Entire multivariate vector-valued functions of bounded $\mathbf{L}$-index: analog of Fricke’s theorem

Keywords: bounded index; bounded $\mathbf{L}$-index in joint variables; entire function; maximum modulus; $\sup$-norm; vector-valued function


We consider a class of vector-valued entire functions $F\colon \mathbb{C}^{n}\rightarrow \mathbb{C}^{p}$. For this class of functions there is introduced a concept of boundedness of $\mathbf{L}$-index in joint variables.

Let $|\cdot|_p$ be a norm in $\mathbb{C}^p$. Let $\mathbf{L}(z)=(l_{1}(z),\ldots,l_{n}(z))$, where $l_{j}(z)\colon \mathbb{C}^{n}\to \mathbb{R}_+$ is a positive continuous function.
An entire vector-valued function $F\colon \mathbb{C}^{n}\rightarrow \mathbb{C}^{p}$ is said to be of
bounded $\mathbf{L}$-index (in joint variables), if there exists $n_{0}\in \mathbb{Z}_{+}$ such that

$\displaystyle \forall z\in G \ \ \forall J \in \mathbb{Z}^n_{+}\colon \quad
\frac{|F^{(J)}(z)|_p}{J!\mathbf{L}^J(z)}\leq \max \left \{\frac{|F^{(K)}(z)|_p}{K!\mathbf{L}^K(z)} \colon K\in \mathbb{Z}^n_{+}, \|K\|\leq n_{0} \right \}.$

We assume the function $\mathbf{L}\colon \mathbb{C}^n\to\mathbb{R}^p_+$ such that $0< \lambda _{1,j}(R)\leq\lambda _{2,j}(R)<\infty$ for any $j\in \{1,2,\ldots, p\}$ and $\forall R\in \mathbb{R}_{+}^{p},$
where $\lambda _{1,j}(R)=\inf\limits_{z_{0}\in \mathbb{C}^{p}} \inf \left \{{l_{j}(z)}/{l_{j}(z_{0})}\colon z\in \mathbb{D}^{n}[z_{0},R/\mathbf{L}(z_{0})]\right \},$ $\lambda _{2,j}(R)$ is defined analogously with replacement $\inf$ by $\sup$.
It is proved the following theorem:

Let $|A|_p=\max\{|a_j|\colon 1\leq j\leq p\}$ for $A=(a_1,\ldots,a_p)\in\mathbb{C}^p$. An entire vector-valued function $F$ has bounded $\mathbf{L}$-index in joint variables if and only if for every $R\in \mathbb{R}^{n}_+$ there exist $n_{0}\in \mathbb{Z}_{+}$, $p_0>0$ such that for all $z_{0}\in \mathbb{C}^{n}$ there exists $K_{0}\in \mathbb{Z}_{+}^{n}$, $\|K_0\|\leq n_{0}$, satisfying inequality

\!\max\!\left \{\frac{|F^{(K)}(z)|_p}{K!\mathbf{L}^{K}(z)} \colon \|K\|\leq n_{0},z\in \mathbb{D}^{n}[z_{0},R/\mathbf{L}(z_{0})]\right \}%\leq \nonumber\\
\leq p_{0}\frac{|F^{(K_0)}(z_0)|_p}{K_0!\mathbf{L}^{K_0}(z_0)},

where $\mathbb{D}^{n}[z_{0},R]=\{z=(z_1,\ldots,z_n)\in \mathbb{C}^{n}\colon |z_1-z_{0,1}|<r_{1},\ldots, |z_n-z_{0,n}|<r_{n}\}$ is the polydisc with $z_0=(z_{0,1},\ldots,z_{0,n}),$\ $R=(r_{1},\ldots,r_{n})$.

This theorem is an analog of Fricke's Theorem obtained for entire functions of bounded index of one complex variable.


1. V.P. Baksa, Analytic vector-valued functions in the unit ball having bounded L-index in joint variables, Carpathian Math. Publ., 11 (2019), No2, 213–227. doi:10.15330/cmp.11.2.213-227
2. V.P. Baksa, A.I. Bandura, O.B. Skaskiv, Analogs of Fricke’s theorems for analytic vector-valued functions in the unit ball having bounded L-index in joint variables, Proceedings of IAMM of NASU, 33 (2019), 16–26. doi: 10.37069/1683-4720-2019-33-1
3. V. Baksa, A. Bandura, O. Skaskiv, Growth estimates for analytic vector-valued functions in the unit ball having bounded L-index in joint variables, Constructive Math. Analysis, 3 (2020), No1, 9–19. doi: 10.33205/cma.650977
4. 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., 2018, 22, No2: 223–234. doi: 10.12697/ACUTM.2018.22.18
5. A. Bandura, O. Skaskiv, L. Smolovyk, Slice holomorphic solutions of some directional differential equations with bounded L-index in the same direction, Demonstr. Math., 52 (2019), No1, 482–489. doi: 10.1515/dema-2019-0043
6. A. Bandura, O. Skaskiv, 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), No4, 483–501. doi:10.24193/subbmath.2018.4.06
7. A. Bandura, O. Skaskiv, Entire functions of several variables of bounded index, Lviv: Publisher I.E.Chyzhykov, 2016, 128 p.
8. A. Bandura, O. Skaskiv, Asymptotic estimates of entire functions of bounded L-index in joint variables, Novi Sad J. Math., 48 (2018) No1, 103–116. doi: 10.30755/NSJOM.06997
9. A. Bandura, N. Petrechko, O. Skaskiv, Maximum modulus in a bidisc of analytic functions of bounded L-index and an analogue of Hayman’s theorem, Mat. Bohemica, 143 (2018), No4, 339–354. doi: 10.21136/MB.2017.0110-16
10. A.I. Bandura, O.B. Skaskiv, V.L. Tsvigun, Some characteristic properties of analytic functions in D × C of bounded L-index in joint variables, Bukovyn. Mat. Zh., 6 (2018), No1-2, 21–31.
11. A. Bandura, O. Skaskiv, Analog of Hayman’s Theorem and its application to some system of linear partial differential equations, J. Math. Phys., Anal., Geom., 15 (2019), No2, 170–191. doi: 10.15407/mag15.02.170
12. Lelong P., Gruman L. Entire functions of several complex variables, Springer Verlag, Berlin-Heidelberg, New York-Tokyo, 1986.
13. W.K. Hayman, Differential inequalities and local valency, Pacific J. Math., 44 (1973), No1, 117–137.
14. B. Lepson, Differential equations of infinite order, hyperdirichlet series and entire functions of bounded index, Proc. Sympos. Pure Math., (1968) 2, 298–307.
15. R.F. Patterson, F. Nuray, A characterization of holomorphic bivariate functions of bounded index, Math. Slov., 67 (2017), No3, 731–736. doi: 10.1515/ms-2017-0005
16. F. Nuray, R.F. Patterson, Vector-valued bivariate entire functions of bounded index satisfying a system of differential equations, Mat. Stud., 49 (2018), No1, 67-74. doi: 10.15330/ms.49.1.67-74
17. M. Sheremeta, Analytic functions of bounded index, Lviv: VNTL Publishers, 1999, 141 p.
18. M. Sheremeta, Geometric properties of analytic solutions of differential equations, Lviv: Publisher I. E. Chyzhykov, 2019, 164 p.
How to Cite
Bandura AI, Baksa VP. Entire multivariate vector-valued functions of bounded $\mathbf{L}$-index: analog of Fricke’s theorem. Mat. Stud. [Internet]. 2020Oct.6 [cited 2021Dec.9];54(1):56-3. Available from: http://matstud.org.ua/ojs/index.php/matstud/article/view/131