Gejun Bao

ON CERTAIN GENERALIZED CLASS OF NON-BAZILEVIˇ C FUNCTIONS

In this paper, a subclass N(‚,fi,A,B) of analytic functions is introduced, which is a generalized class of non-Bazilevic functions. The sub- ordination relations, inclusion relations, distortion theorems and inequality properties are discussed by applying dierential subordination method.

Some Caccioppoli Estimates for Differential Forms

We prove the global Caccioppoli estimate for the solution to the nonhomogeneous -harmonic equation , which is the generalization of the quasilinear equation . We will also give some examples to see that not all properties of functions may be deduced to differential forms.

The Reverse Hölder Inequality for the Solution to p-Harmonic Type System

Some inequalities to -harmonic equation have been proved. The -harmonic equation is a particular form of -harmonic type system only when and . In this paper, we will prove the Poincaré inequality and the reverse Hölder inequality for the solution to the -harmonic type system.

The Obstacle Problem for the

Firstly, we define an order for differential forms. Secondly, we also define the supersolution and subsolution of the -harmonic equation and the obstacle problems for differential forms which satisfy the -harmonic equation, and we obtain the relations between the solutions to -harmonic equation and the solution to the obstacle problem of the -harmonic equation. Finally, as an application of the obstacle problem, we prove the existence and uniqueness of the solution to the -harmonic equation on a bounded domain with a smooth boundary , where the -harmonic equation satisfies where is any given differential form which belongs to .Firstly, we define an order for differential forms. Secondly, we also define the supersolution and subsolution of the Open image in new window -harmonic equation and the obstacle problems for differential forms which satisfy the Open image in new window -harmonic equation, and we obtain the relations between the solutions to Open image in new window -harmonic equation and the solution to the obstacle problem of the Open image in new window -harmonic equation. Finally, as an application of the obstacle problem, we prove the existence and uniqueness of the solution to the Open image in new window -harmonic equation on a bounded domain Open image in new window with a smooth boundary Open image in new window , where the Open image in new window -harmonic equation satisfies Open image in new window where Open image in new window is any given differential form which belongs to Open image in new window .

Some starlikeness criterions for analytic functions.

We determine the condition on , , , and for which implies , where is the class of starlike functions of order . Some results of Obradović and Owa are extended. We also obtain some new results on starlikeness criterions.

The existence of solutions to the nonhomogeneous A -harmonic equation

In this paper, we introduce the obstacle problem about the nonhomogeneous A-harmonic equation. Then, we prove the existence and uniqueness of solutions to the nonhomogeneous A-harmonic equation and the obstacle problem.

Global regularity for solutions of a class of quasilinear elliptic equations

We derive interior and global Hölder estimates for solutions of a class of quasilinear elliptic equations. First, the interior Hölder continuity is obtained by the iteration of an oscillation estimate. Then, the Hölder continuity up to the boundary is established in domains with certain boundary constraints. Last, we prove the global Hölder continuity of solutions provided that their restrictions on boundary are Hölder continuous. The concluding section presents an application to illustrate our main results.MSC:35B65, 35J60.

Poincaré inequalities and the sharp maximal inequalities with Lφ-norms for differential forms

This paper is concerned with the Poincaré inequalities and the sharp maximal inequalities for differential forms with Lφ-norm, where φ satisfies nonstandard growth conditions. These results can be used to estimate the norms of classical operators and analyze integral properties of differential forms.

A note on certain classes of multivalent analytic functions

A theorem involving multivalent analytic functions is considered and then its certain consequences are given:

Weak reverse Hölder inequality of weakly A-harmonic sensors and Hölder continuity of A-harmonic sensors

In this paper, we obtain the weak reverse Hölder inequality of weakly A-harmonic sensors and establish the Hölder continuity of A-harmonic sensors.Mathematics Subject Classification 2010: 58A10 · 35J60