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Entire functions of several variables of bounded index. (English) Zbl 1342.32001
L’viv: Chyslo; L’viv: I.E. Chyzhykov (ISBN 978-966-2645-15-6/pbk). 128 p. (2016).
By B. Lepson [Proc. Symp. Pure Math. 11, 298–307 (1968; Zbl 0199.12902)], an entire function $$f$$, for which there exists a number $$N\in\mathbb{Z}_{+}$$ such that for all $$p\in\mathbb{Z}_{+}$$ and for all $$z\in\mathbb{C}$$ $\frac{|f^{(p)}(z)|}{p!}\leq\max\left\{\frac{|f^{(k)}(z)|}{k!}: 0\leq k\leq N\right\},$ is called a function of bounded index. This concept is closely connected to value distribution. An entire function $$f(z)$$ is said to have bounded value distribution if there exist constants $$p, R$$ such that the equation $$f(z) = w$$ never has more than $$p$$ roots in any disk of radius $$R$$.
W. K. Hayman [Pac. J. Math. 44, 117–137 (1973; Zbl 0248.30026)] proved that this is the case for a particular $$p$$ and some $$R > 0$$ if and only if $$f'(z)$$ is of bounded index. A survey of results on entire functions of bounded index is due to S. M. Shah [Lect. Notes Math. 599, 117–145 (1977; Zbl 0361.30007)].
A. D. Kuzyk and M. N. Sheremeta [Math. Notes 39, 3–8 (1986; Zbl 0603.30034); translation from Mat. Zametki 39, No. 1, 3–13 (1986)] generalized this concept by introducing the class of entire functions of bounded $$l$$-index, where $$l$$ is a positive continuous function. They obtained analogues of known properties of entire functions of bounded index (see the monograph of M. Sheremeta [Analytic functions of bounded index. Lviv: VNTL Publishers (1999; Zbl 0980.30020)] and references therein).
A definition of an entire function of bounded index in several variables was proposed by J. Gopala Krishna and S. M. Shah in their paper [Math. Essays dedicated to A.J. Macintyre, 223–235 (1970; Zbl 0205.09302)]. Afterwards M. T. Bordulyak and M. M. Sheremeta [“Boundedness of the $$L$$-index of an entire function of several variables” (Ukrainian. English summary), Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky 9, 10–13 (1993)] introduced the concept of an entire functions of bounded $$\mathbf L$$-index in joint variables for entire functions of several complex variables. B. C. Chakraborty and R. Chanda [J. Pure Math. 12, 16–21 (1995; Zbl 1053.32501)], B. C. Chakraborty and T. K. Samanta [J. Pure Math. 18, 53–64 (2001; Zbl 1053.30511)] introduced notions of index boundedness and $$L$$-index boundedness for entire functions of several variables. In particular, they established a necessary and sufficient condition for an entire in $$\mathbb{C}^n$$ function to be of $$L$$-bounded index. The multidimensional case is appeared to be much more complicated. All these approaches do not allow to obtain analogues of the one-dimensional characterization of function of bounded index in terms of behaviour the logarithmic derivative outside zero sets.
In the monograph the authors investigate properties of entire functions of bounded $$L$$-index in direction and of bounded $$\mathbf{L}$$-index in joint variables. Their approach is based on the study of so-called slice functions $$g(\tau)=F(a+b\tau)$$, $$\tau\in\mathbb{C}$$, that are the restrictions of the entire function $$F$$ to a complex line $$\{z=a+b\tau: \tau\in\mathbb{C} \}$$, $$a, b \in\mathbb{C}^n$$. It leads to the following definition.
An entire function $$F(z)$$, $$z\in\mathbb{C}^{n}$$, is called a function of bounded $$L$$-index in a direction $$\mathbf{b}\in \mathbb{C}^{n}\setminus\{\mathbf{0}\}$$ if there exists $$m_{0}\in \mathbb{Z}_{+}$$ such that for every $$m\in\mathbb{Z}_{+}$$ and every $$z\in \mathbb{C}^{n}$$ $\frac{1}{m!L^{m}(z)}\left|\frac{\partial^{m}F(z)}{\partial \mathbf{b}^{m}}\right| \leq\max\left\{\frac{1}{k!L^{k}(z)}\left|\frac{\partial^{k}F(z)}{\partial \mathbf{b}^{k}}\right|: 0\leq k \leq m_{0} \right\}, \tag{1}$ where $$\frac{\partial^{0}F(z)}{\partial \mathbf{b}^{0}}:=F(z)$$, $$\frac{\partial F(z)}{\partial \mathbf{b}}:=\sum_{j=1}^{n}\frac{\partial F(z)}{\partial z_{j}}{b_{j}}$$, $$\frac{\partial^{k}F(z)}{\partial \mathbf{b}^{k}}:=\frac{\partial}{\partial \mathbf{b}}\big(\frac{\partial^{k-1}F(z)}{\partial \mathbf{b}^{k-1}}\big)$$, $$k \geq 2.$$
The least such integer $$m_{0}=m_{0}(\mathbf{b})$$ is called the $$L$$-index in the direction $$\mathbf{b}\in\mathbb{C}^{n}\setminus\{\mathbf{0}\}$$ of the entire function $$F(z)$$ and is denoted by $$N_{\mathbf{b}}(F,L)=m_0.$$
In the case $$n=1$$ we obtain the definition of an entire function of one variable of bounded $$l$$-index; in the case $$n=1$$ and $$L(z)\equiv 1$$ it is reduced to the definition of bounded index.
Let $$BLID_{\mathbf{b}}\equiv$$ “a function of bounded $$L$$-index in the direction $$\mathbf{b}$$.
The material of the monograph, besides the exposition results on entire functions of bounded $$L$$-index in joint variables (Chapter 4), contains investigations of the authors. The following is an outline of the contents of the four chapters. 1. Main properties of $$BLID_{\mathbf{b}}$$ (definitions, examples, properties). 2. Characterizations of function of bounded $$L$$-index in direction (local behaviour of directional derivative, analogue of Hayman’s Theorem, maximum and minimum modulus, behaviour of the directional logarithmic derivative, boundedness of value $$L$$-distribution in direction). 3. Applications of $$BLID_{\mathbf{b}}$$ (bounded $$L$$-index in direction of solutions of partial differential equations, bounded $$L$$-index in direction in a bounded domain, growth and bounded $$L$$-index in direction of entire solutions of partial differential equations, bounded $$L$$-index in direction of some composite functions, $$BLID_{\mathbf{b}}$$ and entire functions with “planar” zeros, existence theorems, growth of entire functions of bounded $$L$$-index in direction). 4. Entire functions of bounded $$\mathbf{L}$$-index in joint variables.
I would recommend this book to any person who is interested in the theory of entire functions of several variables.

##### MSC:
 32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces 32A15 Entire functions of several complex variables 32A17 Special families of functions of several complex variables 32-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to several complex variables and analytic spaces 30D20 Entire functions of one complex variable (general theory) 35B08 Entire solutions to PDEs
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