Further study on the Brück conjecture and some non-linear complex differential equations

Abstract

PurposeThe purpose of this current paper is to deal with the study of non-constant entire solutions of some non-linear complex differential equations in connection to Brück conjecture, by using the theory of complex differential equation. The results generalize the results due to Pramaniket al.Design/methodology/approach39B32, 30D35.FindingsIn the current paper, we mainly study the Brück conjecture and the various works that confirm this conjecture. In our study we find that the conjecture can be generalized for differential monomials under some additional conditions and it generalizes some works related to the conjecture. Also we can take the complex numberain the conjecture to be a small function. More precisely, we obtain a result which can be restate in the following way: Letfbe a non-constant entire function such thatσ2(f)<∞,σ2(f)is not a positive integer andδ(0, f)>0. LetM[f]be a differential monomial offof degreeγMandα(z), β(z)∈S(f)be such thatmax{σ(α), σ(β)} <σ(f). IfM[f]+βandfγM−αshare the value 0 CM, thenM[f]+βfγM−α=c,wherec≠0is a constant.Originality/valueThis is an original work of the authors.