## Entire functions that share a polynomial with their derivatives.(English)Zbl 1096.30029

L. A. Rubel and C. C. Yang proved that if a nonconstant entire function $$f$$ and its derivative $$f'$$ share two distinct finite values, then $$f\equiv f'$$. Jank-Mues-Volkmann proved that if a nonconstant entire function $$f$$ and its derivative $$f'$$ share a finite value $$a$$ and $$f=a$$ implies that $$f''=a$$,then $$f\equiv f'$$. A natural question is to consider the case that $$f$$ and $$f'$$ share a small function $$a(z)$$ ($$T(r,a)=S(r,f)$$). In this paper, the author studied the entire functions sharing a polynomial with their derivatives. He proved that if an entire function $$f$$ and its derivative $$f'$$ share a nonconstant polynomial $$Q$$ with degree $$q<k$$ CM, and $$f(z)-Q(z)=0$$ implies that $$f^{(k)}(z)-Q(z)=0$$, then $$f\equiv f'$$. Remark: The statement of Lemma 2 in the paper is not right. It should be stated as “Let $$f$$ be an entire solution of the equation $a_n(z)f^{(n)}+a_{n-1}(z)f^{(n-1)}+\cdots+a_1(z)f'+a_0(z)f=0,(*)$ with polynomial coefficients $$a_0(z),\cdots,a_n(z)$$ such that $$a_0(z)\not\equiv 0$$ and $$a_n(z)\not\equiv 0$$, then $$f$$ is of finite order.” Please note that if $$a_n(z)$$ is not a constant, then the solution of the equation (*) may not be an entire function.

### MSC:

 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory

### Keywords:

Entire function; share value
Full Text:

### References:

 [1] Bank, S.; Laine, I., On the growth of meromorphic solution of linear and algebraic differential equations, Math. scand., 40, 119-126, (1977) · Zbl 0367.34004 [2] Brück, R., On entire functions which share one value CM with their first derivative, Results math., 30, 21-24, (1996) · Zbl 0861.30032 [3] Gamelin, T.W., Complex analysis, Undergraduate texts in math., (2001), Springer New York · Zbl 1315.00024 [4] Gundersen, G.G., Finite order solution of second order linear differential equation, Trans. amer. math. soc., 305, 415-429, (1988) · Zbl 0669.34010 [5] Gundersen, G.G.; Yang, L.Z., Entire functions that share one value with one or two of their derivatives, J. math. anal. appl., 233, 88-95, (1998) · Zbl 0911.30022 [6] Gundersen, G.G., Estimates for the logarithmic derivatives of a meromorphic function, plus similar estimates, J. London math. soc., 37, 88-104, (1988) · Zbl 0638.30030 [7] Hayman, W.K., Meromorphic functions, (1964), Clarendon Oxford · Zbl 0115.06203 [8] Jank, G.; Mues, E.; Volkmann, L., Meromorphe funktionen, die mit ihrer ersten und zweiten ableitung einen endlichen wert teilen, Complex variables theory appl., 6, 51-71, (1986) · Zbl 0603.30037 [9] Laine, I., Nevanlinna theory and complex differential equation, (1993), de Gruyter Berlin [10] Li, P.; Yang, C.C., Uniqueness theorems on entire functions and their derivatives, J. math. anal. appl., 253, 50-57, (2001) · Zbl 0965.30010 [11] Markushevich, A., Theory of functions of a complex variable, vol. 2, (1965), Prentice Hall Englewood Cliffs, NJ, (translated by R. Silverman) · Zbl 0135.12002 [12] Mues, E.; Steinmetz, N., Meromorphe funktionen, die mit ihrer ableitung werte teilen, Manuscripta math., 29, 195-206, (1979) · Zbl 0416.30028 [13] Rubel, L.A.; Yang, C.C., Values shared by an entire function an its derivative, (), 101-103 [14] Wang, J.-P.; Yi, H.-X., Entire functions that share one value CM with their derivatives, J. math. anal. appl., 277, 155-163, (2003) · Zbl 1015.30014 [15] Yi, H.X.; Yang, C.C., Uniqueness theory of meromorphic functions, Pure appl. math. monogr., vol. 32, (1995), Science Press Beijing · Zbl 0799.30019 [16] Zhong, H.L., Entire functions that share one value with their derivatives, Kodai math. J., 18, 250-259, (1995) · Zbl 0840.30013
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