Further study on solutions of some non-linear homogeneous differential equations in connection to Brück conjecture

In this paper, using the theory of complex differential equations, we study the solution of some non-linear complex differential equations in connection to Brück conjecture which generalized some earlier results due to Pramanik, D. C. and Biswas, M., On solutions of some non-linear differential equations in connection to Bruck conjecture, Tamkang J. Math., 48 (2017), No. 4, 365–375; and Wang, H., Yang, L-Z. and Xu, H-Y., On some complex differential and difference equations concerning sharing function, Adv. Diff. Equ., 2014, 2014:274.

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

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 Pramanik et 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 number a in the conjecture to be a small function. More precisely, we obtain a result which can be restate in the following way: Let f be a non-constant entire function such that σ2(f)<∞, σ2(f) is not a positive integer and δ(0, f)>0. Let M[f] be a differential monomial of f of degree γM and α(z), β(z)∈S(f) be such that max{σ(α), σ(β)} <σ(f). If M[f]+β and fγM−α share the value 0 CM, then M[f]+βfγM−α=c,where c≠0 is a constant.Originality/valueThis is an original work of the authors.

Bruck conjecture and certain solution of some differential equation.

We investigate on the famous Br$ \ddot{u} $ck conjecture further, and improved some of the existing results by extending them up to a differential monomial $ M[f] $ sharing small function with certain power of $ f^{d_M} $ of a meromorphic function. We have found the class of the meromorphic function satisfying the relation $ f^{d_M}\equiv M[f] $. For the generalizing our main results further up to a differential polynomial $ P[f] $, some relevant questions finally have been posed for further study in this direction.

Uniqueness of power of ameromorphic functionwith its differential polynomial

In this paper, taking the question of Zhang and Lu into background,we present one theorem which will improve and extend some recentresults related to Bruck Conjecture.

Some uniqueness results related to the Brück conjecture

Let f be a non-constant meromorphic function and let {a=a(z)} ( {\not\equiv 0,\infty} ) be a small function of f. Under certain essential conditions, we obtained a conclusion similar to the Brück Conjecture, when f and its differential polynomial {P[f]} shares a with weight l ( {\geq 0} ). Our result improves and generalizes a recent result of Li, Yang and Liu [N. Li, L. Yang and K. Liu, A further result related to a conjecture of R. Brück, Kyungpook Math. J. 56 2016, 2, 451–464].

On solutions of some non-linear differential equations in connection to Bruck Conjecture

In this paper, we investigate on the non-constant entire solutions of some non-linear complex differential equations in connection to Br\"{u}ck conjecture and prove some results which improve and extend the results of Xu and Yang\bf{[Xu HY, Yang LZ. On a conjecture of R. Br\"{u}ck and some linear differential equations. Springer Plus 2015; 4:748,:1-10, DOI 10.1186/s40064-015-1530-5.]}