42, Ή3. – P. 579–586.

 14. Filevych P.V. On relations between the abscissa of convergence and the abscissa of absolute convergence of random Dirichlet series// Mat. Stud. – 2003. – V.20, Ή1. – P. 33–39.

 15. Filevych P.V. The Baire categories and Wiman’s inequality for entire functions// Mat. Stud. – 2003. – V.20, Ή2. – P. 215–221.

 16. Skaskiv O.B. Random gap series and Wiman’s inequality// Mat. Stud. – 2008. – V.30, Ή1. – P. 101–106. (in Ukrainian)

 17. Skaskiv O.B., Zrum O.V. On an exeptional set in the Wiman inequalities for entire functions// Mat. Stud. – 2004. – V.21, Ή1. – P. 13–24. (in Ukrainian)

 18. Zrum O.V., Skaskiv O.B. On Wiman’s inequality for entire functions of two variables// Mat. Stud. – 2005. – V.23, .2. – P. 149–160. (in Ukrainian)

 19. Zrum O.V., Skaskiv O.B. Wiman’s inequalities for entire functions of two complex variables with rapidly oscillating coefficients// Math. Meth. Fhys.-Mech. Filds. – 2005. – V.48, Ή4. – P. 78–87. (in Ukrainian)

 20. Skaskiv O.B., Zrum O.V. Wiman’s inequalities for entire functions of two complex variables with rapidly oscillating coefficients// Math. Bull. Shevchenko Sci. Soc. – 2006. – V.3. – P. 56–68. (in Ukrainian)

 21. Zadorozhna O.Yu., Skaskiv O.B. On the domains of convergence of the double random Dirichlet series// Mat. Stud. – 2009. – V.32, Ή1. – P. 81–85. (in Ukrainian)

 22. Skaskiv O.B., Zadorozhna O.Yu. On domains of convergence of multiple random Dirichlet series// Mat. Stud. – 2011. – V.36, Ή1. – P. 51–57.

 23. Skaskiv O.B., Kuryliak A.O. Direct analogues of Wiman’s inequality for analytic functions in the unit disc// Carpathian Math. Publications. – 2010. – V.2, Ή1. – P. 109–118. (in Ukrainian)

 24. Skaskiv O.B., Kuryliak A.O. The probability of absence zeros in the disc for some random analytic functions// Math. Bull. Shevchenko Sci. Soc. – 2011. – V.8. – P. 335–352.

 25. Benbourenane D., Korhonen R. On the growth of the logarithmic derivative// Comput. Meth. Func. Theory. – 2001. – V.1, Ή2. – P. 301–310.

 26. Kondratyuk A.A., Kshanovskyy I.P. On the logarithmic derivative of a meromorphic function// Mat. Stud. – 2004. – V.21, Ή1. – P. 98–100.

 27. Hayman W.K., Kennedy P.B. Subharmonic functions. V.1. – London Mathematical Society Monographs, .9, Academic Press, London, 1976.

 28. Levin B.Ja. Lectures on entire functions. – Translations of Mathematical Monographs, vol. 150, American Mathematical Society, Providence, R.I., 1996.

 29. Andrusyak I.V., Filevych P.V. Minimal growth of an entire function with given zeros// Sci. Bull. Chernivtsi Univ., Ser. Math. – 2008. – V.421. – P. 13–19. (in Ukrainian)
Pages 34-51
Volume 37
Issue 1
Year 2012
Journal Matematychni Studii
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