Bitlyan-Gol'dberg type inequality for entire functions and diagonal maximal term

  • A. O. Kuryliak Ivan Franko National University of Lviv, Lviv, Ukraine
  • O. B. Skaskiv Ivan Franko National University of Lviv
  • S. I. Panchuk Ivan Franko National University of Lviv
Keywords: convergence classes; entire functions; several complex variables

Abstract

In the article is obtained an analogue of Wiman-Bitlyan-Gol'dberg type inequality for entire $f\colon\mathbb{C}^p\to \mathbb{C}$ from the class $\mathcal{E}^{p}(\lambda)$ of functions represented by gap power series of the form
$$
f(z)=\sum\limits_{k=0}^{+\infty} P_k(z),\quad
z\in\mathbb{C}^p.
$$
Here $P_0(z)\equiv a_{0}\in\mathbb{C},$ $
P_k(z)=\sum_{\|n\|=\lambda_k} a_{n}z^{n}$ is homogeneous
polynomial of degree $\lambda_k\in\mathbb{Z}_+,$ ànd $ 0=\lambda_0<
\lambda_k\uparrow +\infty$\ $(1\leq k\uparrow +\infty ),$
$\lambda=(\lambda_k)$.\ We consider the exhaustion of the
space\ $\mathbb{C}^{p}$\
by the system $(\mathbf{G}_{r})_{r\geq 0}$ of a bounded complete multiple-circular domains $\mathbf{G}_{r}$
with the center at the point $\mathbf{0}=(0,\ldots,0)\in \mathbb{C}^{p}$. Define $M(r,f)=\max\{|f(z)|\colon z\in\overline{G}_r\}$, $\mu(r,f)=\max\{|P_k(z))|\colon z\in\overline{G}_r\}$.
Let $\mathcal{L}$ be the class of positive continuous functions $\psi\colon \mathbb{R}_{+}\to\mathbb{R}_{+}$ such that $\int_{0}^{+\infty}\frac{dx}{\psi(x)}<+\infty$, $n(t)=\sum_{\lambda_k\leq t}1$ counting function of the sequence $(\lambda_k)$ for $t\geq 0$. The following statement is proved:
{\it If a sequence $\lambda=(\lambda_{k})$ satisfy the condition
\begin{equation*}
(\exists p_1\in (0,+\infty))(\exists t_0>0)(\forall t\geq t_0)\colon\quad n(t+\sqrt{\psi(t)})-n(t-\sqrt{\psi(t)})\leq t^{p_1}
\end{equation*}
for some function $\psi\in \mathcal{L}$,
then for every entire function $f\in\mathcal{E}^{p}(\lambda)$, $p\geq 2$ and for any
$\varepsilon>0$ there exist a constant $C=C(\varepsilon, f)>0$ and a set $E=E(\varepsilon, f)\subset [1,
+\infty)$ of finite logarithmic measure such that the inequality
\begin{equation*}
M(r, f)\leq C m(r,
f)(\ln m(r, f))^{p_1}(\ln\ln m(r, f))^{p_1+\varepsilon}
\end{equation*}
holds for all $ r\in[1,
+\infty]\setminus E$.}

The obtained inequality is sharp in general.
At $\lambda_k\equiv k$, $p=2$ we have $p_1=1/2+\varepsilon$ and the Bitlyan-Gol'dberg inequality (1959) it follows. In the case $\lambda_k\equiv k$, $p=2$ we have $p_1=1/2+\varepsilon$ and from obtained statement we get the assertion on the Bitlyan-Gol'dberg inequality (1959), and at $p=1$ about the classical Wiman inequality it follows.

Author Biography

A. O. Kuryliak, Ivan Franko National University of Lviv, Lviv, Ukraine

Department of Mechanics and Mathematics, Associate Professor

References

Bitlyan I.F., Goldberg A.A. Wiman-Valiron’s theorem for entire functions of several complex variables// Vestn. Leningrad univ. ser. mat., mech. and astr. – 1959. – V.2, No13. – P. 27–41. (in Russian)

O.B. Skaskiv, On the classical Wiman’s inequality for entire Dirichlet series, Visn. Lviv. Univer., ser. mekh.-mat. – 1999. – V.54. – P. 180–182. (in Ukrainian)

M.M. Sheremeta, Wiman-Valiron’s method for entire functions, represented by Dirichlet series, Dokl. USSR Acad. Sci. – 1978. – V.240, No5. – P. 1036–1039. (in Russian)

Skaskiv O.B. Random gap power series and Wiman’s inequality// Mat. Stud. – 2008. – V.30, No1. – P. 101–106. (in Ukrainian)

Valiron G. Functions analytiques. – Paris: Press Univer. de France, 1954.

Wittich H. Neuer Untersuchungen über eindeutige analytische Funktionen. – Berlin-Göttingen-

Heidelberg: Springer-Verlag, 1955.

Kuryliak A.O., Ovchar I.E., Skaskiv O.B. Wiman’s inequality for Laplace integrals// Int. Journal of Math. Analysis. – 2014. – V.8, No8. – P. 381–385.

Schumitsky A. Wiman-Valyron theory for entire functions of several complex variables. – Ph.D. Thesis, Cornell Univ., 1965.

Shumitzky A. A probabilistic approach to the Wiman-Valiron’s theory for entire functions of several complex variables// Complex Variables. – 1989. – V.13. – P. 85–98.

Fenton P.C. Wiman-Valiron theory in two variables// Trans. Amer. Math. Soc. – 1995.– V.347, No11. – P. 4403–4412.

Skaskiv O.B., Trakalo O.M. On classical Wiman’s inequality for entire Dirichlet series// Mat. metods and fys.-mekh. polya. – 2000. – V.43, No3. – P. 34–39. (in Ukrainian)

Zrum O.V., Skaskiv O.B. On Wiman’s inequality for random entire functions of two variables// Mat. Stud. – 2005. – V.23, No2. – P. 149–160. (in Ukrainian)

Skaskiv O.B., Zrum O.V. Wiman-type inequalities for entire functions of two complex variables with rapidly oscillating coefficients// Mat. Metody Phys.-Mekh. Polya. – 2005. – V.48, No4. – P. 78–87. (in

Ukrainian)

Skaskiv O.B., Zrum O.V. Refinement of Fenton’s inequality for entire functions of two complex variables//

Mat. Visn. Nauk. Tov. Im. Shevchenka. – 2006. – V.3. – P. 56–68. (in Ukrainian)

Gopala Krishna J. Generalised inverse and probability techniques and some fundamental growth theorems

in Ck // J. Indian Math. Soc. – 1977. – V.41. – P. 203–219.

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

O.B. Skaskiv, Random gap power series and Wiman’s inequality// Mat. Stud. – 2008. – V.30, No1. – P. 101–106. (in Ukrainian)

Kuryliak A.O., Skaskiv O.B. Wiman’s type inequalities without exceptional sets for random entire functions of several variables// Mat. Stud. – 2012. – V.38, No1. – P. 35–50.

Skaskiv O. B., Kuryliak A. O. Maximum modulus of entire functions of two variables and arguments of coefficients of double power series// Mat. Stud. – 2011. – V.36, No2. – P. 162–175.

Zrum O.V., Kuryliak A.O., Skaskiv O.B. Levy’s phenomenon for entire functions of several variables// arXiv: 1307.6164v1 [math.CV] 23 Jul 2013. – 14 p.

Kuryliak A.O., Skaskiv O.B., Zrum O.V. Levy’s phenomenon for entire functions of several variables// Ufa Math. J. – 2014. – Т.6, No2. – P. 111–120.

Kuryliak A.O., Shapovalovska L.O., Skaskiv O.B. Wiman’s type inequality for some double power series// Mat. Stud. – 2013. – V.39, No2. – P. 134–141.

Kuryliak A.O., Shapovalovska L.O. Wiman’s type inequality for entire functions of several complex variables with rapidly oscillating coefficients// Mat. Stud. – 2015. – V.43, No1. – P. 16–26. doi: 10.15330/ms.43.1.16-26

Kuryliak A.O., Skaskiv O.B., Shapovalovska L.O. Wiman’s inequality for analityc functions in a bydisc// Visn. Lviv. Univ. – 2014. – Vyp.79. –P. 89–96 (2014) (in Ukrainian)

Kuryliak A.O., Shapovalovska L.O., Skaskiv O.B. Wiman’s type inequality for analytic functions in a polydisc// Ukr. Mat. Zh. – 2016. –V.68, No.1. –P. 78–86 (2016) (in Ukrainian)

Kuryliak A.O., Tsvigun V.L. Wiman’s type inequality for multiple power series in an unbounded cylinder domain// Mat. Stud. – 2018. – V.49, No1. – P. 29–51.

Kuryliak A., Skaskiv O., Skaskiv S. Levy’s phenomenon for analytic functions on a polydisk// European Journal of Mathematics. – 2020. – V.6, No1. – P. 138–152 doi.org/10.1007/s40879-019-00363-2

Panchuk S.I., Skaskiv O.B. Lacunary multiple power series and Wiman’s inequality// International conference dedicated to the 120th anniversary of Stefan Banach (Lviv, 17.09—21.09.2012): Abstract of Reports. – Lviv, 2012. — P. 172–173.

Skaskiv O.B. On certain relations between the maximum modulus and the maximal term of an entire Dirichlet series, Math. Notes. – 1999. – V.66, No2. – P. 223–232. Transl. from Mat. Zametky. – 1999.V.66, No2. – P. 282–292.

Skaskiv O.B., Zrum O.V., On an exeptional set in the Wiman inequalities for entire functions// Mat. Stud. – 2004. – V.21, No1. – P. 13–24. (in Ukrainian)

Skaskiv O.B., Filevych P.V., On the size of an exceptional set in the Wiman theorem, Mat. Stud. – 1999. – V.12, No1. – P. 31–36. (in Ukrainian)

Published
2020-12-25
How to Cite
1.
Kuryliak AO, Skaskiv OB, Panchuk SI. Bitlyan-Gol’dberg type inequality for entire functions and diagonal maximal term. Mat. Stud. [Internet]. 2020Dec.25 [cited 2021Jan.24];54(2):135-4. Available from: http://matstud.org.ua/ojs/index.php/matstud/article/view/107
Section
Articles