We show that if a Cantor set E as considered by Garnett in [G2] has positive Hausdorff h-measure for a non-decreasing function h satisfying ∫01 r−3 h(r)2 dr < ∞, then the analytic capacity of E is positive. Our tool will be the Menger three-point curvature and Melnikov’s identity relating it to the Cauchy kernel. We shall also prove some related more general results.

Let $D$ be a Carathéodory domain. For $1\le p\le \infty$, let $L^p\left(D\right)$ be the class of all functions $f$ holomorphic in $D$ such that ${\parallel f\parallel }_{D,p}=[\frac{1}{A}\int {\int }_D^{}{\left|f\left(z\right)\right|}^p{dxdy]}^{1/p}<\infty$, where $A$ is the area of $D$. For $f\in L^p\left(D\right)$, set $$E_n^p\left(f\right)=\underset{t\in {\pi }_n}{inf}{\parallel f-t\parallel }_{D,p};$$${\pi }_n$ consists of all polynomials of degree at most $n$. In this paper we study the growth of an entire function in terms of approximation...

We continue studying the estimation of Bernstein-Walsh type for algebraic polynomials in regions with piecewise smooth boundary.

Let $E={\bigcup }_{n=1}^{\infty }{I}_n$ be the union of infinitely many disjoint closed intervals where $I_n=[a_n$, $b_n]$, $0<a_1<b_1<a_2<b_2<\cdots <b_n<\cdots$, ${lim}_{n\to \infty }{b}_n=\infty .$ Let $\alpha \left(t\right)$ be a nonnegative function and ${\left\{{\lambda }_n\right\}}_{n=1}^{\infty }$ a sequence of distinct complex numbers. In this paper, a theorem on the completeness of the system $\left\{t^{{\lambda }_n}{log}^{m_n}t\right\}$ in $C_0\left(E\right)$ is obtained where $C_0\left(E\right)$ is the weighted Banach space consists of complex functions continuous on $E$ with $f\left(t\right){\mathrm{e}}^{-\alpha \left(t\right)}$ vanishing at infinity.