Uniqueness related to Higher Order Difference Operators of Entire Functions
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In this paper, by using the difference analogue of Nevanlinna's theory, the authors study the shared-value problem concerning two higher order difference operators of a transcendental entire function with finite order. The following conclusion is proved: Let~$f(z)$~be a finite order transcendental entire function such that~$\lambda(f-a(z))<\rho(f)$,~where~$a(z)(\in S(f))$~is an entire function and satisfies~$\rho(a(z))<1$,~and let~$\eta(\in\mathbb{C})$~be a constant such that~$\Delta_{\eta}^{n+1}f(z)\not\equiv0$.~If ~$\Delta_{\eta}^{n+1}f(z)$~and~$\Delta_{\eta}^{n}f(z)$~share~$\Delta_{\eta}^{n}a(z)$~CM,~where~$\Delta_{\eta}^{n}a(z)\in S\left(\Delta_{\eta}^{n+1}f(z)\right)$,~then~\\$f(z)$~has a specific expression~$f(z)=a(z)+Be^{Az}$,~where $A$ and $B$ are two non-zero constants and~$a(z)$~reduces to a constant.
Abstract
In this paper, by using the difference analogue of Nevanlinna's theory, the authors study the shared-value problem concerning two higher order difference operators of a transcendental entire function with finite order. The following conclusion is proved: Let~$f(z)$~be a finite order transcendental entire function such that~$\lambda(f-a(z))<\rho(f)$,~where~$a(z)(\in S(f))$~is an entire function and satisfies~$\rho(a(z))<1$,~and let~$\eta(\in\mathbb{C})$~be a constant such that~$\Delta_{\eta}^{n+1}f(z)\not\equiv0$.~If ~$\Delta_{\eta}^{n+1}f(z)$~and~$\Delta_{\eta}^{n}f(z)$~share~$\Delta_{\eta}^{n}a(z)$~CM,~where~$\Delta_{\eta}^{n}a(z)\in S\left(\Delta_{\eta}^{n+1}f(z)\right)$,~then~\\$f(z)$~has a specific expression~$f(z)=a(z)+Be^{Az}$,~where $A$ and $B$ are two non-zero constants and~$a(z)$~reduces to a constant.
참고문헌 (18)
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