In this paper, we investigate the exponent of convergence of $f^{(i)}-\varphi $ where $f^{(0)}=f\not\equiv 0$ is a solution of linear differential equations with meromorphic coefficients in the unit disc and $\varphi$ is a small function of $f$. Our investigation is based on the behavior of the coefficients in a subset $\Gamma$ of the unit disc such that the set $\Gamma _{0}=\{|z|:z\in\Gamma\}$ is of infinite logarithmic measure. By this investigation we can deduce the fixed points of $f^{(i) } $ by taking $\varphi(z)=z$.