On a conjecture of R. Brück and some linear differential equations

In this paper, we mainly investigate the Brück conjecture concerning entire function f and its differential polynomial \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_1(f)=a_k(z)f^{(k)}+\cdots +a_0(z)f$$\end{document}L1(f)=ak(z)f(k)+⋯+a0(z)f sharing an entire function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha (z)$$\end{document}α(z) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma (\alpha )\le \sigma (f)$$\end{document}σ(α)≤σ(f), by using the theory of complex differential equation.

We use σ 2 (f ) to denote the hyper-order of f(z), which is defined to be (see Yang 2003, 1995) Let f(z) and g(z) be two nonconstant meromorphic functions, for some a ∈ C ∪ {∞}, if the zeros of f (z) − a and g(z) − a (if a = ∞, zeros of f (z) − a and g(z) − a are the poles of f(z) and g(z) respectively) coincide in locations and multiplicities we say that f(z) and g(z) share the value a CM (counting multiplicities) and if coincide in locations only we say that f(z) and g(z) share a IM (ignoring multiplicities). Rubel and Yang (1977) proved the following result.
Theorem 1.1 Rubel and Yang (1977). Let f be a nonconstant entire function. If f and f ′ share two finite distinct values CM, then f ≡ f ′ .
In 1996, Brück proposed the following conjecture Brück (1996): Conjecture 1.1 Brück (1996). Let f be a non-constant entire function. Suppose that σ 2 (f ) is not a positive integer or infinite, if f and f ′ share one finite value a CM, then for some non-zero constant c. Gundersen and Yang (1998) proved that Brück conjecture holds for entire functions of finite order and obtained the following result.
Theorem 1.2 [Gundersen and Yang (1998), Theorem 1]. Let f be a nonconstant entire function of finite order. If f and f ′ share one finite value a CM, then f ′ −a f −a = c for some non-zero constant c.
The shared value problems related to a meromorphic function f and its derivative f (k) have been a more widely studied subtopic of the uniqueness theory of entire and meromorphic functions in the field of complex analysis (see Chen et al. 2014;Li and Yi 2007;Liao 2015;Mues and Steinmetz 1986;Zhang and Yang 2009;Zhang 2005;Zhao 2012). Li and Cao (2008) improved the Brück conjecture for entire function and its derivation sharing polynomials and obtained the following result: Theorem 1.3 Li and Cao (2008). Let Q 1 and Q 2 be two nonzero polynomials, and let P be a polynomial. If f is a nonconstant entire solution of the equation then σ 2 (f ) = deg P, where and in the following, deg P is the degree of P. Mao (2009) studied the problem on Brück conjecture when f (k) is replaced by differential polynomial L(f ) = A k f (k) + · · · + A 1 f ′ + A 0 f in Theorem 1.3.
Theorem 1.4 Mao (2009). Let P(z) be a polynomial, A k (z)(� ≡ 0), . . . , A 0 (z) be polynomials, and f be an entire function of order and hyper-order σ 2 (f ) < 1 2 . If f and L(f) share P CM, then for some constant c � = 0, where, and in the sequel, deg A j denotes the degree of A j (z), k is a positive integer. Chang and Zhu (2009) further investigated the problem related to Brück conjecture and proved that Theorem 1.2 remains valid if the value a is replaced by a function a(z).

Conclusions
Motivated by the above question, the main purpose of this article is to study the growth of solution of differential equation on entire function f and its linear differential polynomial where k is a positive integer, a k (z)(� ≡ 0), a k−1 (z), . . . , a 1 (z) and a 0 (z) are polynomials, and obtain the following theorems.

Some Lemmas
To prove our theorems, we will require some lemmas as follows. If σ (f ) = +∞, then for any given large M > 0 and sufficiently large r n , ν(r n , f ) > r M n . (5) Lemma 3.4 Laine (1993). Let P(z) = b n z n + b n−1 z n−1 + · · · + b 0 with b n � = 0 be a polynomial. Then, for every ε > 0, there exists r 0 > 0 such that for all r = |z| > r 0 the inequalities hold.

Let f(z) and
A(z) be two entire functions with 0 < σ (f ) = σ (A) = σ < +∞, 0 < τ (A) < τ (f ) < +∞, then there exists a set E ⊂ [1, +∞) that has infinite logarithmic measure such that for all r ∈ E and a positive number κ > 0, we have Proof By definition, there exists an increasing sequence {r m } → +∞ satisfying (1 + 1 m )r m < r m+1 and For any given β(τ (A) < β < τ(f )), then there exists some positive integer m 0 such that for all m ≥ m 0 and for any given ε(0 < ε < τ (f ) − β) , we have Thus, there exists some positive integer m 1 such that for all m ≥ m 1 , we have From (6-8), for all m ≥ m 2 = max{m 0 , m 1 } and for any r ∈ [r m , (1 + 1 m )r m ], we have From the definition of type of entire function, for any sufficiently small ε > 0, we have By (9) and (10), set κ = β − τ (A) − ε, for all r ∈ E, we have Thus, this completes the proof of this lemma.