We study the hyper-order of analytic solutions of linear differential equations with analytic coefficients having the same order near a finite singular point. We improve previous results given by S. Cherief and S. Hamouda (2021). We also consider the nonhomogeneous linear differential equations.

The estimations of lower order $\left(R\right)$$\lambda$ in terms of the sequences $\left\{a_n\right\}$ and $\left\{{\lambda }_n\right\}$ for an entire Dirichlet series $f\left(s\right)={\sum }_{n=1}^{\infty }{a}_n{e}^{s\lambda n}$, have been obtained, namely :$$\begin{array}{cc}\hfill \lambda & =\underset{\left\{{\lambda }_{n_p}\right\}}{max}lim\underset{p\to \infty }{inf}\frac{{\lambda }_{n_p}log{\lambda }_{n_{p-1}}}{log|a_{n_p}{|}^{-1}}\hfill \\ & =\underset{\left\{{\lambda }_{n_p}\right\}}{max}lim\underset{p\to \infty }{inf}\frac{({\lambda }_{n_p}-{\lambda }_{n_{p-1}})log{\lambda }_{n_{p-1}}}{log\left|a_{n_{p-1}}\right|a_{n_p}|}.\hfill \end{array}$$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 ).

Let f(z), $z=r{e}^{i\theta }$, be analytic in the finite disc |z| < R. The growth properties of f(z) are studied using the mean values $I_{\delta }\left(r\right)$ and the iterated mean values $N_{\delta ,k}\left(r\right)$ of f(z). A convexity result for the above mean values is obtained and their relative growth is studied using the order and type of f(z).