Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userUniqueness theorems of meromorphic functions with their differential-difference operators in several complex variables
248
0
0.0
(
0
)
Authors
XiaoHuang Huang
Ask ChatGPT about the research
No Arabic abstract
In this paper, we study the uniqueness of meromporphic functions and their difference operators. In particular, We have proved: Let $f$ be a nonconstant entire function on $mathbb{C}^{n}$, let $etain mathbb{C}^{n}$ be a nonzero complex number, and let $a$ and $b$ be two distinct complex numbers in $mathbb{C}^{n}$. If $$varlimsup_{rrightarrowinfty}frac{logT(r,f)}{r}=0,$$ and if $f(z)$ and $(Delta_{eta}^{n}f(z))^{(k)}$ share $a$ CM and share $b$ IM, then $f(z)equiv(Delta_{eta}^{n}f(z))^{(k)}$.
rate research
Read More
In this paper, we study the uniqueness of the differential polynomials of entire functions. We prove the following result: Let $f(z)$ be a nonconstant entire function on $mathbb{C}^{n}$ and $g(z)=b_{-1}+sum_{i=0}^{n}b_{i}D^{k_{i}}f(z)$, where $b_{-1}$ and $b_{i} (i=0ldots,n)$ are small meromorphic functions of $f$, $k_{i}geq0 (i=0ldots,n)$ are integers. Let $a_{1}(z) otequivinfty, a_{2}(z) otequivinfty$ be two distinct small meromorphic functions of $f(z)$. If $f(z)$ and $g(z)$ share $a_{1}(z)$ CM, and $a_{2}(z)$ IM. Then either $f(z)equiv g(z)$ or $a_{1}=2a_{1}=2$, $$f(z)equiv e^{2p}-2e^{p}+2,$$ and $$g(z)equiv e^{p},$$ where $p(z)$ is a non-constant entire function on $mathbb{C}^{n}$.
In this paper, we investigate zeros of difference polynomials of the form $f(z)^nH(z, f)-s(z)$, where $f(z)$ is a meromorphic function, $H(z, f)$ is a difference polynomial of $f(z)$ and $s(z)$ is a small function. We first obtain some inequalities for the relationship of the zero counting function of $f(z)^nH(z, f)-s(z)$ and the characteristic function and pole counting function of $f(z)$. Based on these inequalities, we establish some difference analogues of a classical result of Hayman for meromorphic functions. Some special cases are also investigated. These results improve previous findings.
In this paper, we study the uniqueness of zero-order entire functions and their difference. We have proved: Let $f(z)$ be a nonconstant entire function of zero order, let $q eq0, eta$ be two finite complex numbers, and let $a$ and $b$ be two distinct complex numbers. If $f(z)$ and $Delta_{q,eta}f(z)$ share $a$, $b$ IM, then $fequiv Delta_{q,eta}f$.
In [Israel J. Math, 2014], Grahl and Nevo obtained a significant improvement for the well-known normality criterion of Montel. They proved that for a family of meromorphic functions $mathcal F$ in a domain $Dsubset mathbb C,$ and for a positive constant $epsilon$, if for each $fin mathcal F$ there exist meromorphic functions $a_f,b_f,c_f$ such that $f$ omits $a_f,b_f,c_f$ in $D$ and $$min{rho(a_f(z),b_f(z)), rho(b_f(z),c_f(z)), rho(c_f(z),a_f(z))}geq epsilon,$$ for all $zin D$, then $mathcal F$ is normal in $D$. Here, $rho$ is the spherical metric in $widehat{mathbb C}$. In this paper, we establish the high-dimension
For a Kahler manifold X, we study a space of test functions W* which is a complex version of H1. We prove for W* the classical results of the theory of Dirichlet spaces: the functions in W* are defined up to a pluripolar set and the functional capacity associated to W* tests the pluripolar sets. This functional capacity is a Choquet capacity. The space W* is not reflexive and the smooth functions are not dense in it for the strong topology. So the classical tools of potential theory do not apply here. We use instead pluripotential theory and Dirichlet spaces associated to a current.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
