亚纯函数分担不动点的正规定则（英文）

In this paper, we study the normality criteria of meromorphic functions concerning shared fixed-points, we obtain: Let F be a family of meromorphic functions defined in a domain D. Let n, k ≥ 2 be two positive integers. For every f ∈ F, all of whose zeros have multiplicity at least (nk+2)/(n-1). If f(f(k))nand g(g(k))nshare z in D for each pair of functions f and g, then F is normal.     

SOME NORMALITY CRITERIA OF MEROMORPHIC FUNCTIONS

Let F be a family of functions meromorphic in a domain D, let n ≥ 2 be a positive integer, and let a ≠ 0, b be two finite complex numbers. If, for each f ∈ F, all of whose zeros have multiplicity at least k ＋ 1, and f ＋ a（f^（k））^n≠b in D, then F is normal in D.     

Some Normality Criteria for Families of Meromorphic Functions

Let κ be a positive integer and F be a family of meromorphic functions in a domain D such that for each f ∈ F, all poles of f are of multiplicity at least 2,and all zeros of f are of multiplicity at least κ + 1. Let α and b be two distinct finite complex numbers. If for each f ∈ F, all zeros of f~(κ)-α are of multiplicity at least 2,and for each pair of functions f, g ∈ F, f~(κ)and g~(κ) share b in D, then F is normal in D.     

Normality and shared values concerning differential polynomials

Let F be a family of functions meromorphic in a domain D, let P be a polynomial with either deg P≥3 or deg P = 2 and P having only one distinct zero, and let b be a finite nonzero complex number. If, each pair of functions f and g in F, P (f)f and P (g)g share b in D, then F is normal in D.     

Periodic points and normal families concerning multiplicity

In 1992,Yang Lo posed the following problem:let F be a family of entire functions,let D be a domain in C,and let k 2 be a positive integer.If,for every f∈F,both f and its iteration f~khave no fixed points in D,is F normal in D?This problem was solved by Ess′en and Wu in 1998,and then solved for meromorphic functions by Chang and Fang in 2005.In this paper,we study the problem in which f and f~(k ) have fixed points.We give positive answers for holomorphic and meromorphic functions.(I)Let F be a family of holomorphic functions in a domain D and let k 2 be a positive integer.If,for each f∈F,all zeros of f(z)-z are multiple and f~khas at most k distinct fixed points in D,then F is normal in D.Examples show that the conditions"all zeros of f(z)-z are multiple"and"f~k having at most k distinct fixed points in D"are the best possible.(II)Let F be a family of meromorphic functions in a domain D,and let k 2 and l be two positive integers satisfying l 4 for k=2 and l 3 for k 3.If,for each f∈F,all zeros of f(z)-z have a multiplicity at least l and f~khas at most one fixed point in D,then F is normal in D.Examples show that the conditions"l 3for k 3"and"f~k having at most one fixed point in D"are the best possible.     

关于亚纯函数分担值集正规族的注记(英文)

In the paper,we prove the main result：Let k（≥2）be an integer,and a,b and c be three distinct complex numbers.Let F be a family of functions holomorphic in a domain D in complex plane,all of whose zeros have multiplicity at least k.Suppose that for each f∈F,f（z）and f（k）（z）share the set{a,b,c}.Then F is a normal family in D.     

The normal famility of mermorphic functions

Let F be a family of mermorphic functions in a domain D, and let a, b, c be complex numbers, a ≠ b. If for each f ∈ F, the zeros of f-c are of multiplicity ≥ k ＋ 1, and -↑Ef（k）（a） belong to -↑Ef （a）, -↑Ef（k）（b）belong to -↑Ef （b）, then F is normal in D.     

Complex and p-Adic Meromorphic Functions f′P′( f ),g′P′(g) Sharing a Small Function

Let K be a complete algebraically closed p-adic field of characteristic zero.We apply results in algebraic geometry and a new Nevanlinna theorem for p-adic meromorphic functions in order to prove results of uniqueness in value sharing problems, both on K and on C. Let P be a polynomial of uniqueness for meromorphic functions in K or C or in an open disk. Let f, g be two transcendental meromorphic functions in the whole field K or in C or meromorphic functions in an open disk of K that are not quotients of bounded analytic functions. We show that if f′P′( f) and g′P′(g) share a small function α counting multiplicity, then f = g, provided that the multiplicity order of zeros of P′satisfy certain inequalities. A breakthrough in this paper consists of replacing inequalities n ≥ k+2 or n ≥ k+3 used in previous papers by Hypothesis(G). In the p-adic context, another consists of giving a lower bound for a sum of q counting functions of zeros with(q-1) times the characteristic function of the considered meromorphic function.     

亚纯函数的一个正规定则

We proved:Let F be a family of meromorphic functions in a domain D and a≠0,b∈C.If f′(z)-a(f(z))~2≠b,f≠0 and the poles of f(z)are of multiplicity＞=3 for each f(z)∈F,then F is normal in D.     

分担一个亚纯函数的亚纯函数族的正规定则

Let F be a family of meromorphic functions in D,and let Ψ(≠0) be a meromorphic function in D all of whose poles are simple.Suppose that,for each f ∈F,f≠0 in D.If for each pair of functions {f,g}(?) F,f' and g' share Ψ in D,then F is normal in D.     

Normal Families of Zero-free Meromorphic Functions Ⅱ

Let k, m be two positive integers with m ≤ k and let F be a family of zero-free meromorphic functions in a domain D, let h(z) ≡ 0 be a meromorphic function in D with all poles of h has multiplicity at most m. If, for each f ∈ F, f(k)(z) = h(z) has at most k- m distinct roots(ignoring multiplicity) in D, then F is normal in D. This extends the results due to Chang[1], Gu[3], Yang[11]and Deng[1]etc.     

Quasi-normal Family of Meromorphic Functions Whose Certain Type of Differential Polynomials Have No Zeros

Define the differential operators ?_n for n∈N inductively by ?_1 [f](z)=f(z) and ?_(n+1) [f](z)=f(z)?_n[f](z)+d/dz ?_n[f](z).For a positive integer k≥2 and a positive number δ,let F be the family of functions f meromorphic on domain D■C such that ?_k[f](z)≠0 and |Res(f,a)-j|≥δ for all j∈{0,1,…,k-1} and all simple poles a of f in D.Then F is quasi-normal on D of order 1.     

全纯函数族的正规定则

Let f be a holomorphic function on a domain D Lontaiu in C, and let a be a finite complex number. We denote by -↑Ef/(a) = {z ∈ D : f(z) = a, ignoring multiplicity} the set of all distinct α-points of f. Let F be a family of holomorphic functions on D. If there exist three finite values a, b(≠ O,α) and c(≠ O) such that for every f ∈F, -↑Ef′/(0) Lontain in -↑Ef(α） and -↑Ef′ (b) Lontain in -↑Ef(c), then F is a normal family on D.     

全纯函数的分担值与正规族

Let F be a family of holomorphic functions in a domain D, k be a positive integer, a, b（≠0）, c（≠0） and d be finite complex numbers. If, for each f∈F, all zeros of f-d have multiplicity at least k, f^（k） = a whenever f=0, and f=c whenever f^（k） = b, then F is normal in D. This result extends the well-known normality criterion of Miranda and improves some results due to Chen-Fang, Pang and Xu. Some examples are provided to show that our result is sharp.     

On approximation of smooth functions from null spaces of optimal linear differential operators with constant coefficients

For a real valued function f defined on a finite interval I we consider the problem of approximating f from null spaces of differential operators of the form Ln(ψ) = n ∑ k=0 akψ(k), where the constant coefficients ak ∈ R may be adapted to f . We prove that for each f ∈ C(n)(I), there is a selection of coefficients {a1, ,an} and a corresponding linear combination Sn( f ,t) = n ∑ k=1 bkeλkt of functions ψk(t) = eλkt in the nullity of L which satisfies the following Jackson’s type inequality: f (m) Sn(m )( f ,t) ∞≤ |an|2n|Im|1/1q/ep|λ|λn|n|I||nm1 Ln( f ) p, where |λn| = mka x|λk|, 0 ≤ m ≤ n 1, p,q ≥ 1, and 1p + q1 = 1. For the particular operator Mn(f) = f + 1/(2n) f(2n) the rate of approximation by the eigenvalues of Mn for non-periodic analytic functions on intervals of restricted length is established to be exponential. Applications in algorithms and numerical examples are discussed.     

Degree of L~P Approximation by Operators of Kantorovi(?) Type Satisfying Micchelli Conditions

§1 We see symbols in article, L~∞[a,b]C[a,b], let f(t) be absolute continuous over [a,b], we denote by f∈AC[a,b], L_k~p[a,b]{f:f~(k-1)∈AC[a,b] and f~(k)(t)∈L~p[a,b]}.C_k[a,b]L_k~∞[a,b], W~kL{f:f∈L_k~p[a,b] and ‖f~(k)‖_p≤1}. Let H_n.be set of algebraic polynomials of degree≤n. Let B_n(F) be Bernstein polynomials,P_n(f) be Kantorovi polynomials. We generalize p_n(f). Let T be linear operator C[a,b]AC[a,b],for g(u)∈C[a,b] we have T(g(u),a)=g(a), T(g(u),b)=g(b), let f(t)∈L[a,b], F(u) =integral from n=0 to u(f(t)dt),     

关于分担多项式的全纯函数的正规族(英文)

In this paper,we study normal families of holomorphic function concerning shared a polynomial.Let F be a family of holomorphic functions in a domain D,k（2）be a positive integer,K be a positive number andα（z）be a polynomial of degree p（p 1）.For each f∈F and z∈D,if f and f sharedα（z）CM and｜f（k）（z）｜K whenever f（z）-α（z）=0 in D, then F is normal in D.     

A normal criterion about two families of meromorphic functions concerning shared values

In this paper, we mainly discuss the normality of two families of functions concerning shared values and proved: Let F and G be two families of functions meromorphic on a domain D■C,a1, a2, a3, a4 be four distinct finite complex numbers. If G is normal, and for every f ∈ F , there exists g ∈ G such that f(z) and g(z) share the values a1, a2, a3, a4, then F is normal on D.     

（g, f）-Factorizations Randomly Orthogonal to a Subgraph in Graphs

Let G be a graph with vertex set V(G) and edge set E(G) and let g and f be two integervalued functions defined on V(G) such that 2k - 2 ≤g(x)≤f(x) for all x∈V(G). Let H be a subgraph of G with mk edges. In this paper, it is proved that every (mg m-1,mf-m 1)-graph G has (g, f)-factorizations randomly k-orthogonal to H under some special conditions.     

$c$-半层空间与几乎完全正则空间一点注记

In this paper, we give some characterizations of almost completely regular spaces and c-semistratifiable spaces(CSS) by semi-continuous functions. We mainly show that:(1)Let X be a space. Then the following statements are equivalent:(i) X is almost completely regular.(ii) Every two disjoint subsets of X, one of which is compact and the other is regular closed, are completely separated.(iii) If g, h : X → I, g is compact-like, h is normal lower semicontinuous, and g ≤ h, then there exists a continuous function f : X → I such that g ≤ f ≤ h;and(2) Let X be a space. Then the following statements are equivalent:(a) X is CSS;(b) There is an operator U assigning to a decreasing sequence of compact sets(Fj)j∈N,a decreasing sequence of open sets(U(n,(Fj)))n∈N such that(b1) Fn■U(n,(Fj)) for each n ∈ N;(b2)∩n∈NU(n,(Fj)) =∩n∈NFn;(b3) Given two decreasing sequences of compact sets(Fj)j∈N and(Ej)j∈N such that Fn■Enfor each n ∈ N, then U(n,(Fj))■U(n,(Ej)) for each n ∈ N;(c) There is an operator Φ : LCL(X, I) → USC(X, I) such that, for any h ∈ LCL(X, I),0 Φ(h) h, and 0 Φ(h)(x) h(x) whenever h(x) 0.