Elementary Remarks to the Relative Growth of Series by the System of Mittag-Leffler Functions

Abstract

For a regularly converging in ${\Bbb C}$ series $F_{\varrho}(z)=\sum\limits_{n=1}^{\infty} a_n E_{\varrho}(\lambda_nz)$, where $0<\varrho<+\infty$ and $E_{\varrho}(z)=\sum\limits_{k=0}^{\infty}\frac{z^k}{\Gamma(1+k/\varrho)}$ is the Mittag-Leffler function, it is investigated the asymptotic behavior of the function $E_{\varrho}^{-1} (M_{F_{\varrho}}(r))$, where $M_f(r)=\max\{|f(z)|:\,|z|=r\}$. For example, it is proved that if $\varlimsup\limits_{n\to \infty}\frac{\ln\,\ln\,n}{\ln\,\lambda_n}\le \varrho$ and $a_n\ge 0$ for all $n\ge 1$, then $\varlimsup\limits_{r\to+\infty}\frac{\ln\,E^{-1}_{\varrho}(M_{F_{\varrho}}(r))}{\ln\,r}=\frac{1}{1-\overline{\gamma}\varrho}$, where $\overline{\gamma}=\varlimsup\limits_{n\to\infty}\frac{\ln\,\lambda_n}{\ln\,\ln\,(1/a_n)}$. A similar result is obtained for the Laplace-Stiltjes type integral $I_{\varrho}(r) = \int\limits_{0}^{\infty}a(x)E_{\varrho}(r x) d F(x)$.