Bandura A. Properties of classes of holomorphic functions of bounded index

The objects of investigations are classes of entire and analytic functions in the unit ball or in a polydisc of bounded $\mathbf{L}$-index in joint variables index or of bounded $L$-index in direction. These classes are quite wide. This is proved by the existence theorems. Particularly, for an entire function $F: \mathbb{C}^n\to \mathbb{C}$ there exists a positive continuous function $\mathbf{L}:\mathbb{C}^n\to\mathbb{R}^n_+$ such that $F$ has bounded $\mathbf{L}$-index in direction if and only if $F$ has bounded multiplicity of zero points. For entire and analytic functions of several variables in the unit ball of bounded $\mathbf{L}$-index in joint variables there is established characterizations of these classes, existence theorems and other properties which describe local behavior of the partial derivatives on the skeletons of polydisc or estimates of coefficients of power series expansions. Also we have obtained sharp growth estimates of logarithm of maximum modulus for these functions on the skeleton of polydisc. In this way, we improve some growth estimate for functions of one variable proved by M. M. Sheremeta. There are deduced sufficient conditions of boundedness of the $\mathbf{L}$-index in joint variables for entire and analytic solutions in the unit ball of linear higher-order systems of partial differential equations. For these solutions we find sharp estimates of its growth by parameters depending of the coefficients of the system. Developing a new approach in function theory of bounded index a usage ball exhaustion instead polydisc in multidimensional complex space allowed us to discover new characteristic properties of functions of bounded $\mathbf{L}$-index in joint variables by estimate of maximum modulus of the functions and its derivatives on a sphere. As sufficient conditions there are presented analogs of logarithmic criterion for entire functions of bounded $\mathbf{L}$-index in joint variables. The obtained conditions describe behavior of partial logarithmic derivatives in all variables outside some exceptional sets and a measure of zero zet in polydisc. We also examine characterizations of entire and analytic functions in the unit ball of several variables of bounded $L$-index in a direction. For these classes there are sharp growth estimates of logarithm maximum modulus on a circle. Using logarithmic criterion there is selected a subclass of entire functions of unbounded index in any direction such that they have bounded index in each slice $\{z^0+t\mathbf{b}: t\in\mathbb{C}\}$ for a given $z^0\in\mathbb{C}^n.$ For some functions with described property there is a constructed a positive continuous function such that they have bounded $L$-index in the same direction. We fully prove a hypothesis about weaker sufficient conditions of $L$-index boundedness in direction for entire functions (estimate of maximum modulus, minimum modulus on a circle, logarithmic criterion) by replacement of universal quantifier on existential quantifier of some radius. The sufficient conditions of $l$-index boudedness of entire solutions of equation $w'=f(z,w)$ are presented. This is revealed that the condition „$F$ is holomorphic in $\mathbb{C}^n$“ is essential. It can not be replaced by the condition "$F$ is holomorphic on all slices of the form $z^0+t\mathbf{b}$" in the theory of entire functions of bounded $L$-index in the direction.