Note to the behavior of the maximal term of Dirichlet series absolutely convergent in half-plane

By $S_0(\Lambda)$ denote a class of Dirichlet series $F(s)=\sum_{n=0}^{\infty}a_n\exp\{s\lambda_n\} (s=\sigma+it)$ withan increasing to $+\infty$ sequence $\Lambda=(\lambda_n)$ of exponents ($\lambda_0=0$) and the abscissa of absolute convergence $\sigma_a=0$.We say that $F\in S_0^*(\Lambda)$ if $F\in S_0(\Lambda)$ and $\ln \lambda_n=o(\ln |a_n|)$ $(n\to\infty)$. Let$\mu(\sigma,F)=\max\{|a_n|\exp{(\sigma\lambda_n)}\colon n\ge 0\}$ be the maximal term of Dirichlet series. It is proved that in order that $\ln (1/|\sigma|)=o(\ln \mu(\sigma))$ $(\sigma\uparrow 0)$ for every function $F\in S_0^*(\Lambda)$ it is necessary and sufficient that $\displaystyle \varlimsup\limits_{n\to\infty}\frac{\ln \lambda_{n+1}}{\ln \lambda_n}<+\infty. $For an analytic in the disk $\{z\colon |z|<1\}$ function $f(z)=\sum_{n=0}^{\infty}a_n z^n$ and $r\in (0, 1)$ we put $M_f(r)=\max\{|f(z)|\colon |z|=r<1\}$ and $\mu_f(r)=\max\{|a_n|r^n\colon n\ge 0\}$. Then from hence we get the following statement: {\sl if there exists a sequence $(n_j)$ such that $\ln n_{j+1}=O(\ln n_{j})$ and $\ln n_{j}=o(\ln |a_{n_{j}}|)$ as $j\to\infty$, then the functions $\ln \mu_f(r)$ and $\ln M_f(r)$ are or not are slowly increasing simultaneously.

Wiman's type inequality for analytic and entire functions and $h$-measure of an exceptional sets

Let $\mathcal{E}_R$ be the class of analytic functions $f$ represented by power series of the form $f(z)=\sum\limits\limits_{n=0}^{+\infty}a_n z^n$ with the radius of convergence $R:=R(f)\in(0;+\infty].$ For $r\in [0, R)$ we denote the maximum modulus by $M_f(r)=\max\{|f(z)|\colon$ $ |z|=r\}$ and the maximal term of the series by $\mu_f(r)=\max\{|a_n| r^n\colon n\geq 0\}$. We also denote by $\mathcal{H}_R$, $R\leq +\infty$, the class of continuous positive functions, which increase on $[0;R)$ to $+\infty$, such that $h(r)\geq2$ for all $r\in (0,R)$ and $ \int^R_{r_{0}} h(r) d\ln r =+\infty $ for some $r_0\in(0,R)$. In particular, the following statements are proved. $1^0.$ If $h\in \mathcal{H}_R$ and $f\in \mathcal{E}_R,$ then for any $\delta>0$ there exist $E(\delta,f,h):=E\subset(0,R)$, $r_0 \in (0,R)$ such that $$ \forall\ r\in (r_0,R)\backslash E\colon\ M_f(r)\leq h(r) \mu_f(r) \big\{\ln h(r)\ln(h(r)\mu_f(r))\big\}^{1/2+\delta}$$ and $$\int\nolimits_E h(r) dr < +\infty. $$ $2^0.$ If we additionally assume that the function $f\in \mathcal{E}_R$ is unbounded, then $$ \ln M_f(r)\leq(1+o(1))\ln (h(r)\mu_f(r)) $$ holds as $r\to R$, $r\notin E$. Remark, that assertion $1^0$ at $h(r)\equiv \text{const}$ implies the classical Wiman-Valiron theorem for entire functions and at $h(r)\equiv 1/(1-r)$ theorem about the Kövari-type inequality for analytic functions in the unit disc. From statement $2^0$ in the case that $\ln h(r)=o(\ln\mu_f(r))$, $r\to R$, it follows that $ \ln M_f(r)=(1+o(1))\ln \mu_f(r) $ holds as $r\to R$, $r\notin E$.

Hadamard compositions of Gelfond-Leont’ev-Sǎlǎgean and Gelfond-Leont’ev-Ruscheweyh derivatives of functions analytic in the unit disk

For analytic functions $$f(z)=z+\sum\limits_{k=2}^{\infty}f_kz^k \mbox{ and } g(z)=z+\sum\limits_{k=2}^{\infty}g_kz^k$$ in the unit disk properties of the Hadamard compositions $D^n_{l,[S]}f*D^n_{l,[S]}g$ and $D^n_{l,[R]}f*D^n_{l,[R]}g$ of their Gelfond-Leont'ev-S$\check{\text{a}}$l$\check{\text{a}}$gean derivatives $$D^n_{l,[S]}f(z)=z+\sum\limits_{k=2}^{\infty}\left(\frac{l_1l_{k-1}}{l_k}\right)^nf_kz^k$$ and Gelfond-Leont'ev-Ruscheweyh derivatives$$D^n_{l,[R]}f(z)=z+\sum\limits_{k=2}^{\infty}\frac{l_{k-1}l_n}{l_{n+k-1}}f_kz^k$$ are investigated. For study, generalized orders are used. A connection between the growth of the maximal term of the Hadamard composition of Gelfond-Leont'ev-S$\check{\text{a}}$l$\check{\text{a}}$gean derivatives or Gelfond-Leont'ev-Rusche\-weyh derivatives and the growth of the maximal term of these derivatives of Hadamard composition is established. Similar results are obtained in terms of the classical order and the lower order of the growth.

Bitlyan-Gol'dberg type inequality for entire functions and diagonal maximal term

In the article is obtained an analogue of Wiman-Bitlyan-Gol'dberg type inequality for entire $f\colon\mathbb{C}^p\to \mathbb{C}$ from the class $\mathcal{E}^{p}(\lambda)$ of functions represented by gap power series of the form$$f(z)=\sum\limits_{k=0}^{+\infty} P_k(z),\quadz\in\mathbb{C}^p.$$Here $P_0(z)\equiv a_{0}\in\mathbb{C},$ $P_k(z)=\sum_{\|n\|=\lambda_k} a_{n}z^{n}$ is homogeneouspolynomial of degree $\lambda_k\in\mathbb{Z}_+,$ ànd $ 0=\lambda_0<\lambda_k\uparrow +\infty$\ $(1\leq k\uparrow +\infty ),$$\lambda=(\lambda_k)$.\ We consider the exhaustion of thespace\ $\mathbb{C}^{p}$\by the system $(\mathbf{G}_{r})_{r\geq 0}$ of a bounded complete multiple-circular domains $\mathbf{G}_{r}$with the center at the point $\mathbf{0}=(0,\ldots,0)\in \mathbb{C}^{p}$. Define $M(r,f)=\max\{|f(z)|\colon z\in\overline{G}_r\}$, $\mu(r,f)=\max\{|P_k(z))|\colon z\in\overline{G}_r\}$.Let $\mathcal{L}$ be the class of positive continuous functions $\psi\colon \mathbb{R}_{+}\to\mathbb{R}_{+}$ such that $\int_{0}^{+\infty}\frac{dx}{\psi(x)}<+\infty$, $n(t)=\sum_{\lambda_k\leq t}1$ counting function of the sequence $(\lambda_k)$ for $t\geq 0$. The following statement is proved:{\it If a sequence $\lambda=(\lambda_{k})$ satisfy the condition\begin{equation*}(\exists p_1\in (0,+\infty))(\exists t_0>0)(\forall t\geq t_0)\colon\quad n(t+\sqrt{\psi(t)})-n(t-\sqrt{\psi(t)})\leq t^{p_1}\end{equation*}for some function $\psi\in \mathcal{L}$,then for every entire function $f\in\mathcal{E}^{p}(\lambda)$, $p\geq 2$ and for any$\varepsilon>0$ there exist a constant $C=C(\varepsilon, f)>0$ and a set $E=E(\varepsilon, f)\subset [1,+\infty)$ of finite logarithmic measure such that the inequality\begin{equation*}M(r, f)\leq C m(r,f)(\ln m(r, f))^{p_1}(\ln\ln m(r, f))^{p_1+\varepsilon}\end{equation*}holds for all $ r\in[1,+\infty]\setminus E$.}The obtained inequality is sharp in general.At $\lambda_k\equiv k$, $p=2$ we have $p_1=1/2+\varepsilon$ and the Bitlyan-Gol'dberg inequality (1959) it follows. In the case $\lambda_k\equiv k$, $p=2$ we have $p_1=1/2+\varepsilon$ and from obtained statement we get the assertion on the Bitlyan-Gol'dberg inequality (1959), and at $p=1$ about the classical Wiman inequality it follows.

Growth estimates for the maximal term and central exponent of the derivative of a Dirichlet series

Let $A\in(-\infty,+\infty]$, $\Phi:[a,A)\to\mathbb{R}$ be a continuous function such that $x\sigma-\Phi(\sigma)\to-\infty$ as $\sigma\uparrow A$ for every $x\in\mathbb{R}$, $\widetilde{\Phi}(x)=\max\{x\sigma -\Phi(\sigma):\sigma\in [a,A)\}$ be the Young-conjugate function of $\Phi$, $\overline{\Phi}(x)=\widetilde{\Phi}(x)/x$ and $\Gamma(x)=(\widetilde{\Phi}(x)-\ln x)/x$ for all sufficiently large $x$, $(\lambda_n)$ be a nonnegative sequence increasing to $+\infty$, and $F(s)=\sum\limits\limits_{n=0}^\infty a_ne^{s\lambda_n}$ be a Dirichlet series such that its maximal term $\mu(\sigma,F)=\max\{|a_n|e^{\sigma\lambda_n}:n\ge0\}$ and central index $\nu(\sigma,F)=\max\{n\ge0:|a_n|e^{\sigma\lambda_n}=\mu(\sigma,F)\}$ are defined for all $\sigma<A$. It is proved that if $\ln\mu(\sigma,F)\le(1+o(1))\Phi(\sigma)$ as $\sigma\uparrow A$, then the inequalities $$ \varlimsup_{\sigma\uparrow A}\frac{\mu(\sigma,F')}{\mu(\sigma,F)\overline{\Phi}\,^{-1}(\sigma)}\le1,\qquad \varlimsup_{\sigma\uparrow A}\frac{\lambda_{\nu(\sigma,F')}}{\Gamma^{-1}(\sigma)}\le1, $$ hold, and these inequalities are sharp.

The growth of the maximal term of Dirichlet series

Let $\Lambda$ be the class of nonnegative sequences $(\lambda_n)$ increasing to $+\infty$, $A\in(-\infty,+\infty]$, $L_A$ be the class of continuous functions increasing to $+\infty$ on $[A_0,A)$, $(\lambda_n)\in\Lambda$, and $F(s)=\sum a_ne^{s\lambda_n}$ be a Dirichlet series such that its maximum term $\mu(\sigma,F)=\max_n|a_n|e^{\sigma\lambda_n}$ is defined for every $\sigma\in(-\infty,A)$. It is proved that for all functions $\alpha\in L_{+\infty}$ and $\beta\in L_A$ the equality$$\rho^*_{\alpha,\beta}(F)=\max_{(\eta_n)\in\Lambda}\overline{\lim_{n\to\infty}}\frac{\alpha(\eta_n)}{\beta\left(\frac{\eta_n}{\lambda_n}+\frac{1}{\lambda_n}\ln\frac{1}{|a_n|}\right)}$$ holds, where $\rho^*_{\alpha,\beta}(F)$ is the generalized $\alpha,\beta$-order of the function $\ln\mu(\sigma,F)$, i.e. $\rho^*_{\alpha,\beta}(F)=0$ if the function $\mu(\sigma,F)$ is bounded on $(-\infty,A)$, and $\rho^*_{\alpha,\beta}(F)=\overline{\lim_{\sigma\uparrow A}}\alpha(\ln\mu(\sigma,F))/\beta(\sigma)$ if the function $\mu(\sigma,F)$ is unbounded on $(-\infty,A)$.

Generalized types of the growth of Dirichlet series

Let $A\in(-\infty,+\infty]$ and $\Phi$ be a continuously on $[\sigma_0,A)$ function such that $\Phi(\sigma)\to+\infty$ as $\sigma\to A-0$. We establish a necessary and sufficient condition on a nonnegative sequence $\lambda=(\lambda_n)$, increasing to $+\infty$, under which the equality$$\overline{\lim\limits_{\sigma\uparrow A}}\frac{\ln M(\sigma,F)}{\Phi(\sigma)}=\overline{\lim\limits_{\sigma\uparrow A}}\frac{\ln\mu(\sigma,F)}{\Phi(\sigma)},$$holds for every Dirichlet series of the form $F(s)=\sum_{n=0}^\infty a_ne^{s\lambda_n}$, $s=\sigma+it$, absolutely convergent in the half-plane ${Re}\, s<A$, where $M(\sigma,F)=\sup\{|F(s)|:{Re}\, s=\sigma\}$ and $\mu(\sigma,F)=\max\{|a_n|e^{\sigma\lambda_n}:n\ge 0\}$ are the maximum modulus and maximal term of this series respectively.