Positivity preserving transformations for -binomial coefficients

Several new transformations for -binomial coefficients are found, which have the special feature that the kernel is a polynomial with nonnegative coefficients. By studying the group-like properties of these positivity preserving transformations, as well as their connection with the Bailey lemma, many new summation and transformation formulas for basic hypergeometric series are found. The new -binomial transformations are also applied to obtain multisum Rogers-Ramanujan identities, to find new representations for the Rogers-Szegö polynomials, and to make some progress on Bressoud's generalized Borwein conjecture. For the original Borwein conjecture we formulate a refinement based on new triple sum representations of the Borwein polynomials.

     

On a refinement of a theorem of Landau on Koebe domains

We state and prove a refinement of a classical theorem due to Landau on the Koebe domains for certain families of holomorphic functions introduced by A. W. Goodman. Our geometric approach in this article enables us to derive several statements of interest, which would not be produced via the methods in Goodman's paper, as immediate corollaries of the proof of the main theorem.     

A Schwarz Lemma for Harmonic Mappings Between the Unit Balls in Real Euclidean Spaces

The authors prove a Schwarz lemma for harmonic mappings between the unit balls in real Euclidean spaces. Roughly speaking, this result says that under a harmonic mapping between the unit balls in real Euclidean spaces, the image of a smaller ball centered at origin can be controlled. This extends the related result proved by Chen in complex plane.     

GROWTH AND DISTORTION THEOREMS FOR ALMOST STARLIKE MAPPINGS OF COMPLEX ORDER λ

In this article, the sharp growth theorem for almost starlike mappings of complex order λ is given firstly. Secondly, distortion theorem along a unit direction is also established as the application of the growth theorem. In particular, using our results can reduce to some well-known results.     

Extensions of the sewing lemma with applications

We give several extensions of the sewing lemma of Feyel and de La Pradelle and show how these results generalize Young’s integration theory in a simple and natural way. For illustrative purposes, we apply the lemma to integrals involving discontinuous functions of a fractional Brownian motion with the Hurst index $H>1∕2.$     

Decomposition of group rings and the converse of Schur’s lemma

In this work, we give necessary and su?cient conditions for a group ring (of finite group) to satisfy the converse of Schur’s lemma for group rings of finite groups. Rings considered here are commutative or noncommutative perfect. Some cases of twisted group rings are studied. Also we introduce the notion of semi-CSL ring.     

A generalization of Gale's lemma

In this work, we present a generalization of Gale's lemma. Using this generalization, we introduce two sharp combinatorial lower bounds for and , the two classic topological lower bounds for the chromatic number of a graph G.     

广义山路引理的一些应用

具有(CPS)型条件的Ghoussoub-Preiss广义山路引理是经典的AmbrosettiRabinowitz山路引理的一个推广,本文将应用它来研究给定能量的具有对称性或非对称性势能的二阶哈密尔顿系统周期解的存在性.