The growth at infinity of a sequence of entire functions of bounded orders

In this paper we shall consider the growth at infinity of a sequence $(P_n)$ of entire functions of bounded orders. Our results extend the results in cite{trong-tuyen2} for the growth of entire functions of genus zero. Given a sequence of entire functions of bounded orders $P_n(z)$, we found a nearly optimal condition, given in terms of zeros of $P_n$, for which $(k_n)$ that we have begin{eqnarray*} limsup_{ntoinfty}|P_n(z)|^{1/k_n}leq 1 end{eqnarray*} for all $zin mathbb C$ (see Theorem ref{theo5}). Exploring the growth of a sequence of entire functions of bounded orders lead naturally to an extremal function which is similar to the Siciaks extremal function (See Section 6).