Entire functions sharing one finite value CM with their shifts and difference operators
Boudaoud Miloudi
Received April 8, 2025. Published online November 11, 2025.
Abstract: We investigate the uniqueness of an entire function $f(z)$ sharing one finite value CM with its shift and its difference operator, in each case we find that $f(z)=h(z)\beta^{\alpha z/c}$, where $ \alpha\not=0,$ $\beta>1$ and $h(z)$ is a periodic entire function of period $c$. Here, we say that two entire functions $f(z)$ and $g(z)$ share a value $a$ CM if $f(z)-a$ and $g(z)-a$ have the same zeros with same multiplicities.
References: [1] B. Chen, Z. Chen, S. Li: Uniqueness theorems on entire functions and their difference operators or shifts. Abstr. Appl. Anal. 2012 (2012), Article ID 906893, 8 pages. DOI 10.1155/2012/906893 | MR 2926877 | Zbl 1258.30010
[2] B. Chen, S. Li: Uniqueness problems on entire functions that share a small function with their difference operators. Adv. Difference Equ. 2014 (2014), Article ID 311, 11 pages. DOI 10.1186/1687-1847-2014-311 | MR 3360566 | Zbl 1417.30018
[3] A. El Farissi, Z. Latreuch, A. Asiri: On the uniqueness theory of entire functions and their difference operators. Complex Anal. Oper. Theory 10 (2016), 1317-1327. DOI 10.1007/s11785-015-0514-3 | MR 3532314 | Zbl 1354.30020
[4] A. El Farissi, Z. Latreuch, B. Belaïdi, A. Asiri: Entire functions that share a small function with their difference operators. Electron. J. Differ. Equ. 2016 (2016), Article ID 32, 13 pages. MR 3466503 | Zbl 1332.30050
[5] J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo: Uniqueness of meromorphic functions sharing values with their shifts. Complex Var. Elliptic Equ. 56 (2011), 81-92. DOI 10.1080/17476930903394770 | MR 2774582 | Zbl 1217.30029
[6] J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, J. Zhang: Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity. J. Math. Anal. Appl. 355 (2009), 352-363. DOI 10.1016/j.jmaa.2009.01.053 | MR 2514472 | Zbl 1180.30039
[7] Z. Latreuch, A. El Farissi, B. Belaïdi: Entire functions sharing small functions with their difference operators. Electron. J. Differ. Equ. 2015 (2015), Article ID 132, 10 pages. MR 3358504 | Zbl 1320.30058
[8] S. Li, Z. Gao: Entire functions sharing one or two finite values CM with their shifts or difference operators. Arch. Math. 97 (2011), 475-483. DOI 10.1007/s00013-011-0324-4 | MR 2860575 | Zbl 1243.30068
[9] C.-C. Yang, H.-Y. Yi: Uniqueness Theory of Meromorphic Functions. Mathematics and its Applications 557. Kluwer Academic, Dordrecht (2003). MR 2105668 | Zbl 1070.30011
[10] J. Zhang, L. Liao: Entire functions sharing some values with their difference operators. Sci. China, Math. 57 (2014), 2143-2152. DOI 10.1007/s11425-014-4848-5 | MR 3247539 | Zbl 1329.30020
