25 research outputs found
Some criteria of boundedness of the L-index in direction for slice holomorphic functions of several complex variables
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 {z⁰ + tb : t ∈ C} for every z⁰ ∈ Cⁿ and for a given direction b ∈ Cⁿ \ {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 : Cⁿ → C is the necessary and sufficient condition for the existence of a positive continuous function L : Cⁿ → R₊ such that F has a bounded L-index in direction.The authors are thankful to Professor S. Yu. Favorov (Kharkiv) for the formulation of interesting problem
Direct analogues of Wiman's inequality for analytic functions in the unit disc
Let be an analytic function on \{z:|z|<1\},\ h\in H and . Ifthen Wiman's inequality is true for all , where $h-\mbox{meas}\ E<+\infty.
Groups associated with braces
We construct the group associated with a brace and investigate the properties of
On the abscises of the convergence of multiple Dirichlet series
For multiple Dirichlet series of the form we establish relations between domains of the convergence , absolutely convergence and of the domain of the existence of the maximal term of the series as follows: where by condition \liminf\limits_{\|n\|\to\infty}\frac{(\gamma-1)\ln\,|a_{(n)}|+\delta_0\|\lambda_{(n)}\|}{\ln\|n\|}>p; where by condition $\liminf\limits_{\|n\|\to\infty}\frac{(\gamma-1)\ln\,|a_{(n)}|+(\delta,\lambda_{(n)})}{\ln\,n_1+...+\ln\,n_p}>1.