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.

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.

Reviewer: Igor Chyzhykov (Lviv)

##### 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 |