Related papers: On harmonic functions and the Schwarz lemma

We compare the distribution function and the maximum of solutions of nonlinear elliptic equations defined in general domains with solutions of similar problems defined in a ball using Schwarz symmetrization. As an application, we prove the…

We evaluate binomial series with harmonic number coefficients, providing recursion relations, integral representations, and several examples. The results are of interest to analytic number theory, the analysis of algorithms, and…

We call the solution of a kind of second order homogeneous partial differential equation as real kernel alpha-harmonic mappings. In this paper, the representation theorem, the Lipschitz continuity, the univalency and the related problems of…

For $-1\leq B<A\leq 1$, let $\mathcal{C}(A,B)$ denote the class of normalized Janowski convex functions defined in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$ that satisfy the subordination relation $1+zf''(z)/f'(z)\prec…

We derive some consequences of the Liouville theorem for plurisubharmonic functions of L.-F. Tam and the author. The first result provides a nonlinear version of the complex splitting theorem (which splits off a factor of $\mathbb{C}$…

For an infinite penny graph, we study the finite-dimensional property for the space of harmonic functions, or ancient solutions of the heat equation, of polynomial growth. We prove the asymptotically sharp dimensional estimate for the above…

We establish sharp $L^p$ integral mean estimates for $(\alpha,\beta)$-harmonic functions on the unit disk. Explicit bounds for the functions and their partial derivatives are obtained in terms of boundary data, by means of the associated…

Any three circles theorem for discrete harmonic functions must contain an inherent error term. In this paper we find the sharp error term in an $L^2$-three circles theorem for harmonic functions defined in $\Zb^2$. The proof is highly…

We investigate conditions for logarithmic complete monotonicity of product ratios of gamma and q-gamma functions whose arguments are linear functions of the variable. We give necessary and sufficient conditions in terms of nonnegativity of…

We study noncommutative versions of holomorphic and harmonic functions on the unit disk.

In this paper, two sharp inequalities for bounding the psi function $\psi$ and the harmonic numbers $H_n$ are established respectively, some results in [I. Muqattash and M. Yahdi, \textit{Infinite family of approximations of the Digamma…

The explicit formulas expressing harmonic sums via alternating Euler sums (colored multiple zeta values) are given, and some explicit evaluations are given as applications.

We discuss the behavior of harmonic functions on Riemannian cones as defined below and Lioville's theorem.

A mathematically correct approach to study theories with infinite-dimensional groups of symmetries is presented. It is based on quasi-invariant measures on the groups. In this paper, the properties of the measure on the group of…

We introduce different classical characteristics used to regularize a subharmonic function and compare them. As an application we give a complete proof of a useful characterization of the modulus of continuity of such functions in terms of…

In this paper, we give a new generalization of the Bohr inequality in refined form both for bounded analytic functions, and for sense-preserving harmonic functions with analytic part being bounded.

We obtain a sharp estimate on the norm of the differential of a harmonic map from the unit disc $\mathbb D$ in $\mathbb C$ into the unit ball $\mathbb B^n$ in $\mathbb R^n$, $n\ge 2$, at any point where the map is conformal. In dimension…

We give an example of a sequence of positive harmonic functions on $\mathbb{Z}^d$, $d\geq 2$, that converges pointwise to a non-harmonic function.

We modify the proof of the basic lemma of a paper of Saks and Zygmund on additive functions of rectangles.

In this paper we prove a compactness theorem for a sequence of harmonic maps which are defined on a converging sequence of Riemannian manifolds.