The quasiconformal subinvariance property of John domains in \protect \IR^n and its application

The main aim of this paper is to give a complete solution to one of the open problems, raised by Heinonen from 1989, concerning the subinvariance of John domains under quasiconformal mappings in \IR^n. As application, the quasisymmetry of quasiconformal mappings is discussed.Comment: 61 pages, under revision for Mathematische. Annale

Injectivity of sections of convex harmonic mappings and convolution theorems

In the article the authors consider the class ${\mathcal H}_0$ of sense-preserving harmonic functions $f=h+\overline{g}$ defined in the unit disk $|z|<1$ and normalized so that $h(0)=0=h'(0)-1$ and $g(0)=0=g'(0)$, where $h$ and $g$ are analytic in the unit disk. In the first part of the article we present two classes $\mathcal{P}_H^0(\alpha)$ and $\mathcal{G}_H^0(\beta)$ of functions from ${\mathcal H}_0$ and show that if $f\in \mathcal{P}_H^0(\alpha)$ and $F\in\mathcal{G}_H^0(\beta)$, then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters $\alpha$ and $\beta$ are satisfied. In the second part we study the harmonic sections (partial sums) $$s_{n, n}(f)(z)=s_n(h)(z)+\overline{s_n(g)(z)},$$ where $f=h+\overline{g}\in {\mathcal H}_0$, $s_n(h)$ and $s_n(g)$ denote the $n$-th partial sums of $h$ and $g$, respectively. We prove, among others, that if $f=h+\overline{g}\in{\mathcal H}_0$ is a univalent harmonic convex mapping, then $s_{n, n}(f)$ is univalent and close-to-convex in the disk $|z|< 1/4$ for $n\geq 2$, and $s_{n, n}(f)$ is also convex in the disk $|z|< 1/4$ for $n\geq2$ and $n\neq 3$. Moreover, we show that the section $s_{3,3}(f)$ of $f\in {\mathcal C}_H^0$ is not convex in the disk $|z|<1/4$ but is shown to be convex in a smaller disk.Comment: 16 pages, 3 figures; To appear in Czechoslovak Mathematical Journa

Targeting EGF-receptor(s) - STAT1 axis attenuates tumor growth and metastasis through downregulation of MUC4 mucin in human pancreatic cancer.

Transmembrane proteins MUC4, EGFR and HER2 are shown to be critical in invasion and metastasis of pancreatic cancer. Besides, we and others have demonstrated de novo expression of MUC4 in ~70-90% of pancreatic cancer patients and its stabilizing effects on HER2 downstream signaling in pancreatic cancer. Here, we found that use of canertinib or afatinib resulted in reduction of MUC4 and abrogation of in vitro and in vivo oncogenic functions of MUC4 in pancreatic cancer cells. Notably, silencing of EGFR family member in pancreatic cancer cells decreased MUC4 expression through reduced phospho-STAT1. Furthermore, canertinib and afatinib treatment also inhibited proliferation, migration and survival of pancreatic cancer cells by attenuation of signaling events including pERK1/2 (T202/Y204), cyclin D1, cyclin A, pFAK (Y925) and pAKT (Ser473). Using in vivo bioluminescent imaging, we demonstrated that canertinib treatment significantly reduced tumor burden (P=0.0164) and metastasis to various organs. Further, reduced expression of MUC4 and EGFR family members were confirmed in xenografts. Our results for the first time demonstrated the targeting of EGFR family members along with MUC4 by using pan-EGFR inhibitors. In conclusion, our studies will enhance the translational acquaintance of pan-EGFR inhibitors for combinational therapies to combat against lethal pancreatic cancer

Impaired expression of protein phosphatase 2A subunits enhances metastatic potential of human prostate cancer cells through activation of AKT pathway.

BACKGROUND: Protein phosphatase 2A (PP2A) is a dephosphorylating enzyme, loss of which can contribute to prostate cancer (PCa) pathogenesis. The aim of this study was to analyse the transcriptional and translational expression patterns of individual subunits of the PP2A holoenzyme during PCa progression. METHODS: Immunohistochemistry (IHC), western blot, and real-time PCR was performed on androgen-dependent (AD) and androgen-independent (AI) PCa cells, and benign and malignant prostate tissues for all the three PP2A (scaffold, regulatory, and catalytic) subunits. Mechanistic and functional studies were performed using various biochemical and cellular techniques. RESULTS: Through immunohistochemical analysis we observed significantly reduced levels of PP2A-A and -B\u27γ subunits (P CONCLUSION: We conclude that loss of expression of scaffold and regulatory subunits of PP2A is responsible for its altered function during PCa pathogenesis

On the Bohr inequality

The Bohr inequality, first introduced by Harald Bohr in 1914, deals with finding the largest radius $r$, $0<r<1$, such that $\sum_{n=0}^\infty |a_n|r^n \leq 1$ holds whenever $|\sum_{n=0}^\infty a_nz^n|\leq 1$ in the unit disk $\mathbb{D}$ of the complex plane. The exact value of this largest radius, known as the \emph{Bohr radius}, has been established to be $1/3.$ This paper surveys recent advances and generalizations on the Bohr inequality. It discusses the Bohr radius for certain power series in $\mathbb{D},$ as well as for analytic functions from $\mathbb{D}$ into particular domains. These domains include the punctured unit disk, the exterior of the closed unit disk, and concave wedge-domains. The analogous Bohr radius is also studied for harmonic and starlike logharmonic mappings in $\mathbb{D}.$ The Bohr phenomenon which is described in terms of the Euclidean distance is further investigated using the spherical chordal metric and the hyperbolic metric. The exposition concludes with a discussion on the $n$-dimensional Bohr radius