Marijan Markovic
In this paper we give a generalization and improvement of the Pavlović result on the characterization of continuously differentiable functions in the Bloch space on the unit ball in $\mathbb{R}^m$. Then we derive a Holland--Walsh type theorem for analytic normal mappings on the unit disk.
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In this paper we prove a normality criterion for the families of meromorphic functions involving sharing of functions. Our result generalizes some of the earlier results on Gu's normality criterion.
Extension of classical Mandelbrojt's criterion for normality of a family of holomorphic zero-free functions of several complex variables is given. We show that a family of holomorphic functions of several complex variables whose corresponding Levi form are uniformly bounded away from zero is normal.
We prove an extension of one direction in Marty's theorem where the spherical derivative is replaced by a more general quantity involving higher derivatives.
Normality criterion for logharmonic mapping
本文针对对数调和映射建立了正规性的充要条件，包括Marty准则、Zalcman-Pang引理和Lohwater-Pommerenke定理，并展示了引理的应用。
Normality criterion for a family of holomorphic curves that partially share wandering hyperplanes with their derivatives, and holomorphic functions lifted to curves in $P^2(\mathbb{C})$
本研究推广了已有复射影空间全纯曲线族正规性结果，得到亚纯函数族分担游荡函数的正规定则，还通过将全纯函数表示为到P²(C)的映射得到新定则，可构造Bloch原理逆命题的反例。
Let ${\cal F}$ be a family of meromorphic functions on a domain $D$. We present a quite general sufficient condition for ${\cal F}$ to be a normal family. This criterion contains many known results as special cases. The overall idea is that certain comparatively weak conditions on ${\cal F}$ locally lead to somewhat stronger conditions, which in turn lead to even stronger conditions on the limit function $g$ in the famous Zalcman Lemma. Ultimately, the defect relations for $g$ force normality of ${\cal F}$.
A Criterion for quasi normality in $\mathbb{C}^n$
本文针对取值于完备复Hermitian流形的全纯函数族，提出了一个Zalcman型重正规化结果，以此作为判定其在$\mathbb{C}^n$中拟正规性的准则。
We show that the family of all holomorphic functions $f$ in a domain $D$ satisfying $$\frac{|f^{(k)}|}{1+|f|}(z)\le C \qquad \mbox{ for all } z\in D$$ (where $k$ is a natural number and $C>0$) is quasi-normal. Furthermore, we give a general counterexample to show that for $α>1$ and $k\ge2$ the condition $$\frac{|f^{(k)}|}{1+|f|^α}(z)\le C \qquad \mbox{ for all } z\in D$$ does not imply quasi-normality.
In this note we give some variations on an extension of Caratheodory's normality criterion due to Grahl and Nevo.
A normality criterion for a family of meromorphic functions
本文提出了一类亚纯函数在特定条件下的正常性准则，涉及齐次微分多项式和全纯函数的交互作用。
In this paper, we continue to discuss normality based on a single\linebreak holomorphic function. We obtain the following result. Let $\CF$ be a family of functions holomorphic on a domain $D\subset\mathbb C$. Let $k\ge2$ be an integer and let $h(\not\equiv0)$ be a holomorphic function on $D$, such that $h(z)$ has no common zeros with any $f\in\CF$. Assume also that the following two conditions hold for every $f\in\CF$:\linebreak %{enumerate} [(a)] (a) $f(z)=0\Longrightarrow f'(z)=h(z)$ and %[(b)] (b) $f'(z)=h(z)\Longrightarrow|f^{(k)}(z)|\le c$, where $c$ is a constant. Then $\CF$ is normal o…
A normality Criterion for a Family of Meromorphic Functions
本文研究了Schwick亚纯函数族正规性准则的相反情形，探讨当导数条件存在零点时函数族的正规性问题。
Normality and Montel's Theorem
本文提出并证明了一个亚纯函数族的正规性准则，该准则涉及零点重数及族中成员共享全纯函数，并在某种意义上推广了Montel的正规性判别法。
In this paper, we first generalize a value distribution result of Lahiri and Dewan [4] and as an application of this result we prove a normality criterion using partial sharing of small functions. Further, in sequel normality criteria of Hu and Meng [3] and Ding, Ding and Yuan [1] are improved and generalized when the domain $D:=\{z:|z|<R,0<R\leq\infty\}$.
Normality and Sharing Functions
本文证明了涉及全纯函数共享的亚纯函数族正规性准则，推广了H. H. Chen、M. L. Fang、M. Han和Y. Gu的部分结果。
In this paper, we prove two normality criteria for a family of meromorphic functions. The first criterion extends a result of Fang and Zalcman[Normal families and shared values of meromorphic functions II, Comput. Methods Funct. Theory, 1(2001), 289 - 299] to a bigger class of differential polynomials whereas the second one leads to some counterexamples to the converse of the Bloch's principle.
In this article we prove a sufficient condition of quasi-normality in higher dimension for a family of meromorphic mappings in which each pair of functions of family shares some moving hypersurfaces. We also prove a normality criterion concerning shared moving hypersurfaces.
On $φ$-Normality of Harmonic Mappings and Their Families
本文为调和映射及其族建立了$\varphi$-正规性的Lappan型准则，涉及高阶球面导数与测试值数量，并推广至多项式情形。
In this paper we prove some normality criteria for a family of meromorphic functions concerning shared analytic functions, which extend or generalized some result obtained by Y. F. Wang, M. L. Fang~\cite{WF} and J. Qui, T. Zhu ~\cite{QZ}.
In the present paper, we introduced the extended bicomplex plane $\bar{\mathbb{T}}$, its geometric model: the bicomplex Riemann sphere, and the bicomplex chordal metric that enables us to talk about the convergence of the sequences of bicomplex meromorphic functions. Hence the concept of the normality of a family of bicomplex meromorphic functions on bicomplex domains emerges. Besides obtaining a normality criterion for such families, the bicomplex analog of the Montel theorem for meromorphic functions and the Fundamental Normality Tests for families of bicomplex holomorphic functions and bico…
Let $f$ be a transcendental meromorphic function defined in the complex plane $\mathbb{C}$, and $φ(\not\equiv 0,\infty)$ be a small function of $f$. In this paper, We give a quantitative estimation of the characteristic function $T(r, f)$ in terms of $N\left(r,\frac{1}{M[f]-φ(z)}\right)$ as well as $\ol{N}\left(r,\frac{1}{M[f]-φ(z)}\right)$, where $M[f]$ is the differential monomial, generated by $f$.\par Moreover, we prove one normality criterion: Let $\mathscr{F}$ be a family of analytic functions on a domain $D$ and let $k(\geq1)$, $q_{0}(\geq 3)$, $q_{i}(\geq0)$ $(i=1,2,\ldots,k-1)$, $q_{k…
Essential normality of quotient submodules over strongly pseudoconvex finite manifolds
该研究针对满足性质（S）的复流形中强拟凸域上的希尔伯特商子模，证明特定解析子簇对应的Bergman-Sobolev商子模的p-本质正规性条件，部分证实几何Arveson-Douglas猜想并解决相关公开问题，还给出K-同调等领域的应用。
We establish a criterion for local boundedness and hence normality of a family $\F$ of analytic functions on a domain $D$ in the complex plane whose corresponding family of derivatives is locally bounded. Furthermore we investigate the relation between domains of normality of a family $\F$ of meromorphic functions and its corresponding Schwarzian derivative family. We also establish some criterion for the Schwarzian derivative family of a family $\F$ of analytic functions on a domain $D$ in the complex plane to be a normal family.
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 $D\subset \mathbb C,$ and for a positive constant $ε$, if for each $f\in \mathcal F$ there exist meromorphic functions $a_f,b_f,c_f$ such that $f$ omits $a_f,b_f,c_f$ in $D$ and $$\min\{ρ(a_f(z),b_f(z)), ρ(b_f(z),c_f(z)), ρ(c_f(z),a_f(z))\}\geq ε,$$ for all $z\in D$, then $\mathcal F$ is normal in $D$. Here, $ρ$ is the spherical metric in $\widehat{\mathbb C}$. In this paper, we establish t…
In this paper, as an application of Zalcman's lemma in $\mathbb{C}^n$, we give a sufficient condition for normality of holomorphic functions of several complex variables, which generalizes previous known one-dimensional criterion of A.J. Lohwater and Ch. Pommerenke \cite[Theorem 1]{MR0338381}.
The biholomorphic invariance of essential normality on bounded symmetric domains
本文通过新积分公式和代数方法，证明了有界对称域上Hilbert模的p-本质正规性在双全纯映射及自同构乘子下保持不变，并推广了无穷本质正规性与超刚性的等价结果。
A Function-Sharing Criterion for Normal Functions
本文通过单位圆盘上三个边界互异的亚纯函数共享同一函数 $f$ 的条件，证明了 $f$ 的正规性，并进一步给出正规函数的另一判定准则。
Avoidance Criteria for Normal Holomorphic Curves on Complex Projective Space
研究全纯曲线族在复射影空间中避开足够多移动超曲面的条件，并探讨共享超平面的全纯曲线族的正常性条件。
Normality Criteria for Differential Monomials and the Sharpness of Lappan-type Theorems
本文研究微分单项的正常性准则，证明通过更高阶导数可减少目标集点数，并推广了Pang-Zalcman定理以更广泛的应用范围。
A Normal Criterion Concerning Sequence of Functions and their Differential Polynomials
本文证明了开单位圆盘内亚纯函数序列的一个正规定则：在极点重数、非零性及微分多项式与扰动全纯函数差值零点数受限条件下，该函数序列正规。
We give a local criterion in terms of a residue current for strong holomorphicity of a meromorphic function on an arbitrary pure-dimensional analytic variety. This generalizes a result by A Tsikh for the case of a reduced complete intersection.
We present a constructive criterion for flatness of a morphism of analytic spaces X -> Y or, more generally, for flatness over Y of a coherent sheaf of modules on X. The criterion is a combination of a simple linear-algebra condition "in codimension zero" and a condition "in codimension one" which can be used together with the Weierstrass preparation theorem to inductively reduce the fibre dimension of the morphism.
In this paper, we obtain a new characterization for univalent harmonic mappings and obtain a structural formula for the associated function which defines the analytic $Φ$-like functions in the unit disk. The new criterion stated in this article for the injectivity of harmonic mappings implies the well-known results of Kas'yanyuk \cite{Kas59} and Brickman \cite{Brick73} for analytic functions, but with a simpler proof than theirs. A number of consequences of the characterization, and examples are also presented. Further investigation provides a new method to construct univalent harmonic mapping…
We give a solvability criterion for a special case of the $μ$-synthesis problem. That is, we prove the necessity and sufficiency of a condition for the existence of an analytic $2 \times 2$ matrix-valued function on the disc subject to a bound on the structured singular value and satisfying a finite set of interpolation conditions. To do this we prove a realization theorem for analytic functions from the disc to the tetrablock. We also obtain a solvability criterion for the problem of analytic interpolation from the disc to the tetrablock.
We deal with a problem of the explicit reconstruction of any holomorphic function $f$ on $\mathbb{C}^2$ from its restricions on a union of complex lines. The validity of such a reconstruction essentially depends on the mutual repartition of these lines, condition that can be analytically described. The motivation of this problem comes also from possible applications in mathematical economics and medical imaging.
Normality arguments are applied to study the oscillation of solutions of $f''+Af=0$, where the coefficient $A$ is analytic in the unit disc $\mathbb{D}$ and $\sup_{z\in\mathbb{D}} (1-|z|^2)^2|A(z)| < \infty$. It is shown that such differential equation may admit a non-normal solution having prescribed uniformly separated zeros.
We prove that, if F is a coherent sheaf of modules over the source of a morphism f:X->Y of complex-analytic spaces, where Y is smooth, then the stalk of F at a point x in X is flat over R, the local ring of the target at f(x) if and only if the n-fold analytic tensor power of this stalk over R (where n = dim R) has no vertical elements. The result implies that if F is a finite module over a morphism f:X->Y of complex algebraic varieties, where Y is smooth and n=dim Y, then the stalk of F at x is R-flat if and only if its n-fold tensor power is a torsionfree R-module. The latter generalizes a c…
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