scholarly journals Growth Estimates for Analytic Vector-Valued Functions in the Unit Ball Having Bounded $\mathbf{L}$-index in Joint Variables

2020 ◽  
Vol 3 (1) ◽  
pp. 9-19
Author(s):  
Vita BAKSA ◽  
Andriy BANDURA ◽  
Oleh SKASKIV
Keyword(s):  
Unit Ball ◽  
Analytic Vector ◽  
Vector Valued Functions ◽  
Vector Valued

Analogs of Fricke's theorems for analytic vector-valued functions in the unit ball having bounded L-index in joint variables.

2019 ◽  
Vol 33 ◽  
pp. 16-26 ◽  
Author(s):  
Vitalina Baksa ◽  
Andriy Bandura ◽  
Oleg Skaskiv
Keyword(s):  
Unit Ball ◽  
Vector Function ◽  
Sufficient Conditions ◽  
Maximum Modulus ◽  
Function Class ◽  
Analytic Vector ◽  
Continuous Vector ◽  
Necessary And Sufficient ◽  
Vector Valued Functions ◽  
Vector Valued

In this paper, we present necessary and sufficient conditions of boundedness of $\mathbb{L}$-index in joint variables for vector-functions analytic in the unit ball, where $\mathbf{L}=(l_1,l_2): \mathbb{B}^2\to\mathbb{R}^2_+$ is a positive continuous vector-function, $\mathbb{B}^2=\{z\in\mathbb{C}^2: |z|=\sqrt{|z_1|^2+|z_2|^2}\le 1\}.$ Particularly, we deduce analog of Fricke's theorems for this function class, give estimate of maximum modulus on the skeleton of bidisc. The first theorem concerns sufficient conditions. In this theorem we assume existence of some radii, for which the maximum of norm of vector-function on the skeleton of bidisc with larger radius does not exceed maximum of norm of vector-function on the skeleton of bidisc with lesser radius multiplied by some costant depending only on these radii. In the second theorem we show that boundedness of $\mathbf{L}$-index in joint variables implies validity of the mentioned estimate for all radii.

Analogs of Hayman’s Theorem and of logarithmic criterion for analytic vector-valued functions in the unit ball having bounded L-index in joint variables

2020 ◽  
Vol 70 (5) ◽  
pp. 1141-1152
Author(s):  
Vita Baksa ◽  
Andriy Bandura ◽  
Oleh Skaskiv
Keyword(s):  
Unit Ball ◽  
Logarithmic Derivative ◽  
Sufficient Conditions ◽  
Analytic Vector ◽  
Exceptional Set ◽  
Partial Derivatives ◽  
Definition Of ◽  
Vector Valued Function ◽  
Vector Valued Functions ◽  
Vector Valued

AbstractIn this paper, we present necessary and sufficient conditions of boundedness of L-index in joint variables for vector-valued functions analytic in the unit ball $\begin{array}{} \mathbb{B}^2\! = \!\{z\!\in\!\mathbb{C}^2: |z|\! = \!\small\sqrt{|z_1|^2+|z_2|^2}\! \lt \! 1\}, \end{array} $ where L = (l1, l2): 𝔹2 → $\begin{array}{} \mathbb{R}^2_+ \end{array} $ is a positive continuous vector-valued function.Particularly, we deduce analog of Hayman’s theorem for this class of functions. The theorem shows that in the definition of boundedness of L-index in joint variables for vector-valued functions we can replace estimate of norms of all partial derivatives by the estimate of norm of (p + 1)-th order partial derivative. This form of criteria could be convenient to investigate analytic vector-valued solutions of system of partial differential equations because it allow to estimate higher-order partial derivatives by partial derivatives of lesser order. Also, we obtain sufficient conditions for index boundedness in terms of estimate of modulus of logarithmic derivative in each variable for every component of vector-valued function outside some exceptional set by the vector-valued function L(z).

scholarly journals On analytic vector-valued functions of a real variable

1951 ◽  
Vol 12 (1) ◽  
pp. 108-111 ◽  
Author(s):  
A. Alexiewicz ◽  
W. Orlicz
Keyword(s):  
Real Variable ◽  
Analytic Vector ◽  
Vector Valued Functions ◽  
Vector Valued

scholarly journals On vector-valued Hardy martingales and a generalized Jensen's inequality

2003 ◽  
Vol 2003 (9) ◽  
pp. 527-532
Author(s):  
Annela R. Kelly ◽  
Brian P. Kelly
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Analytic Functions ◽  
Jensen’S Inequality ◽  
Monotonicity Property ◽  
Dual Group ◽  
Analytic Vector ◽  
Jensen's Inequality ◽  
Vector Valued Functions ◽  
Vector Valued

We establish a generalized Jensen's inequality for analytic vector-valued functions on𝕋Nusing a monotonicity property of vector-valued Hardy martingales. We then discuss how this result extends to functions on a compact abelian groupG, which are analytic with respect to an order on the dual group. We also give a generalization of Helson and Lowdenslager's version of Jensen's inequality to certain operator-valued analytic functions.

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