An entire function sharing a polynomial with its derivatives. (English) Zbl 1388.30036

Lahiri, Indrajit; Kaish, Imrul

Boll. Unione Mat. Ital. 10, No. 2, 229-240 (2017).

The paper deals with unicity theorems for entire functions. Let \(f\) be an entire function and \(a=a(z)\) a polynomial of degree \(p\). We let \(E(a;f)\) denote the set of zeros of \(f-a\) counted with multiplicities. We also denote by \(\overline{E}(a;f)\) the set of distinct zeros of \(f-a\). In the case where \(f\) is a transcendental function, we set \(\text{deg}\,f=+\infty\). We define the counting function \(n_A(r, f, a)\) as the number of zeros of \(f\) in \(A\cap \left\{|z|<r \right\}\) with multiplicities, and let \[ N_A(r, f, a)=\int_0^r (n_A(t, f, a)-n_A(0, f, a))dt/t. \] Suppose that \(n \geq p+2\) is a positive integer. Let \(A=\overline{E}(a;f)\setminus \overline{E}(a;f')\) and \(B=\overline{E}(a;f')\setminus (\overline{E}(a;f^{(n)})\cap \overline{E}(a;f^{(n+1)}))\). Then the authors prove the following unicity theorem: If \[ N_A(r, a, f)+N_B(r, a, f')=S(r, f) \] and each common zero of \(f-a\) and \(f'-a\) has the same multiplicity, then \(f=\lambda e^z\) for some constant \(\lambda \neq 0\).

Reviewer: Yoshihiro Aihara (Fukushima)

MSC:

30D35

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

Keywords:

entire functions; uniqueness theorems

References:

[1]	Chang, J., Fang, M.: Uniqueness of entire functions and fixed points. Kodai Math. J. 25, 309-320 (2002) · Zbl 1031.30015
[2]	Gamelin, T.W.: Complex analysis, undergraduate texts in mathematics. Springer, New York (2001)
[3]	Gundersen, G.: Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates. J. Lond. Math. Soc. 37(2), 88-104 (1988) · Zbl 0638.30030
[4]	Gundersen, G.: Finite order solution of second order linear differential equations. Trans. Am. Math. Soc. 305, 415-429 (1988) · Zbl 0669.34010
[5]	Hayman, W.K.: Meromorphic functions. The Clarendon Press, Oxford (1964) · Zbl 0115.06203
[6]	Jank, G., Mues, E., Volkmann, L.: Meromorphe Funktionen, die mit ihrer ersten und zweiten Ableitung einen endlichen Wert teilen. Complex Var. Theory Appl. 6, 51-71 (1986) · Zbl 0603.30037
[7]	Lahiri, I., Das, S.: Entire functions sharing polynomials with derivatives, Ann. Univ. Ferrara. doi:10.1007/s11565-015-0236-5 · Zbl 1352.30031
[8]	Laine, I.: Nevanlinna theory and complex differential equations. Walter de Gruyter, Berlin, New York (1993) · Zbl 0784.30002
[9]	Markushevich, A.I.: Theory of Functions of a Complex Variable, vol. II. Prentice-Hall Inc., USA (1965) (translated by R.A. Silverman) · Zbl 0135.12002
[10]	Yang, C.C., Yi, H.X.: Uniqueness theory of meromorphic functions. Science Press and Kluwer Academic Publishers (2003) · Zbl 1070.30011
[11]	Zhong, H.: Entire functions that share one value with their derivatives. Kodai Math. J. 18, 250-259 (1995) · Zbl 0840.30013
[12]	Zhu, J.: The general form of Hayman’s inequality and fixed points of meromorphic functions. Kexue Tongbao, vol. 33, no. 4, pp. 265-269 (1988) · Zbl 0683.30029