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In this paper, we use the idea of normal family to investigate the problem of entire function that share entire function with its ﬁrst derivative.

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In this paper it is shown that the functions f(z)=ez are the only nonconstant entire (meromorphic) functions which share two (three) distinct finite values with their derivative.

In this paper we estimate the size of the ρn’s in the famous L. Zalcman’s Lemma. With it, we obtain a uniqueness theorem for entire functions and their first derivatives,
which improves and generalizes the related results of Rubel and Yang and of Li and Yi. Some examples are provided to show
the sharpness of our result. As an application, we prove that R. Brück’s conjecture is true for a class of functions.

It is shown that if a nonconstant meromorphic function f and its derivative f' share two finite values (counting multiplicities), the f(z) = ce^z.

The Brück conjecture states that if a nonconstant entire function \$f\$ with hyper-order \${\it\sigma}_{2}(f)\in [0,+\infty )\setminus \mathbb{N}\$ shares one finite value \$a\$ (counting multiplicities) with its derivative \$f^{\prime }\$, then \$f^{\prime }-a=c(f-a)\$, where \$c\$ is a nonzero constant. The conjecture has been established for entire functions with order \${\it\sigma}(f)<+\infty\$ and hyper-order \${\it\sigma}_{2}(f)<{\textstyle \frac{1}{2}}\$. The purpose of this paper is to prove the Brück conjecture for the case \${\it\sigma}_{2}(f)=\frac{1}{2}\$ by studying the infinite hyper-order solutions of the linear differential equations \$f^{(k)}+A(z)f=Q(z)\$. The shared value \$a\$ is extended to be a ‘small’ function with respect to the entire function \$f\$.

In this paper, we study the uniqueness problem of entire functions sharing polynomials IM with their first derivative. As an application, we generalize Brück’s conjecture from sharing value CM to sharing polynomial IM for a class of functions. In fact, we prove a result as follows: Let \({a({\not\equiv} 0)}\) be a polynomial and \({n \geq 2}\) be an integer, let f be a transcendental entire function, and let \({F = f^n}\). If F and F′ share a IM, then \({f(z) = Ae^{z/n},}\) where A is a nonzero constant. It extends some previous related theorems.

In this paper, we investigate the conjecture of R. Brück, and prove that the conjecture of R. Brück holds for entire functions of infinite order and hyper order less than 1/2.

In this paper, we prove the following result: Let f(z) be a transcendental entire function, Q(z) ≢ 0 be a small function of f(z), and n ≥ 2 be a positive integer. If f n(z) and (f n(z))′ share Q(z) CM, then f(z) = ce1/nz, where c is a nonzero constant. This result extends Lv's result from the case of polynomial to small entire function. © 2014 Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg.

If ƒ is a non-constant entire function which is sharing two distinct values a, b ∈ ℂ with ƒ′, then a result of Mues and Steinmetz states that ƒ′ = ƒ. In this note we consider the case that ƒ and ƒ′ share only one value counting multiplicity, where appropriate restrictions on the growth of ƒ are assumed.

In this paper, we prove the following TheoremLetf(z)be a transcendental meromorphic function onC, all of whose zeros have multiplicity at leastk+1(k⩾2), except possibly finitely many, and all of whose poles are multiple, except possibly finitely many, and let the functiona(z)=P(z)exp(Q(z))≢0, where P and Q are polynomials such thatlim¯r→∞(T(r,a)T(r,f)+T(r,f)T(r,a))=∞. Then the functionf(k)(z)−a(z)has infinitely many zeros.

In this paper, we will first establish a uniqueness theorem for meromorphic functions f
1, f
2, and f
3 that satisfy the functional equation f
1 +f
2 +f
3 =1, and as applications, we give some results about the existence of meromorphic solutions of f
n
+g
n
+h
n
=1 and two results related to a conjecture of R. Brück.
Keywords.Functional equations-meromorphic functions-uniqueness
Mathematics Subject Classification (2000).30D35-30D20

We prove two uniqueness theorems for entire functions of finite order that share one finite value with one or two of their derivatives.

In this paper, we give a general estimate result of Gol’dberg’s theorem concerning finite growth order of meromorphic solutions of first-order algebraic differential equations by using the normal family theory. It is an extending result of corresponding theorem for Bergweiler and Barsegian and so on. We also give some examples to show that our result is sharp in special case.

A note on the Brück conjecture

- F Lü

F. Lü, A note on the Brück conjecture, Bull. Korean Math. Soc. 48(5) (2011), 951-957.