We obtain a Clunie type theorem for a rather general form of functional equations involving differential polynomials. Our theorems can give a much sharper estimate on the error term of the proximity function of solutions of differential equations and functional equations than the upper bound obtained by Clunie, Doeringer, He-Xiao, Korhonen and etc. In particular, our theorem can also be applied to study various types of Painlevé differential equations.
Chung-Chun Yang. Zhuan Ye. "Estimates of the proximate function of differential polynomials." Proc. Japan Acad. Ser. A Math. Sci. 83 (4) 50 - 55, April 2007. https://doi.org/10.3792/pjaa.83.50