# Entire functions of bounded index in frame

• A.I. Bandura Ivano-Frankivsk National Tecnical University of OIl and Gas
Keywords: bounded index; bounded $L$-index in direction; entire function; maximum modulus; frame; bounded index in frame

### Abstract

We introduce a concept of entire functions having bounded index in a variable direction, i.e. in a frame.
An entire function $F\colon\ \mathbb{C}^n\to \mathbb{C}$ is called a function of bounded frame index in a frame $\mathbf{b}(z)$,
if~there exists $m_{0} \in\mathbb{Z}_{+}$ such that for every $m \in\mathbb{Z}_{+}$ and for all $z\in \mathbb{C}^{n}$
one has
$\displaystyle \frac{|{\partial^{m}_{\mathbf{b}(z)}F(z)}|}{m!} \leq\max_{0\leq k \leq m_{0}} \frac{|{\partial^{k}_{\mathbf{b}(z)}F(z)}|}{k!},$
where $\partial^{0}_{\mathbf{b}(z)}F(z)=F(z),$
$\partial^{1}_{\mathbf{b}(z)}F(z)=\sum_{j=1}^n \frac{\partial F}{\partial z_j}(z)\cdot b_j(z),$
$\partial^{k}_{\mathbf{b}(z)}F(z)=\partial_{\mathbf{b}(z)}(\partial^{k-1}_{\mathbf{b}(z)}F(z))$ for $k\ge 2$
and $\mathbf{b}\colon\ \mathbb{C}^n\to\mathbb{C}^n$ is a entire vector-valued function.
There are investigated properties of these functions. We established analogs of propositions known for entire functions
of bounded index in direction. The main idea of proof is usage the slice $\{z+t\mathbf{b}(z)\colon\ t\in\mathbb{C}\}$ for given $z\in\mathbb{C}^n.$
We proved the following criterion (Theorem 1) describing local behavior of modulus $\partial_{\mathbf{b}(z)}^kF(z+t\mathbf{b}(z))$ on the circle $|t|=\eta$:

An entire~function
$F\colon\ \mathbb{C}^n\to\mathbb{C}$ is of bounded frame index in the frame $\mathbf{b}(z)$ if and only if
for each $\eta>0$ there exist
$n_{0}=n_{0}(\eta)\in \mathbb{Z}_{+}$ and $P_{1}=P_{1}(\eta)\geq 1$
such that for every $z\in \mathbb{C}^{n}$ there exists $k_{0}=k_{0}(z)\in \mathbb{Z}_{+},$\
$0\leq k_{0}\leq n_{0},$ for which inequality
$\max\{|\partial_{\mathbf{b}(z)}^{k_{0}} F(z+t\mathbf{b}(z))|: |t|\leq\eta \}\leq P_{1}|\partial_{\mathbf{b(z)}}^{k_{0}}F(z)|$
holds.

### References

Bandura A., Skaskiv O., Boundedness of the L-index in a direction of entire solutions of second order partial differential equation, Acta Comment. Univ. Tartu. Math., 22 (2018), no.2, 223–234, doi: 10.12697/ACUTM.2018.22.18

Bandura A. I., Skaskiv O. B., Entire functions of bounded L-index in direction, Mat. Stud., 27 (2007), no. 1, 30–52 (in Ukrainian)

Bandura A., Skaskiv O. Entire functions of several variables of bounded index, Lviv: Publ. I.E. Chyzhykov, 2016, 128 p.

Bandura A., Skaskiv O. Asymptotic estimates positive integrals and entire functions, Lviv-Ivano-Frankivsk, Publ. O.M. Goliney, 2015, 108 p. (in Ukrainian)

Bandura A., Skaskiv O. Analytic functions in the unit Ball. Bounded L-index in joint variables and solutions of systems of PDE’s. – Beau-Bassin: LAP Lambert Academic Publishing, 2017. – 100 p.

Bandura A., Skaskiv O. Functions analytic in the Unit ball having bounded L-index in a direction, Rocky Mountain J. Math., 49 (2019), no. 4, 1063–1092. doi: 10.1216/RMJ-2019-49-4-1063

Bandura, A.I.; Skaskiv, O.B. Sufficient sets for boundedness L-index in direction for entire functions, Mat. Stud., 30 (2008), 177–182.

Bandura A., Skaskiv O., Slice holomorphic functions in several variables with bounded L-index in direction, Axioms, 2019, 8 (2019), no. 3, Article ID 88, doi: 10.3390/axioms8030088

Bandura A., Skaskiv O. Asymptotic estimates of entire functions of bounded L-index in joint variables, Novi Sad J. Math., 48 (2018), no. 1, 103–116. doi: 10.30755/NSJOM.06997

Bandura A., Petrechko N., Skaskiv O. Maximum modulus in a bidisc of analytic functions of bounded L-index and an analogue of Hayman’s theorem, Mat. Bohemica, 143 (2018), no.4, 339–354. doi: 10.21136/MB.2017.0110-16

Bandura A. I., Skaskiv O. B., Tsvigun V. L. The functions of bounded L-index in the collection of variables analytic in D × C, J. Math. Sci., 246 (2020), no. 2, 256–263. https://doi.org/10.1007/s10958-020-04735-y

Bandura A., Skaskiv O. Sufficient conditions of boundedness of L-index and analog of Hayman’s Theorem for analytic functions in a ball, Stud. Univ. Babeş-Bolyai Math. 63 (2018), 483–501. doi: 10.24193/subbmath.2018.4.06

Bandura, A.I. A modified criterion of boundedness of L-index in direction, Mat. Stud. 39 (2013), 99–102.

Fricke, G.H. Entire functions of locally slow growth, J. Anal. Math. 28 (1975), 101–122. doi: 10.1007/BF02786809

Fricke, G.H. Functions of bounded index and their logarithmic derivatives, Math. Ann. 206 (1973), 215–223. doi: 10.1007/BF01429209

Kuzyk A. D., Sheremeta M. N. Entire functions of bounded l-distribution of values, Math. Notes, 39 (1986), no. 1, 3–8, doi: 10.1007/BF01647624

Lepson B. Differential equations of infinite order, hyperdirichlet series and entire functions of bounded index, Proc. Sympos. Pure Math., 2 (1968), 298–307

Macdonnell J. J., Some convergence theorems for Dirichlet-type series whose coefficients are entire functions of bounded index, Doctoral dissertation, Catholic University of America, Washington, USA, 1957.

Nuray F., Patterson R. F. Vector-valued bivariate entire functions of bounded index satisfying a system of differential equations, Mat. Stud., 49 (2018), no. 1, 67–74, doi: 10.15330/ms.49.1.67-74

Nuray F., Patterson R.F. Multivalence of bivariate functions of bounded index, Le Matematiche, 70 (2015), 225–233. doi: 10.4418/2015.70.2.14

Sheremeta M., Analytic functions of bounded index, Lviv, VNTL Publishers, 1999.

Published
2020-12-25
How to Cite
1.
Bandura A. Entire functions of bounded index in frame. Mat. Stud. [Internet]. 2020Dec.25 [cited 2021Oct.16];54(2):193-02. Available from: http://matstud.org.ua/ojs/index.php/matstud/article/view/147
Issue
Section
Articles