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.

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.

On the equivalence of the sum and the maximal term of the Dirichlet series absolutely convergent in the half-plane

For absolutely convergent in the half-plane ${zcolon {mRe,}z<0}$ Dirichlet series $F(z)=sumlimits_{n=0}^{+infty}a_ne^{zlambda_n},$ where $0leqlambda_nuparrow +infty (0leq nuparrow+infty),$ we establish conditions on the coefficients of itsNewton majorant, sufficient for the relation $F(x+iy)=(1+o(1))a_{u(x)}e^{(x+iy)lambda_{u(x)}}$ to hold as$xo -0$ outside some set $E$ of zero logarithmic density in thepoint $0,$ uniformly by $yin{mathbb R}$