Wiman’s type inequality for multiple power series in an unbounded cylinder domain

A. O. Kuryliak; V. L. Tsvigun

doi:10.15330/ms.49.1.29-51

In this paper we prove some analogues of Wiman’s inequality for analytic $f(z)$ and random analytic functions $f(z,t)$ on $\mathbb T=\mathbb D^l\times\mathbb C^{p-l}$, $l\in\mathbb N,\ 1\leq l\le p$, $I=\{1,\ldots, l\},\ J=\{l+1,\ldots, p\}$ of the form $f(z)=\sum_{\|n\|=0}^{+\infty}a_nz^n$, $f(z,t)=\sum_{\|n\|=0}^{+\infty}a_nZ_n(t)z^n$ respectively. Here $Z=(Z_n)$ is a multiplicative system of random variables on the Steinhaus probability space, uniformly bounded by the number 1. In particular, there are proved the following statements: For every $\delta>0$ there exist sets $E_1=E_1(\delta,f),\ E_2=E_2(\delta,f)\subset [0,1)^l\times(1,+\infty)^{p-l}$ of asymptotically finite logarithmic measure, such that the inequalities \begin{gather*} M_f(r)\leq\mu_f(r)\prod_{i\in I}\frac{1}{(1-r_i)^{1+\delta}}\ln^{{p/2}+\delta}\Bigl(\mu_f(r)\prod_{i\in I}\frac{1}{1-r_i}\Bigl)\Bigl (\prod_{j\in J}\ln r_j\Bigl)^{p+\delta},\\ M_f(r,t)\leq\mu_f(r)\prod_{i\in I}\frac{1}{(1-r_i)^{1/2+\delta}}\ln^{{p/4}+\delta}\Bigl(\mu_f(r)\prod_{i\in I}\frac{1}{1-r_i}\Bigl)\Bigl (\prod_{j\in J}\ln r_j\Bigl)^{ p/2+\delta}. \end{gather*} hold for all $r\in T\setminus E_1$ and for all $r\in T\setminus E_2$ a.s. in $t$, respectively. Also is proved sharpness of the obtained inequalities.

1. Wiman A. Uber dem Zusammenhang zwischen dem Maximalbetrage einer analytischen Function und dem grossten Gliede der zugehorigen Taylorilbr schen Reihe// Acta Math. - 1914. - V.37. - P. 305-326.

2. Polya G., Szego G. Aufgaben und Lehrsatze aus der Analysis, V.2. - Berlin, Springer, 1925.

3. Valiron G. Fonctions analytiques. . Paris: Press. Univer. de France, 1954.

4. Wittich H. Neuere Untersuchungen uber eindeutige analytische Funktionen. - Berlin-Gottingen- Heidelberg: Springer, 1955. - 164 s.

5. Goldberg A.A., Levin B.Y., Ostrovskii I.V., Entire and meromorphic functions. - Itogi Nauki i Tekhniki. Seriya VINITI. - 1990. - V.85. - P. 5-186. (in Russian)

6. Skaskiv O.B., Filevych P.V., On the size of an exceptional set in the Wiman theorem, Mat. Stud. - 1999. - V.12, №1. - P.31-36. (in Ukrainian)

7. Skaskiv O.B., Zrum O.V. On an exeptional set in the Wiman inequalities for entire functions, Mat. Stud. - 2004. - V.21, №1. - 13-24. (in Ukrainian)

8. Skaskiv O.B., Bandura A.I., Asymptotic estimates of positive integrals and entire functions. - Lviv. Ivano-Frankivsk: LNU-INFTUNG, 2015. - 108 p. (in Ukrainian)

9. Erd.os P., Renyi A. On random entire function// Zastosowania mat. - 1969. - V.10. - P. 47-55.

10. Steele J.M., Sharper Wiman inequality for entire functions with rapidly oscillating coefficients// J. Math. Anal. and Appl. - 1987. - V.123. - P. 550-558.

11. K.ovari T. On the maximum modulus and maximal term of functions analytic in the unit disc// J. London Math. Soc. - 1966. - V.41. - P. 129-137.

12. Suleymanov N.M. Wiman-Valirons type inequalities for power series with bounded radii of convergence and its sharpness// DAN SSSR. - 1980. - V.253, №4. - P. 822-824. (in Russian)

13. Skaskiv O.B., Kuryliak A.O. Direct analogues of WimanЃfs inequality for analytic functions in the unit disk// Carpathian Math. Publ. - 2010. - V.2, №1. - P. 109-118. (in Ukrainian)

14. Gopala Krishna J., Nagaraja Rao I.H. Generalised inverse and probability techniques and some fundamental growth theorems in Ck// J. Indian Math. Soc. - 1977. - V.41. - P. 203-219.

15. Fenton P.C. Wiman-Valiron theory in two variables// Trans. Amer. Math. Soc. - 1995. - V.347, №11. - P. 4403-4412.

16. Kuryliak A.O., Skaskiv O.B. Wimans type inequalities without exceptional sets for random entire functions of several variables// Mat. Stud. - 2012. - V.38, №1. - P. 35-50.

17. Schumitzky A. Wiman-Valiron theory for functions of several complex variables. - Ph. D. Thesis: Cornel. Univ., 1965.

18. Skaskiv O.B., Zrum O.V. WimanЃfs type inequality for entire functions of two complex variables with rapidly oscilating coefficient// Mat. metods and fys.-mekh. polya. - 2005. - V.48, №4. - P. 78-87. (in Ukrainian)

19. Skaskiv O.B., Zrum O.V. On inprovement of Fentons inequality for entire functions of two complex variables// Math. Bull. Shevchenko Sci. Soc. - 2006. - V.3. - P. 56-68. (in Ukrainian)

20. Skaskiv O.B., Trakalo O.M. On classical Wimans inequality for multiple entire Dirichlet series// Mat. metods and fys.-mekh. polya. - 2000. - V.43, №3. - P. 34-39. (in Ukrainian)

21. Zrum O.V., Skaskiv O.B. On Wimans inequality for random entire functions of two variables// Mat. Stud. - 2005. - V.23, №2. - P. 149-160. (in Ukrainian)

22. Kuryliak A.O., Skaskiv O.B. Wimans type inequalities without exceptional sets for random entire functions of several variables// Mat. Stud. - 2012. - V.38, №1. - P. 35-50.

23. Kuryliak A.O., Skaskiv O.B., Zrum O.V. Levys phenomenon for entire functions of several variables// Ufa Math. J. - 2014. - V.6, №2 - P. 118-127.

24. Kuryliak A.O., Skaskiv O.B., Skaskiv S.R. Levys phenomenon for analytic functions in the polydisc// ArXiv:160204756v1 [math.CV] 15 Feb 2016. - 14 p.

25. Kuryliak A.O., Shapovalovska L.O., Skaskiv O.B. Wimans type inequality for analytic functions in the polydisc// Ukr. Mat. J. - 2016. - V.68. - P. 78-86.

26. Kuryliak A.O., Skaskiv O.B., Skaskiv S.R. Analogues of Wimans inequality and Levys phenomenon for analytic functions in bidisc// Buk. Mat. J. - 2015. - V.3, №3-4. - P. 102-110. (in Ukrainian)

27. Kuryliak A.O., Shapovalovska L.O., Skaskiv O.B. Wimans type inequality for some double power series// Mat. Stud. - 2013. - V.39, №2. - P. 134-141.

28. Kuryliak A.O., Shapovalovska L.O. Wimans type inequality for entire functions of several complex varibles with rapidly oscillating coefficients// Mat. Stud. - 2015. - V.43, №1. - P. 16-26.

29. Kuryliak A., Skaskiv O., Tsvigun V. Levys phenomenon for analytic functions in $\mathbb{D}\times\mathbb{C}$// Mat. Stud. - 2016. - V.46, №2. - P. 121-129.

30. Levy P. Sur la croissance de fonctions entiere // Bull. Soc. Math. France. - 1930. - V.58. - P. 29-59; P. 127-149.

31. P.V. Filevych. 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)

32. Filevych P.V. Wiman-Valiron type inequalities for entire and random entire functions of finite logarithmic order// Sib. Mat. Zhurn. - 2003. - V.42, №3. - P. 683-694. (in Russian) English translation in: Siberian Math. J. - 2003. - V.42, №3. - P. 579-586.

33. Kuryliak A.O., Skaskiv O.B. Ineguality of Wiman type for analytic in a disc functions and the Baire categories// Nauk. Visn. Cherniv. Natiomal Univ. Ser.: Mat. - 2011. - V.1, №4. - P. 73-79. (in Ukrainian)

34. Kuryliak A.O., Skaskiv O.B., Chyzhykov I.E. Baire categories and Wiman’s inequality for analytic functions// Bull. Soc. Sc. et des letters de Lodz. – 2012. – V.62, №3. – P. 17–33.

35. Godwin H.J. On generalizations of Tchebycheff’s inequality// J. Amer Stat. Assoc. – 1955. – V.50. – P. 923–945.

36. Savage I.R. Probability inequalities of the Tchebycheff type// J. Research National Bureau of Stand. – 1961. – V.65B, №3. – P. 211–222.

	Wiman’s type inequality for multiple power series in an unbounded cylinder domain
Author	A. O. Kuryliak¹, V. L. Tsvigun² andriykuryliak@gmail.com¹, 12lvan.n@ukr.net² Ivan Franko National University of Lviv, Lviv, Ukraine
Abstract	In this paper we prove some analogues of Wiman’s inequality for analytic $f(z)$ and random analytic functions $f(z,t)$ on $\mathbb T=\mathbb D^l\times\mathbb C^{p-l}$, $l\in\mathbb N,\ 1\leq l\le p$, $I=\{1,\ldots, l\},\ J=\{l+1,\ldots, p\}$ of the form $f(z)=\sum_{\\|n\\|=0}^{+\infty}a_nz^n$, $f(z,t)=\sum_{\\|n\\|=0}^{+\infty}a_nZ_n(t)z^n$ respectively. Here $Z=(Z_n)$ is a multiplicative system of random variables on the Steinhaus probability space, uniformly bounded by the number 1. In particular, there are proved the following statements: For every $\delta>0$ there exist sets $E_1=E_1(\delta,f),\ E_2=E_2(\delta,f)\subset [0,1)^l\times(1,+\infty)^{p-l}$ of asymptotically finite logarithmic measure, such that the inequalities \begin{gather} M_f(r)\leq\mu_f(r)\prod_{i\in I}\frac{1}{(1-r_i)^{1+\delta}}\ln^{{p/2}+\delta}\Bigl(\mu_f(r)\prod_{i\in I}\frac{1}{1-r_i}\Bigl)\Bigl (\prod_{j\in J}\ln r_j\Bigl)^{p+\delta},\\ M_f(r,t)\leq\mu_f(r)\prod_{i\in I}\frac{1}{(1-r_i)^{1/2+\delta}}\ln^{{p/4}+\delta}\Bigl(\mu_f(r)\prod_{i\in I}\frac{1}{1-r_i}\Bigl)\Bigl (\prod_{j\in J}\ln r_j\Bigl)^{ p/2+\delta}. \end{gather} hold for all $r\in T\setminus E_1$ and for all $r\in T\setminus E_2$ a.s. in $t$, respectively. Also is proved sharpness of the obtained inequalities.
Keywords	maximum modulus; maximal term; multiple power series; Wiman’s type inequality
DOI	doi:10.15330/ms.49.1.29-51
Reference	1. Wiman A. Uber dem Zusammenhang zwischen dem Maximalbetrage einer analytischen Function und dem grossten Gliede der zugehorigen Taylorilbr schen Reihe// Acta Math. - 1914. - V.37. - P. 305-326. 2. Polya G., Szego G. Aufgaben und Lehrsatze aus der Analysis, V.2. - Berlin, Springer, 1925. 3. Valiron G. Fonctions analytiques. . Paris: Press. Univer. de France, 1954. 4. Wittich H. Neuere Untersuchungen uber eindeutige analytische Funktionen. - Berlin-Gottingen- Heidelberg: Springer, 1955. - 164 s. 5. Goldberg A.A., Levin B.Y., Ostrovskii I.V., Entire and meromorphic functions. - Itogi Nauki i Tekhniki. Seriya VINITI. - 1990. - V.85. - P. 5-186. (in Russian) 6. Skaskiv O.B., Filevych P.V., On the size of an exceptional set in the Wiman theorem, Mat. Stud. - 1999. - V.12, №1. - P.31-36. (in Ukrainian) 7. Skaskiv O.B., Zrum O.V. On an exeptional set in the Wiman inequalities for entire functions, Mat. Stud. - 2004. - V.21, №1. - 13-24. (in Ukrainian) 8. Skaskiv O.B., Bandura A.I., Asymptotic estimates of positive integrals and entire functions. - Lviv. Ivano-Frankivsk: LNU-INFTUNG, 2015. - 108 p. (in Ukrainian) 9. Erd.os P., Renyi A. On random entire function// Zastosowania mat. - 1969. - V.10. - P. 47-55. 10. Steele J.M., Sharper Wiman inequality for entire functions with rapidly oscillating coefficients// J. Math. Anal. and Appl. - 1987. - V.123. - P. 550-558. 11. K.ovari T. On the maximum modulus and maximal term of functions analytic in the unit disc// J. London Math. Soc. - 1966. - V.41. - P. 129-137. 12. Suleymanov N.M. Wiman-Valirons type inequalities for power series with bounded radii of convergence and its sharpness// DAN SSSR. - 1980. - V.253, №4. - P. 822-824. (in Russian) 13. Skaskiv O.B., Kuryliak A.O. Direct analogues of WimanЃfs inequality for analytic functions in the unit disk// Carpathian Math. Publ. - 2010. - V.2, №1. - P. 109-118. (in Ukrainian) 14. Gopala Krishna J., Nagaraja Rao I.H. Generalised inverse and probability techniques and some fundamental growth theorems in Ck// J. Indian Math. Soc. - 1977. - V.41. - P. 203-219. 15. Fenton P.C. Wiman-Valiron theory in two variables// Trans. Amer. Math. Soc. - 1995. - V.347, №11. - P. 4403-4412. 16. Kuryliak A.O., Skaskiv O.B. Wimans type inequalities without exceptional sets for random entire functions of several variables// Mat. Stud. - 2012. - V.38, №1. - P. 35-50. 17. Schumitzky A. Wiman-Valiron theory for functions of several complex variables. - Ph. D. Thesis: Cornel. Univ., 1965. 18. Skaskiv O.B., Zrum O.V. WimanЃfs type inequality for entire functions of two complex variables with rapidly oscilating coefficient// Mat. metods and fys.-mekh. polya. - 2005. - V.48, №4. - P. 78-87. (in Ukrainian) 19. Skaskiv O.B., Zrum O.V. On inprovement of Fentons inequality for entire functions of two complex variables// Math. Bull. Shevchenko Sci. Soc. - 2006. - V.3. - P. 56-68. (in Ukrainian) 20. Skaskiv O.B., Trakalo O.M. On classical Wimans inequality for multiple entire Dirichlet series// Mat. metods and fys.-mekh. polya. - 2000. - V.43, №3. - P. 34-39. (in Ukrainian) 21. Zrum O.V., Skaskiv O.B. On Wimans inequality for random entire functions of two variables// Mat. Stud. - 2005. - V.23, №2. - P. 149-160. (in Ukrainian) 22. Kuryliak A.O., Skaskiv O.B. Wimans type inequalities without exceptional sets for random entire functions of several variables// Mat. Stud. - 2012. - V.38, №1. - P. 35-50. 23. Kuryliak A.O., Skaskiv O.B., Zrum O.V. Levys phenomenon for entire functions of several variables// Ufa Math. J. - 2014. - V.6, №2 - P. 118-127. 24. Kuryliak A.O., Skaskiv O.B., Skaskiv S.R. Levys phenomenon for analytic functions in the polydisc// ArXiv:160204756v1 [math.CV] 15 Feb 2016. - 14 p. 25. Kuryliak A.O., Shapovalovska L.O., Skaskiv O.B. Wimans type inequality for analytic functions in the polydisc// Ukr. Mat. J. - 2016. - V.68. - P. 78-86. 26. Kuryliak A.O., Skaskiv O.B., Skaskiv S.R. Analogues of Wimans inequality and Levys phenomenon for analytic functions in bidisc// Buk. Mat. J. - 2015. - V.3, №3-4. - P. 102-110. (in Ukrainian) 27. Kuryliak A.O., Shapovalovska L.O., Skaskiv O.B. Wimans type inequality for some double power series// Mat. Stud. - 2013. - V.39, №2. - P. 134-141. 28. Kuryliak A.O., Shapovalovska L.O. Wimans type inequality for entire functions of several complex varibles with rapidly oscillating coefficients// Mat. Stud. - 2015. - V.43, №1. - P. 16-26. 29. Kuryliak A., Skaskiv O., Tsvigun V. Levys phenomenon for analytic functions in $\mathbb{D}\times\mathbb{C}$// Mat. Stud. - 2016. - V.46, №2. - P. 121-129. 30. Levy P. Sur la croissance de fonctions entiere // Bull. Soc. Math. France. - 1930. - V.58. - P. 29-59; P. 127-149. 31. P.V. Filevych. 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) 32. Filevych P.V. Wiman-Valiron type inequalities for entire and random entire functions of finite logarithmic order// Sib. Mat. Zhurn. - 2003. - V.42, №3. - P. 683-694. (in Russian) English translation in: Siberian Math. J. - 2003. - V.42, №3. - P. 579-586. 33. Kuryliak A.O., Skaskiv O.B. Ineguality of Wiman type for analytic in a disc functions and the Baire categories// Nauk. Visn. Cherniv. Natiomal Univ. Ser.: Mat. - 2011. - V.1, №4. - P. 73-79. (in Ukrainian) 34. Kuryliak A.O., Skaskiv O.B., Chyzhykov I.E. Baire categories and Wiman’s inequality for analytic functions// Bull. Soc. Sc. et des letters de Lodz. – 2012. – V.62, №3. – P. 17–33. 35. Godwin H.J. On generalizations of Tchebycheff’s inequality// J. Amer Stat. Assoc. – 1955. – V.50. – P. 923–945. 36. Savage I.R. Probability inequalities of the Tchebycheff type// J. Research National Bureau of Stand. – 1961. – V.65B, №3. – P. 211–222.
Pages	29-51
Volume	49
Issue	1
Year	2018
Journal	Matematychni Studii
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Table of content of issue	html