In this paper we prove the following theorem: % Let $f(z)$ be a meromorphic function of infinite order. If $ \sum \limits_{a \neq \infty} \delta (a,f) + \delta (\infty, f) = 2,$ then for each positive integer $k$, we have % $K(f^{(k)}) = \frac{2k(1 - \delta(\infty, f))} {1 + k - k\delta (\infty, f)},$ % where % $ K(f^{(k)}) = \lim \limits_{r \rightarrow \infty } (N(r, 1 / f^{(k)}) + N(r,f^{(k)})) / T(r,f^{(k)})$ exists. % This result improves the results by S. K. Singh and V. N. Kulkarni [1] and Mingliang Fang [2].

In this paper, we study the monotonicity of some weighted differences between any two terms in chains of Jensen's inequalities for convex functions. The problem is reduced to the solvability of some weight equations or weight inequalities. Our proofs are based on the classical Jensen's inequality and some elementary identity involved combinatorial numbers.

We establish a sharp inequality for multilinear Littlewood-Paley operator. As application, we obtain the weighted norm inequalities and $L\log L$ type endpoint estimate for the multilinear operator.

In this paper, we investigate the entire functions that share a polynomial with one of their derivatives and prove several theorems which generalize the main results given by L. Z. Yang in [14].

Let $\CF$ be a family of functions meromorphic on the plane domain $D$, all of whose zeros are multiple. Suppose that $f^{(k)}(z)\ne1$ for all $f\in \CF$ and $z\in D.$ Then if $\CF$ is quasinormal on $D$, it is quasinormal of order 1 there.

We study an analytically irreducible algebroid germ $(X, 0)$ of complex singularity by considering the filtrations of its analytic algebra, and their associated graded rings, induced by the {\it divisorial valuations} associated to the irreducible components of the exceptional divisor of the normalized blow-up of the normalization $(\bar{X}, 0)$ of $(X, 0)$, centered at the point $0 \in \bar{X}$. If $(X, 0)$ is a quasi-ordinary hypersurface singularity, we obtain that the associated graded ring is a $\C$-algebra of finite type, namely the coordinate ring of a non necessarily normal affine toric variety of the form $Z^\Gamma = \mbox{\rm Spec} \C [\Gamma]$, and we show that the semigroup $\Gamma$ is an analytical invariant of $(X, 0)$. This provides another proof of the analytical invariance of the {\it normalized characteristic monomials} of $(X, 0)$. If $(X, 0)$ is the algebroid germ of non necessarily normal toric variety, we apply the same method to prove a local version of the isomorphism problem for algebroid germs of non necessarily normal toric varieties (solved by Gubeladze in the algebraic case).

Reductive models $((\frak g,\frak h),H)$ for Cartan geometries are showed to fall into two classes, symmetric and non symmetric type, according to the existence or non existence of a mutation $\frak g'=\frak h\oplus\frak m$ where the $H$-module $\frak m$ is an abelian subalgebra. Sasakian structures are showed to be Cartan geometries having a model of non symmetric type and other examples of models of this type are exhibited. Reductive models for which no Cartan space forms exist are constructed. The phenomenon of non-existence of Cartan space forms pertains to models of non symmetric type.