Fast growing solutions to linear differential equations with entire coefficients having the same ρϕ-order

This paper deals with the growth of solutions of a class of higher order linear differential equations \[f^{(k)}+A_{k-1}(z)f^{(k-1)}+ \ldots +A_{1}(z)f^{\prime}+A_{0}(z)f=0; k \geq 2\] when most coefficients $A_{j} (z) (j = 0, \ldots, k-1)$ have the same $\rho_{\varphi}$-order with each other. By using the concept of $\tau_{\varphi}$-type, we obtain some results which indicate growth estimate of every non-trivial entire solution of the above equations by the growth estimate of the coefficient $A_{0} (z)$. We improve and generalize some recent results due to Chyzhykov-Semochko and the author.