Some inequalities related to differential monomials

Region of variability for certain classes of univalent functions satisfying differential inequalities

In this paper we determine the region of variability for certain subclasses of univalent functions satisfying differential inequalities. In the final section we graphically illustrate the region of variability for several sets of parameters.

Asymptotic expansions and summability with respect to an analytic germ

In a previous article of the authors with M. Canalis-Durand, monomial asymptotic expansions, Gevrey asymptotic expansions and monomial summability were introduced and applied to certain systems of singularly perturbed differential equations. In the present work, we extend this concept, introducing (Gevrey) asymptotic expansions and summability with respect to a germ of an analytic function in several variables - this includes polynomials. The reduction theory of singularities of curves and monomialization of germs of analytic functions are crucial to establish properties of the new notions, fo…

Isoperimetric type inequalities for mappings induced by weighted Laplace differential operators

The main purpose of this paper is to establish some isoperimetric type inequalities for mappings induced by the weighted Laplace differential operators. The obtained results of this paper provide improvements and extensions of the corresponding known results.

微分单项的正常性准则及Lappan型定理的精确性

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。

Differential Inequalities, Normality and Quasi-Normality

We prove that if D is a domain in C, alpha>1 and c>0, then the family F of functions meromorphic in D such that |f'(z)|/(1+|f(z)|^alpha)>c for every z in D is normalin D. For alpha=1, the same assumptions imply quasi-normality but not necessarily normality.

Quasi-normality induced by differential inequalities

We show that the family ${\cal F}_k$ of all meromorphic functions $f$ in a domain $D$ satisfying $$\frac{|f^{(k)}|}{1+|f|}(z)\ge C \qquad \mbox{ for all } z\in D$$ (where $k$ is a natural number and $C>0$) is quasi-normal. The proof relies mainly on the Zalcman-Pang rescaling method.

Uniqueness in Rough Almost Complex Structures and Differential Inequalities

We prove that for almost complex structures of Hölder class at least 1/2, any J-holomorphic disc, that is constant on some non empty open set, is constant. This is in striking contrast with well known, trivial, non-uniqueness results. We also investigate uniqueness questions (do vanishing on some open set, or vanishing to infinite order, or having a non isolated zero, imply vanishing) in connection with differential inequalities that arise in the theory of almost complex manifolds. The case of vector valued functions is different from the case of scalar valued functions.

On classes of meromorphic locally univalent functions defined by differential inequalities

n this article we consider functions meromorphic in the unit disk. We give an elementary proof for a condition that is sufficient for the univalence of such functions which also contains some known results. We include few open problems for further research.

Differential inequalities and quasi-normal families

We show that a family ${\cal F}$ of meromorphic functions in a domain $D$ satisfying $$\frac{|f^{(k)}|}{1+|f^{(j)}|^α}(z)\ge C \qquad \mbox{for all} z\in D \mbox{and all} f\in {\cal F}$$ (where $k$ and $j$ are integers with $k>j\ge 0$ and $C>0$, $α>1$ are real numbers) is quasi-normal. Furthermore, if all functions in ${\cal F}$ are holomorphic, the order of quasi-normality of ${\cal F}$ is at most $j-1$. The proof relies on the Zalcman rescaling method and previous results on differential inequalities constituting normality.

分数阶微分与积分算子的Bohr型不等式

Bohr-Type Inequalities for Fractional Differential and Integral Operators

本文在分数阶微积分框架下，利用Riemann-Liouville算子为解析函数建立广义Bohr半径，统一并推广了经典结果，且数值验证了半径随分数阶阶数递减的规律。

Some classes of order α for second-order differential inequalities

For analytic functions f(z) in the open unit disk U with f(0)=f'(0)-1=0, S. S. Miller and P. T. Mocanu (Integral Transform. Spec. Funct. 19(2008)) have considered some sufficient problems for starlikeness. The object of the present paper is to discuss some sufficient problems for f(z) to be in some classes of order α.

Differential inequalities and a Marty-type criterion for quasi-normality

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.

Generalized algebraic Morse inequalities and jet differentials

We give a fully algebraic proof of an important theorem of Demailly, stating the existence of many Green-Griffiths jet differentials on a complex projective manifold of general type. To this end, we introduce a new algebraic version of the Morse inequalities, which we then use in our proof as an algebraic counterpart to Demailly's and Bonavero's holomorphic Morse inequalities.

Algebraic differential equations associated to some polynomials

We compute the Gauss-Manin differential equation for any period of a polynomial in \ $\C[x_{0},\dots, x_{n}]$ \ with \ $(n+2)$ \ monomials. We give two general factorizations theorem in the algebra \ $\C< z, (\frac{\partial}{\partial z})^{-1}>$ \ for such a differential equations.

Complement to algebraic differential equations ...

We complete the study of some periods of polynomials in (n+1) variables with (n+2) monomials in computing the behavior of these periods in the natural parameter for such a polynomial.

紧Kähler流形上扭曲微分形式的热核估计与Sobolev型不等式

Estimates of heat kernels and Sobolev-type inequalities for twisted differential forms on compact Kähler manifolds

本文将Sobolev型不等式从函数情形推广至紧Kähler流形上的扭曲微分形式，建立了全纯向量丛取值微分形式的热核估计，并以此证明了消失定理与$\bar\partial$-算子的$L^{q,p}$-估计。

Some nonlinear differential inequalities and an application to Hölder continuous almost complex structures

We consider some second order quasilinear partial differential inequalities for real valued functions on the unit ball and find conditions under which there is a lower bound for the supremum of nonnegative solutions that do not vanish at the origin. As a consequence, for complex valued functions $f(z)$ satisfying $\partial f/\partial\bar z=|f|^α$, $0<α<1$, and $f(0)\ne0$, there is also a lower bound for $\sup|f|$ on the unit disk. For each $α$, we construct a manifold with an $α$-Hölder continuous almost complex structure where the Kobayashi-Royden pseudonorm is not upper semicontinuous.