On the growth of analytic functions represented by the Dirichlet series on semistrips (Q1335988)

The author applies the Wiman-Valiron method and Turán's lemma to study the growth in semi-strips of the Dirichlet series \[ F(z)= \sum^ \infty_{n= 0} a_ n\exp (z\lambda_ n)\quad (0= \lambda_ 0< \lambda_ n \uparrow+\infty,\;n\uparrow+ \infty) \] whose abscissa of absolute convergence is zero. Let \[ \ln \mu(\sigma,F)= \max_ n\{\ln| a_ n|+ \lambda_ n\sigma\},\quad\ln a^*_ n= \sup\{\ln \mu(\sigma,F)- \lambda_ n\sigma)\quad(\sigma> 0). \] He proves that under the condition \[ \sum^ \infty_{n= 0} 1/(\ln a^*_ n)^ \land<\infty,\tag{1} \] \[ \ln M(\sigma,F,S_ 2)\geq \ln M(\sigma,F,S_ 1)\geq \ln(M(\sigma,F,S_ 2)= o(\mu(\sigma,F)))+ o(\ln\mu(\sigma, F))\quad (\sigma\uparrow 0) \] holds outside a set \(E\) of finite logarithmic measure whenever there exists \(q> 0\) such that \(|\sigma|^ q \ln\mu(\sigma,F)\uparrow +\infty\) \((\sigma\uparrow 0)\), where \(x^ \land= \max\{x,1\}\), \[ M(\sigma,F,S_ j)= \sup\{| F(\sigma+ it)|: t\in I_ j\}, \] \(I_ j= [t_ j- a_ j,t_ j+ a_ j]\), \(I_ 1\subset I_ 2\). The author obtains also some more general results, some estimation of \(a_ n \exp(\sigma\lambda_ n)\) and shows that the condition (1) is close to the best possible one in a certain sense. This condition is introduced to replace the conditions imposed on \(\{\lambda_ n\}\) in previous analogous results. In fact, (1) implies that \(n= o(\lambda_ n)\) \((n\to \infty)\). But no restriction on the step \((\lambda_{n+1}- \lambda_ n)\) is assumed.