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Entire function sharing a small function with its mixed-operators. (English) Zbl 1418.30028

Summary: In this article, we investigate the uniqueness problem on a transcendental entire function \(f(z)\) with its linear mixed-operators \(Tf\), where \(T\) is a linear combination of differential-difference operators \(D^{\nu}_{\eta}:=f^{(\nu)}(z+\eta)\) and shift operators \(E_{\zeta}:=f(z+\zeta)\), where \(\eta,\nu,\zeta\) are constants. We obtain that if a transcendental entire function \(f(z)\) satisfies \(\lambda(f-\alpha)<\sigma(f)<+\infty\), where \(\alpha(z)\) is an entire function with \(\sigma(\alpha)<1\), and if \(f\) and \(Tf \) share one small entire function \(a(z)\) with \(\sigma (a)<\sigma(f)\), then \(\frac{Tf-a(z)}{f(z)-a(z)}=\tau\), where \(\tau\) is a non-zero constant. Furthermore, we obtain the value \(\tau\) and the expression of \(f\) by imposing additional conditions.

MSC:

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
39A10 Additive difference equations
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