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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.

Let f be a nonconstant entire function and a and b be two distinct complex numbers. Let a 0 ,a 1 ,⋯,a k (k≥1) be constants with a k ¬≡0. Let g=a 0 f+a 1 f (1) +⋯+a k f (k) . If f and g share a and b, ignoring multiplicities, the authors prove φ=(f ' (f-g))/(f-a)(f-b) is a constant satisfying a 0 φ+a 1 φ 2 +⋯+a k φ k+1 ≡0. When a j =0 for 0≤j≤k=1 and a k =1, the result verifies a conjecture of Günter Frank. The paper includes a more general theorem and an examination of solutions for the linear differential equation f ' (f-g)-φ(f-a)(f-b)=0 when g=f ' and φ is a nonconstant entire function. These interesting results follow from expert use of the Nevanlinna theory including judicious application of the well-known Clunie theorem on differential polynomials (cf [J. Lond. Math. Soc. 37, 17-27 (1962; Zbl 0104.29504)]).

1 Basic Nevanlinna theory.- 2 Unicity of functions of finite (lower) order.- 3 Five-value, multiple value and uniqueness.- 4 The four-value theorem.- 5 Functions sharing three common values.- 6 Three-value sets of meromorphic functions.- 7 Functions sharing one or two values.- 8 Functions sharing values with their derivatives.- 9 Two functions whose derivatives share values.- 10 Meromorphic functions sharing sets.

In this paper, when an entire function $f$ and the linear combination of its derivatives $L(f)$ with small functions as its coefficients share one value CM and another value IM is studied. We also resolved the question when an entire function $f$ and its derivative $f^{\prime}$ share two values CM jointly. Some of the results remain to be valid if $f$ is meromorphic and satisfying $N(r,f)=o(T(r,f))$ as $r\rightarrow\infty$ and the values $a,$ $b$ are replaced by small functions of $f(z)$ .

Lecture notes on sharing values of entire and meromorphic functions

- G Frank

G. Frank, Lecture notes on sharing values of entire and meromorphic functions, Workshop
in Complex Analysis at Tianjing, China, 1991.

On the Nevanlinna characteristics of some meromorphic functions, Theory of Functions

- A Z Mohon
- Ko