Growth properties of solutions to higher order complex linear differential equations with analytic coefficients in the annulus

In this paper, by using the Nevanlinna value distribution theory of meromorphic functions on an annulus, we deal with the growth properties of solutions of the linear differential equation $ f^{\left( k\right) }+B_{k-1}\left( z\right) f^{\left( k-1\right) }+\cdots +B_{1}\left( z\right) f^{\prime }+B_{0}\left( z\right) f=0$, where $k\geq 2$ is an integer and $B_{k-1}\left( z\right),\dots,B_{1}\left( z\right) ,B_{0}\left( z\right) $ are analytic on an annulus. Under some conditions on the coefficients, we obtain some results concerning the estimates of the order and the hyper-order of solutions of the above equation. The results obtained extend and improve those of Wu and Xuan in [16].