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Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity. (English) Zbl 1180.30039

In this paper, shared value problems are studied for a meromorphic function $f\left(z\right)$ in the plane and its shift $f\left(z+c\right)$, where $c\in ℂ$. Among the others, the authors show in Theorem 2 that if $f\left(z\right)$ is of finite order and shares two values $CM$ (counting multiplicities) and one value $IM$ (ignoring multiplicities) with its shift $f\left(z+c\right)$, then $f\left(z\right)\equiv f\left(z+c\right)$, that is, $f$ should be a periodic function with period $c$. The three values can be also taken as $c$-periodic functions small with respect to the function $f$. As a new criterion for elliptic functions, their Theorem 10 gives the condition that $f$ shares three values “$2CM+1IM$” with two shifts in different directions. This is a direct application of Theorem 2. Those theorems improve some of the results given in a recent paper due to the first four authors [Uniqueness of meromorphic functions sharing values with their shifts, Complex Var. Elliptic Equ., in press] where all the shared values are considered mainly together with multiplicity, that is, under $CM$.

Moreover, under the assumptions that $f\left(z\right)$ is a meromorphic function of order less than 2 and that $f\left(z\right)$ and its shift share both a value $a\in ℂ$ and $\infty$ $CM$, their Theorem 1 confirms a shifted analogue of a well-known conjecture by R. Brück [Result. Math. 30, 21–24 (1996; Zbl 0861.30032)] concerning the value sharing of an entire function $f\left(z\right)$ with its derivative ${f}^{\text{'}}\left(z\right)$. The authors remark with an entire function of order 2 that this order restriction cannot be relaxed any further.

The main tools of this research are different analogons of important results in the classical Nevanlinna theory such as the lemma on the logarithmic derivative and the second main theorems. These are obtained by R. Halburd and R. Korhonen [Ann. Acad. Sci. Fenn., Math. 31, No. 2, 463–478 (2006; Zbl 1108.30022), J. Math. Anal. Appl. 314, No. 2, 477–487 (2006; Zbl 1085.30026), Proc. Lond. Math. Soc. (3) 94, No. 2, 443–474 (2007; Zbl 1119.39014)], and also independently by Y.-M. Chiang and S.-J. Feng [Ramanujan J. 16, No. 1, 105–129 (2008; Zbl 1152.30024)], all of which are valid for meromorphic functions of finite order. The authors merge them skillfully with some results in uniqueness theory of meromorphic functions such as those by [G. Brosch, Eindeutigkeitssätze für meromorphe Funktionen. (Uniqueness theorems for meromorphic functions). Aachen: RWTH Aachen, Math.-Naturwiss. Fak., Diss. 77 S. (1989; Zbl 0694.30027)] or by G. Jank and N. Terglane [Math. Pannonica 2, No. 2, 37–46 (1991; Zbl 0747.30022)].

The authors offer also some more alternative improvements of their previous theorems.

##### MSC:
 30D35 Distribution of values (one complex variable); Nevanlinna theory