Search

Using recent estimates of maximum modulus for partial derivatives of the analytic functions with bounded L-index in joint variables we describe maximum modulus of these functions at the polydisc skeleton with given radii by the maximum modulus with lesser radii. Such a description is sufficient and necessary condition of boundedness of L-index in joint variables for functions which are analytic in a complete Reinhardt domain. The vector-valued function L is a positive and continuous function in the domain and its values at a point is greater than reciprocal of distance from the point to the boundary of the Reinhardt domain multiplied by some constant. [ABSTRACT FROM AUTHOR]

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 [...]

There is introduced a concept of index for the Fueter regular function of the quaternionic variables. There are considered three approaches (Fueter, Sudbery and Mariconda) constructing the Fueter regular function from a holomorphic function of complex variable. Using Mariconda's approach there are constucted some analogs of such elementary functions as the exponent, the sine and the cosine. For the Mariconda analogs we proved that they have bounded index and their indices equal 1, 2, 2, respectively. Using recent results on sum of entire functions whose derivatives are of bounded index it is established that the Fueter regular function constructed by Mariconda's approach is of bounded index, if the derivatives of its addends have bounded index. Also there was examined a function of the form H(q) = f1(x0 + ix1) + jf2(x2 + ix3), where f1 and f2 are entire functions of complex variable. For the function H it is proved its Fueter regularity and index boundedness if the first order derivatives of f1 and f2 have bounded index. Moreover, the index of the function H does not exceed the maximum of indices of the functions f'1 and f'2 increased by 1. [ABSTRACT FROM AUTHOR]

In the paper, we formulate some open problems related to the best description of the values of the exceptional sets in Wiman's inequality for entire functions and in the Erdos-Macintyre type theorems for entire multiple Dirichlet series. At the same time, we clarify the statement of one I.V. Ostrovskii problem on Wiman's inequality. We also prove three propositions and one theorem. On the one hand, in a rather special case, these results give the best possible description of the values of the exceptional set in the Erdos-Macintyre-type theorem. On the second hand, they indicate the possible structure of the best possible description in the general case. [ABSTRACT FROM AUTHOR]

We study a composition of two functions belonging to a class of slice holomorphic functions in the whole n-dimensional complex space. The slice holomorphy in the space means that for some fixed direction b ∈ Cn\ {0} and for every point z0 ∈ Cn the function is holomorphic on its restriction on the slice {z0 + tb : t ∈ C}: An additional assumption on joint continuity for these functions allows to construct an analog of theory of entire functions having bounded index. The analog is applicable to study properties of slice holomorphic solutions of directional differential equations, describe local behavior and value distribution. In particular, we found conditions providing boundedness of L-index in the direction b for a function ..., where f : Cn → C is a slice entire function, ... : Cn → C is a slice entire function, L : Cn → ℝ+ is a continuous function. The obtained results are also new in one-dimensional case, i.e. for n = 1, m = 1: They are deduced using new approach in this area analog of logarithmic criterion. For a class of nonvanishing outer functions in the composition the sufficient conditions obtained by logarithmic criterion are weaker than the conditions by the Hayman theorem. [ABSTRACT FROM AUTHOR]

Let b ∈ Cn / {0} be a fixed direction. We consider slice holomorphic functions of several complex variables in the unit ball, i.e. we study functions which are analytic in intersection of every slice {z0 + tb : t ∈ C} with the unit ball Bn = {z ∈ C |z| : = √|z|¹² + ... + |zn|² < 1} for any z0 ∈ Bn. For this class of functions we consider the concept of boundedness of L-index in the direction b, where L : Bn → R+ is a positive continuous function such that L(z) > β|b|/1-|z| and β > 1 is some constant. For functions from this class we deduce analog of Hayman's Theorem. It is criterion useful in applications to differential equations. We introduce a concept of function having bounded value L-distribution in direction for the slice holomorphic functions in the unit ball. It is proved that slice holomorphic function in the unit ball has bounded value L-distribution in a direction if and only if its directional derivative has bounded L-index in the same direction. Other propositions concern existence theorems. We show that for any slice holomorphic function F with bounded multiplicities of zeros on any slice in the fixed direction there exists such a positive continuous function L that the function F has bounded L-index in the direction. [ABSTRACT FROM AUTHOR]

We introduce a concept of asymptotic vector of an entire curve with linearly independent components and without common zeros and investigate a relationship between the asymptotic vectors and the Picard exceptional vectors. A non-zero vector a = (a1, a2,..., ap) ∈ Cp is called an asymptotic vector for the entire curve ⃗ G(z) = (g¹(z), g2(z),..., gp(z)) if there exists a continuous curve L: R+ → C given by an equation z = z (t), 0 ≤ t < ∞, |z (t)| < ∞, z (t) → ∞ as t → ∞ such that lim z→∞ z∈L ⃗ G(z)⃗a/∥ ⃗ G(z)∥ = limt→∞ ⃗ G(z(t))⃗a/⃗ ∥G(z(t))∥ = 0, where ∥ ⃗ G(z)∥ = 􀀀 |g1(z)|² +... + |gp(z)|² |1/2, ⃗ G(z)⃗a = g1(z) · ¯a1 + g2(z) · ¯a2 +... + gp(z) · ¯ap. A non-zero vector ⃗a = (a1, a2,..., ap) ∈ Cp is called a Picard exceptional vector of an entire curve ⃗ G(z) if the function ⃗ G(z)⃗a has a finite number of zeros in {|z| < ∞}. We prove that any Picard exceptional vector of transcendental entire curve with linearly independent components and without common zeros is an asymptotic vector. Here we demonstrate that the exceptional vectors in the sense of Borel or Nevanlina and, moreover, in the sense of Valiron do not have to be asymptotic. For this goal we use an example of meromorphic function of finite positive order, for which ∞ is no asymptotic value, but it is the Nevanlinna exceptional value. This function is constructed in known Goldberg and Ostrovskii's monograph "Value Distribution of Meromorphic Functions". Other our result describes sufficient conditions providing that some vectors are asymptotic for transcendental entire curve of finite order with linearly independent components and without common zeros. In this result, we require that the order of the Nevanlinna counting function for this curve and for each such a vector is less than order of the curve. At the end of paper we formulate three unsolved problems concerning asymptotic vectors of entire curve. [ABSTRACT FROM AUTHOR]

We have obtained a description of structure of the sets of Picard and Borel exceptional vectors for transcendental entire curve in some sense. We consider only p-dimensional entire curves with linearly independent components without common zeros. In particular, the set of Borel exceptional vectors together with the zero vector is a finite union of subspaces in Cp of dimension at most p - 1. Moreover, the sum of their dimensions does not exceed p if any pairwise intersection of the subspaces contains only the zero vector. A similar result is also valid for the set of Picard exceptional vectors. Another result shows that the structure of the set of Borel exceptional vectors for an entire curve of integer order differs somewhat from the structure of such a set for an entire curve of non-integer order. For a transcendental entire curve ⃗G: C → Cp with linearly independent components and without common zeros having non-integer or zero order the set of Borel exceptional vectors together with the zero vector is a subspace in Cp of dimension at most p - 1. However, the set of Picard exceptional vectors does not possess this property. We propose two examples of entire curves. The first example shows the set of Borel exceptional vectors together with the zero vector for p-dimensional entire curve of integer order is union of subspaces of dimension at most p - 1 such that the total sum of these dimensions does not exceed p and intersection of any pair of these subspaces contains only zero vector. The set of Picard exceptional vectors for the curve has the same property. In the second example, we construct a q-dimensional entire curve of non-integer order for which the set of Borel exceptional vectors together with the zero vector is a subspace in Cq of dimension at most q - 1 and the set of Picard exceptional vectors together with the zero vector do not have the property. This set is a union of some subspaces. [ABSTRACT FROM AUTHOR]

We study the following question: "Let f: ... be an entire function of bounded l-index, ... an be entire function, ... be a continuous function. What is a positive continuous function ... and a direction ... such that the composite function ... has bounded L-index in the direction b" In the present paper, early known result on boundedness of L-index in direction for the composition of entire functions ... is modified. We replace a condition that a directional derivative of the inner function in a direction b does not equal zero. The condition is replaced by a construction of greater function L(z) for which ... has bounded L-index in a direction. We relax the condition ... It is replaced by the condition... is the l-index of the function f. The described result is an improvement of previous one. [ABSTRACT FROM AUTHOR]

We introduce a concept of entire functions having bounded index in a variable direction, i.e. in a frame. An entire function F : Cn C is called a function of bounded frame index in a frame b(z), if there exists m0 Є Z+ such that for every m Є Z+ and for all z Є Cn one has |dm)F(z)| |db()F(z)| (z) < max >, where db(0 2 and b: Cn Cn is a entire vector-valued function. There are investigated properties of these functions. We established analogs of propositions known for entire functions of bounded index in direction. The main idea of proof is usage the slice {z + tb(z): t Є C} for given z Є Cn. We proved the following criterion (Theorem 1) describing local behavior of modulus db(z)F(z + tb(z)) on the circle |t| = η: An entire function F : Cn C is of bounded frame index in the frame b(z) if and only if for each η > 0 there exist n0 = n0(n) Є Z+ and Pi = Ρ-ί(η) > 1 such that for every z Є Cn there exists k0 = k0(z) Є Z+, 0 < k0 < n0, for which inequality max db0z) F (z + tb(z))| |t|< η} < Pi|db0z)F(z)| holds. [ABSTRACT FROM AUTHOR]

We consider a class of vector-valued entire functions F: ℂ n → ℂp. For this class of functions there is introduced a concept of boundedness of l-index in joint variables. Let / ⋅ /p be a norm in ℂp. Let L(z) = (l1(z),..., ln(z)), where lj(z). ℂn → ℝ+ is a positive continuous function. An entire vector-valued function F: ℂn → ℂp is said to be of bounded L-index (in joint variables), if there exists n0 ∈ ℤ+ such that.... We assume the function L: ℂ n → ℝ+p such that 0 < λ1,j (R) ≤ λ 2,j (R) < ∞ for any j ∈ {1, 2,...,p} and ∀R ∈ ℝ+p, where λ 1,j (R) = inf inf {lj (z)/lj (z0): z ∈ ⅅn [z0, R/L(z0)]}, λ 2,j (R) is defined analogously with replacement inf by sup. It is proved the following theorem: Let /A/p = max max{/aj/ : 1 ≤ j ≤ p} for A = (a1,..., ap) ∈ ℂp. An entire vector-valued function F has bounded L-index in joint variables if and only if for every ℝ ∈ ℝ + n there exist n0 ∈ ℤ+, p0 > 0 such that for all z0 ∈ ℂn there exists K0 ∈ ℤ+n, ... This theorem is an analog of Fricke's Theorem obtained for entire functions of bounded index of one complex variable. [ABSTRACT FROM AUTHOR]

It is proved that the structure of the set of Borel exceptional vectors for transcendental entire curve something likes structure of the set of Nevanlinna deficient vectors for entire curve of finite order. Particularly, any admissible system of Borel exceptional vectors for every p-dimensional transcendental entire curve with linearly independent components and without common zeros cannot have more than p vectors. [ABSTRACT FROM AUTHOR]

In this paper we propose an approach to introduce a concept of bounded index in an infinite-dimensional space. Our object of investigation is the space ℓ∞ equipped with the norm ∥x∥∞ = sup{|xn|: n ∈ N}. We consider entire curves from C to ℓ∞ and prove a proposition indicating connection between of the l-index boundedness of every component of the curve and the l-index boundedness of the curve. Moreover, we obtain sufficient conditions of the l-index boundedness of entire curves in the space. They describe local behavior of norm of derivatives of the entire curves on the discs. Also, there is posed a problem on necessary conditions of the l-index boundedness of entire curves in infinite-dimensional spaces. [ABSTRACT FROM AUTHOR]

We partially reinforce some criteria of L-index boundedness in direction for functions analytic in the unit ball. These results describe local behavior of directional derivatives on the circle, estimates of maximum modulus, minimum modulus of analytic function, distribution of its zeros and modulus of directional logarithmic derivative of analytic function outside some exceptional set. Replacement of universal quantifier on existential quantifier gives new weaker sufficient conditions of L-index boundedness in direction for functions analytic in the unit ball. The results are also new for analytic functions in the unit disc. The logarithmic criterion has applications in analytic theory of differential equations. This is convenient to investigate index boundedness for entire solutions of linear differential equations. It is also apllicable to infinite products. Auxiliary class of positive continuous functions in the unit ball (so-denoted Qb(Bn)) is also considered. There are proved some characterizing properties of these functions. The properties describe local behavior of these functions in the polydisc neighborhood of every point from the unit ball. [ABSTRACT FROM AUTHOR]

In the paper, we present sufficient conditions of index boundedness in joint variables for a sum of entire functions. Our propositions are generalizations of Pugh's result obtained for entire functions of one variable of bounded index. Also we give an example of two functions of bounded index in joint variables such that its sum are not bounded index in joint variables. Thus, a class of entire functions of bounded index in joint variables is not closed under the operation of addition of the functions. [ABSTRACT FROM AUTHOR]

We present a generalization of concept of bounded value L-distribution in direction b∈Cn\{0} for analytic functions in the unit ball Bn⊂Cn, where L:Bn→R+ is a continuous function. It is established a connection between the class of analytic functions in the unit ball of bounded L-index in the direction b and the class of analytic functions in the unit ball of bounded value L-distribution in the same direction. The main result of the paper is the following: An analytic function F:Bn→C is a function of bounded value L-distribution in a direction b∈Cn\{0} if and only if its directional derivative ∂F/∂b has bounded L-index in the same direction b. [ABSTRACT FROM AUTHOR]

We partially proved a conjecture from Mat. Stud. 47 (2017), no.2, 207--210: for an entire function f the function H(z)=f(1/(1-z)n), n∈N, is of bounded l-index in C∖{0} with l(|z|)=β/(1-|z|)n+1, β>1, if and only if f is of bounded index. Also the boundedness of l-M-index of the function H is investigated. For arbitrary entire functions f and g the boundedness of the l-M-index of the function F(z)=f(g(z)) is studied with respect to boundedness of the M-index of a function f with l(r)=M′g(r), Mg(r)=max{|g(z)|:|z|=r}. [ABSTRACT FROM AUTHOR]

We have generalized some criteria of boundedness of L-index in joint variables for analytic functions in the unit ball, where L:Bn→R+n is a continuous vector-function, Bn is the unit ball in Cn. One of propositions gives an estimate of the coefficients of power series expansions by a dominating homogeneous polynomial for analytic functions in the unit ball. Also we provide growth estimates of these functions. They describe the behavior of maximum modulus of analytic function on a skeleton in a polydisc by behavior of the function L. Most of our results are based on polydisc exhaustion of the unit ball. Nevertheless, we have generalized criteria of boundedness of L-index in joint variables which describe local behavior of partial derivatives on sphere in Cn. The proposition uses a ball exhaustion. An analog of Hayman's theorem is applied to investigation of boundedness of L-index in joint variables for analytic solutions in the unit ball of some linear higher-order systems of PDE's. There were found sufficient conditions providing the boundedness. Growth estimates of analytic solutions in the unit ball are also obtained. [ABSTRACT FROM AUTHOR]

In the present paper we give an answer to the following question: Let f:C → C be an entire function of bounded l-index, Φ:Cn → C be an entire function, n≥2, l:C→R+ be a continuous function. What are a positive continuous function L:Cn→R+ and a direction b∈Cn\{0} such that the composite function f(Φ(z)) has bounded L-index in the direction b? [ABSTRACT FROM AUTHOR]

We generalized some criteria of boundedness of L-index in joint variables for analytic in a bidisc functions, where L(z) = (l1(z1,z2), i2(z1,z2)), lj : D2 -> R+ is a continuous function, j Є {1,2}, D2 is a bidisc {(Z1,z2) Є C2 : |z11 < 1, |z2| < 1}. We obtained propositions, which describe a behaviour of power series expansion on a skeleton of a bidisc. The power series expansion is estimated by a dominating homogeneous polynomial with a degree that does not exceed some number, depending only from radii of a bidisc. Replacing universal quantifier by existential quantifier for radii of a bidisc, we also proved sufficient conditions of boundedness of L-index in joint variables for analytic functions, which are weaker than necessary conditions. [ABSTRACT FROM AUTHOR]

A concept of boundedness of L-index in joint variables (see in Bordulyak M.T. The space of entire in CN functions of bounded L-index, Mat. Stud. 4 (1995), 53-58; Bandura A. I., Bordulyak M. T., Skaskiv O. B. Sufficient conditions of boundedness of L-index in joint variables, Mat. Stud. 45 (2016), 12-26) is generalised for an analytic function in a polydisc. We proved criteria of boundedness of L-index in joint variables which describe local behaviour of partial derivative. Some improvements of known sufficient conditions of boundednees of L-index in joint variables are obtained. [ABSTRACT FROM AUTHOR]