Some notes on harmonic and holomorphic functions

A note on harmonic continuation of characteristic function

We propose a necessary and sufficient condition for a real-valued function on the real line to be a characteristic function of a probability measures. The statement is given in terms of harmonic functions and completely monotonic functions.

Notes on Stein-Sahi representations and some problems of non $L^2$ harmonic analysis

We discuss one natural class of kernels on pseudo-Riemannian symmetric spaces.

Fixed Point Indices and Invariant Periodic Sets of Holomorphic Systems

This note presents a method to study center families of periodic orbits of complex holomorphic differential equations near singularities, based on some iteration properties of fixed point indices. As an application of this method, we will prove Needham's theorem in a more general version.

A Note on the Weighted Harmonic-Geometric-Arithmetic Means Inequalities

In this note, we derive non trivial sharp bounds related to the weighted harmonic-geometric-arithmetic means inequalities, when two out of the three terms are known. As application, we give an explicit bound for the trace of the inverse of a symmetric positive definite matrix and an inequality related to the coefficients of polynomials with positive roots.

A Note on Gradually Varied Functions and Harmonic Functions

Any constructive continuous function must have a gradually varied approximation in compact space. However, the refinement of domain for $σ-$-net might be very small. Keeping the original discretization (square or triangulation), can we get some interesting properties related to gradual variation? In this note, we try to prove that many harmonic functions are gradually varied or near gradually varied; this means that the value of the center point differs from that of its neighbor at most by 2. It is obvious that most of the gradually varied functions are not harmonic.This note discusses some of…

A note on mean-value properties of harmonic functions on the hypercube

For functions defined on the $n$-dimensional hypercube $I_n (r) = \{\bm{x} \in \mathbb{R}^n ~\vert~ \vert x_i \vert \le r,~ i = 1, 2, \ldots , n\}$ and harmonic therein, we establish certain analogues of Gauss surface and volume mean-value formulas for harmonic functions on the ball in $\mathbb{R}^n$ and their extensions for polyharmonic functions. In particular, our results contribute to the best one-sided $L^1$-approximation by harmonic functions on $I_n (r)$.

Brennan猜想对全纯函数半群成立

Brennan's conjecture holds for semigroups of holomorphic functions

本文针对全纯函数连续半群这一特殊情形，结合经典复分析技术与Békollé-Bonami权和积分算子谱的近期结果，给出了Brennan猜想的一个简短证明。