Sufficient conditions of boundedness of L-index in joint variables

A. I. Bandura, M. T. Bordulyak, O. B. Skaskiv
Department of Higher Mathematics Ivano-Frankivsk National Technical University of Oil and Gas; Department of Function Theory and Theory of Probability Ivan Franko National University of Lviv
A concept of boundedness of $\mathbf{L}$-index in joint variables (see in Bordulyak M.T. The space of entire in $\mathbb{C}^n$ functions of bounded $L$-index, Mat. Stud., 4 (1995), 53-58. (in Ukrainian)) is generalised for $\mathbf{L}(z)=(l_1(z),$ $\ldots,$ $l_{n}(z)),$ $z\in\mathbb{C}^n.$ We proved criteria of boundedness of $\mathbf{L}$-index in joint variables and established a connection between the classes of entire functions of bounded $l_j$-index in each direction $\mathbf{e}_j$ and functions of bounded $\mathbf{L}$-index in joint variables. We deduce new sufficient conditions of boundedness of $\mathbf{L}$-index in joint variables. The obtained restrictions describe the behaviour of logarithmic derivative in each variable and the distribution of zeros.
entire function; bounded L-index in joint variables; bounded L-index in direction; directional logarithmic derivative; distibution of zeros
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