Abstract In 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 B2={z∈C2:|z|=|z1|2+|z2|2<1}, $\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 → R+2 $\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).

中文翻译：

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海曼定理和单位球中解析向量值函数的对数准则的类似物在关节变量中具有有界 L 指数

摘要 在本文中，我们给出了单位球 B2={z∈C2:|z|=|z1|2+|z2|2 中解析向量值函数的联合变量中 L-index 有界的充要条件<1}, $\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 → R+2 $\begin{array}{} \mathbb{R}^2_+ \end{array} $ 是一个正连续向量值函数。特别地，我们推导出此类函数的海曼定理的类比。该定理表明，在向量值函数的联合变量中 L 指数的有界定义中，我们可以用 (p + 1) 阶偏导数的范数估计代替所有偏导数的范数估计。这种形式的准则可以方便地研究偏微分方程组的解析向量值解，因为它允许通过较低阶的偏导数来估计较高阶的偏导数。此外，我们根据向量值函数 L(z) 的某些异常集之外的向量值函数的每个分量的对数导数模的估计，获得了索引有界性的充分条件。