The Relation Between Solutions of a class of HigherOrder Differential Equations With Periodic Coefficients andFunctions of Small Growth

Linear differential equations ~$f^{(k)}+[P_{k-1}(\mathrm{e}^{z})+Q_{k-1}(\mathrm{e}^{-z})]f^{(k-1)}+\cdots+[P_{0}(\mathrm{e}^{z})+Q_{0}(\mathrm{e}^{-z})]f=0$~and ~$f^{(k)}+[P_{k-1}(\mathrm{e}^{z})+Q_{k-1}(\mathrm{e}^{-z})]f^{(k-1)}+\cdots+[P_{0}(\mathrm{e}^{z})+Q_{0}(\mathrm{e}^{-z})]f=R_{1}(\mathrm{e}^{z})+R_{2}(\mathrm{e}^{-z})$~ where ~$P_{j}(z)$~,~$Q_{j}(z)$~$(j=0，1,2，\cdots，k-1)$~and~$R_{i}(z)(i=1，2)$~are polynomials in z are investigated. The relationship between solutions and their 1th derivatives and small functions are studied.