On the Gelfand-Hille theorems
In the paper we consider the growth of entire solution of a nonlinear differential equation and improve some existing results.
The estimations of lower order in terms of the sequences and for an entire Dirichlet series , have been obtained, namely :One of these estimations improves considerably the estimations earlier obtained by Rahman (Quart. J. Math. Oxford, (2), 7, 96-99 (1956)) and Juneja and Singh (Math. Ann., 184(1969), 25-29 ).