Search

In the paper we investigate slice holomorphic functions F : ℂn → ℂ having bounded L-index in a direction, i.e. these functions are entire on every slice {z0 + tb : t ∈ℂ} for an arbitrary z0 ∈ℂn and for the fixed direction b ∈ℂn \ {0}, and (∃m0 ∈ ℤ+) (∀m ∈ ℤ+) (∀z ∈ ℂn) the following inequality holds

We investigate the slice holomorphic functions of several complex variables that have a bounded L-index in some direction and are entire on every slice {[z.sup.0] + tb : t [member of] C} for every [z.sup.0] [member of] [C.sup.n] and for a given direction b [member of] [C.sup.n] \ {0}. For this class of functions, we prove some criteria of boundedness of the L-index in direction describing a local behavior of the maximum and minimum moduli of a slice holomorphic function and give estimates of the logarithmic derivative and the distribution of zeros. Moreover, we obtain analogs of the known Hayman theorem and logarithmic criteria. They are applicable to the analytic theory of differential equations. We also study the value distribution and prove the existence theorem for those functions. It is shown that the bounded multiplicity of zeros for a slice holomorphic function F : [C.sup.n] [right arrow] C is the necessary and sufficient condition for the existence of a positive continuous function L : [C.sup.n] [right arrow] [R.sub.+] such that F has a bounded L-index in direction. Keywords. Bounded index, bounded L-index in direction, slice function, holomorphic function, maximum modulus, minimum modulus, bounded l-index, existence theorem, distribution of zeros, logarithmic derivative, directional derivative., 1. Introduction The paper is a sequel of [1]. There, the concept of L-index boundedness in direction for slice entire functions of several complex variables was introduced, and some criteria [...]

For a given direction b ∈ C n \ {0} we study non-homogeneous directional linear higher-order equations whose all coefficients belong to a class of joint continuous functions which are holomorphic on intersection of all directional slices with a unit ball. Conditions are established providing boundedness of L-index in the direction with a positive continuous function L satisfying some behavior conditions in the unit ball. The provided conditions concern every solution belonging to the same class of functions as the coefficients of the equation. Our considerations use some estimates involving a directional logarithmic derivative and distribution of zeros on all directional slices in the unit ball. [ABSTRACT FROM AUTHOR]

We give negative answer to the question of Bordulyak and Sheremeta for more general classes of entire functions than in the original formulation: Does index boundedness in joint variables for an entire function F imply index boundedness in the variable zj for the function F? This question is addressed for entire functions of bounded L-index in joint variables and entire functions of bounded L-index in direction. We also present a class of analytic functions in the unit ball which has bounded L-index in joint variables and has unbounded l-index in the variables z1 and z2 for any positive continuous function l : B2 → C.

We obtain the sufficient conditions of boundedness of L-index in joint variables for analytic functions in the unit ball, where L : [C.sup.n] [right arrow] [R.sup.n.sub.+] is a continuous positive vector-function. They give an estimate of the maximum modulus of an analytic function by its minimum modulus on a skeleton in a polydisc and describe the behavior of all partial logarithmic derivatives outside some exceptional set and the distribution of zeros. The deduced results are also new for analytic functions in the unit disc of bounded index and l-index. They generalize known results by G. H. Fricke, M. M. Sheremeta, A. D. Kuzyk, and V. O. Kushnir. Keywords. Analytic function, unit ball, bounded L-index in joint variables, maximum modulus, partial derivative, minimum modulus, distribution of zeros, skeleton of polydisc., 1. Introduction A concept of an entire function of bounded index arose in the analytic theory of differential equations. It appeared in the work by B. Lepson [24]. An entire [...]

The concept of boundedness of the L-index in joint variables (see Mat. Stud., 45, 12-26 (2016), dx.doi.org/10.15330/ms.45.1.12-26) is generalized for a function analytic in a ball. The criteria of boundedness of the L-index in joint variables, which describe the local behavior of partial derivatives on the skeleton of a polydisc, are proved. Keywords. Analytic function, ball, bounded L-index in joint variables, maximum modulus, partial derivative., 1. Introduction The concept of an entire function of bounded index appeared in the work by B. Lepson [23]. An entire function f is said to be of bounded index, [...]

Our results concern growth estimates for vector-valued functions of L-index in joint variables which are analytic in the unit ball. There are deduced analogs of known growth estimates obtained early for functions analytic in the unit ball. Our estimates contain logarithm of sup-norm instead of logarithm modulus of the function. They describe the behavior of logarithm of norm of analytic vector-valued function on a skeleton in a bidisc by behavior of the function L: These estimates are sharp in a general case. The presented results are based on bidisc exhaustion of a unit ball. [ABSTRACT FROM AUTHOR]

We used the analog of known Hayman's theorem to study the boundedness of L-index in joint variables of entire solutions of some linear higher-order systems of PDE's and found sufficient conditions providing the boundedness, where L(z) = (l1(z), ..., ln(z)), lj: Cn → ℝ+ is a continuous function j ∈ {1, ..., n}. Growth estimates of these solutions are also obtained. We proposed the examples of systems of PDE's which prove the exactness of these estimates for entire solutions. The obtained results are new even for the one-dimensional case because of the weakened restrictions imposed on the positive continuous function l. [ABSTRACT FROM AUTHOR]

We generalize some criteria of boundedness of L-index in joint variables for analytic in an unit ball functions. Our propositions give an estimate maximum modulus of the analytic function on a skeleton in polydisc with the larger radii by maximum modulus on a skeleton in the polydisc with the lesser radii. An analog of Hayman's Theorem for the functions is obtained. Also we established a connection between class of analytic in ball functions of bounded lj -index in every direction 1j, j ∊ {1, … , n} and class of analytic in ball of functions of bounded L-index in joint variables, where L(z) = (l1(z), … , ln(z)), lj : Bn ∊ R+ is continuous function, 1j = (0, … , 0, 1 j..th place 0, … , 0) ∊ Rn+, z ∊ Cn. [ABSTRACT FROM AUTHOR]

We generalize some criteria of boundedness of L-index in joint variables for in a bidisc analytic functions. Our propositions give an estimate the maximum modulus on a skeleton in a bidisc and an estimate of (p + 1)th partial derivative by lower order partial derivatives (analogue of Hayman's theorem). [ABSTRACT FROM AUTHOR]

Let b ∈ C n \ { 0 } be a fixed direction. We consider slice holomorphic functions of several complex variables in the unit ball, i.e., we study functions that are analytic in the intersection of every slice { z 0 + t b : t ∈ C } with the unit ball B n = { z ∈ C : | z | : = | z | 1 2 + ... + | z n | 2 < 1 } for any z 0 ∈ B n . For this class of functions, there is introduced a concept of boundedness of L-index in the direction b , where L : B n → R + is a positive continuous function such that L (z) > β | b | 1 − | z | , where β > 1 is some constant. For functions from this class, we describe a local behavior of modulus of directional derivatives on every 'circle' { z + t b : | t | = r / L (z) } with r ∈ (0 ; β ] , t ∈ C , z ∈ C n . It is estimated by the value of the function at the center of the circle. Other propositions concern a connection between the boundedness of L-index in the direction b of the slice holomorphic function F and the boundedness of l z -index of the slice function g z (t) = F (z + t b) with l z (t) = L (z + t b). In addition, we show that every slice holomorphic and joint continuous function in the unit ball has a bounded L-index in direction in any domain compactly embedded in the unit ball and for any continuous function L : B n → R +. [ABSTRACT FROM AUTHOR]