Для регулярно збіжного в ${\Bbb C}$ ряду
$$
  F_{\varrho}(z)=\sum\limits_{n=1}^{\infty}a_nE_{\varrho}(\lambda_nz),
$$
де $0<\varrho<+\infty$ і $\displaystyle E_{\varrho}(z)=\sum\limits_{k=0}^{\infty}\frac{z^k}{\Gamma(1+k/\varrho)}$ -- функція Міттаг-Леффлера, досліджено асимптотичне поводження функції $E_{\varrho}^{-1}(M_{F_{\varrho}}(r))$, де  $M_f(r)=\max\{|f(z)|:\,|z|=r\}$. Доведено, наприклад, що якщо $a_n\ge 0$  для всіх $n\ge 1$ і
$$
  \varlimsup\limits_{n\to \infty}\frac{\ln\,\ln\,n}{\ln\,\lambda_n}\le \varrho,
$$
то
$$
\varlimsup\limits_{r\to+\infty}\frac{\ln\,E^{-1}_{\varrho}(M_{F_{\varrho}}(r))}{\ln\,r}=\frac{1}{1-\overline{\gamma}\varrho},
$$
де
$$
  \overline{\gamma}=\varlimsup\limits_{n\to\infty}\frac{\ln\,\lambda_n}{\ln\,\ln\,(1/a_n)}.
$$

Подібний результат отримано для інтегралу типу Лапласа-Стілтьєса
$$
I_{\varrho}(r) = \int\limits_{0}^{\infty}a(x)E_{\varrho}(r x) d F(x).
$$

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