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Register a new userHigher dimensional generalizations of some theorems on normality of meromorphic functions
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Tran Van Tan
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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
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In this article, we prove some normality criteria for a family of meromorphic functions having zeros with some multiplicity. Our main result involves sharing of a holomorphic function by certain differential polynomials. Our results generalize some of the results of Fang and Zalcman and Chen et al to a great extent.
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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.
Let ${b_{j}}_{j=1}^{k}$ be meromorphic functions, and let $w$ be admissible meromorphic solutions of delay differential equation $$w(z)=w(z)left[frac{P(z, w(z))}{Q(z,w(z))}+sum_{j=1}^{k}b_{j}(z)w(z-c_{j})right]$$ with distinct delays $c_{1}, ldots, c_{k}inmathbb{C}setminus{0},$ where the two nonzero polynomials $P(z, w(z))$ and $Q(z, w(z))$ in $w$ with meromorphic coefficients are prime each other. We obtain that if $limsup_{rrightarrowinfty}frac{log T(r, w)}{r}=0,$ then $$deg_{w}(P/Q)leq k+2.$$ Furthermore, if $Q(z, w(z))$ has at least one nonzero root, then $deg_{w}(P)=deg_{w}(Q)+1leq k+2;$ if all roots of $Q(z, w(z))$ are nonzero, then $deg_{w}(P)=deg_{w}(Q)+1leq k+1;$ if $deg_{w}(Q)=0,$ then $deg_{w}(P)leq 1.$par In particular, whenever $deg_{w}(Q)=0$ and $deg_{w}(P)leq 1$ and without the growth condition, any admissible meromorphic solution of the above delay differential equation (called Lenhart-Travis type logistic delay differential equation) with reduced form can not be an entire function $w$ satisfying $overline{N}(r, frac{1}{w})=O(N(r, frac{1}{w}));$ while if all coefficients are rational functions, then the condition $overline{N}(r, frac{1}{w})=O(N(r, frac{1}{w}))$ can be omitted. Furthermore, any admissible meromorphic solution of the logistic delay differential equation (that is, for the simplest special case where $k=1$ and $deg_{w}(P/Q)=0$ ) satisfies that $N(r,w)$ and $T(r, w)$ have the same growth category. Some examples support our results.
We give conditions characterizing holomorphic and meromorphic functions in the unit disk of the complex plane in terms of certain weak forms of the maximum principle. Our work is directly inspired by recent results of John Wermer, and by the theory of the projective hull of a compact subset of complex projective space developed by Reese Harvey and Blaine Lawson.
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