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 $f(z)=\sum_{n=0}^{\infty} a_n z^n$ be an analytic function on \{z:|z|<1\},\ h\in H and $\Omega_f(r)= \sum_{n=0}^{\infty} |a_n| r^n$. If$$\beta_{fh}=\liminf\limits_{r\to1}\frac{\ln\ln\Omega_f(r)}{\ln h(r)}=+\infty,$$then Wiman's inequality $M_f(r)\leq \mu_f(r) \ln^{1/2+\delta}\mu_f(r)$ is true for all $r\in (r_0, 1)\backslash E(\delta)$, where $h-\mbox{meas}\ E&lt;+\infty.

Groups associated with braces

We construct the group $H(A)$ associated with a brace $A$ and investigate the properties of $H(A)$

On the abscises of the convergence of multiple Dirichlet series

For multiple Dirichlet series of the form $F(s)=\sum_{\|n\|=0}^\infty a_{(n)}\exp\{(\lambda_{(n)},s)\}$ we establish relations between domains of the convergence $G_c$, absolutely convergence $G_a$ and of the domain of the existence of the maximal term $G_{\mu}$ of the series as follows: $\gamma G_{c}\subset G_{a}+\delta_0 e_{1},\ \gamma G_{\mu}\subset G_{a}+\delta_0 e_{1},$ where $e_{1}=(1,...,1)\in \mathbb{R}^p,\;\; \delta_0\in \mathbb{R},$ by condition \liminf\limits_{\|n\|\to\infty}\frac{(\gamma-1)\ln\,|a_{(n)}|+\delta_0\|\lambda_{(n)}\|}{\ln\|n\|}>p; $\gamma G_c\subset G_a+\delta; \;\; \gamma G_{\mu}\subset G_a+\delta,$ where $\delta\in\mathbb{R}^{p},$ by condition $\liminf\limits_{\|n\|\to\infty}\frac{(\gamma-1)\ln\,|a_{(n)}|+(\delta,\lambda_{(n)})}{\ln\,n_1+...+\ln\,n_p}&gt;1.