Related papers: Some sharp Schwarz type estimates and their applic…
Let $\alpha>-1$ and assume that $f$ is $\alpha-$harmonic mapping defined in the unit disk that belongs to the Hardy class $h^p$ with $p\ge 1$. We obtain some sharp estimates of the type $|f(z)|\le g(|r|) \|f^\ast\|_p$ and $|Df(z)|\le…
We prove a sharp version of the Hopf boundary point lemma for Black-Scholes type equations. We also investigate the existence and the regularity of the spatial derivative of the solutions at the spatial boundary.
The purpose of this paper is twofold. One is to enrich from a geometrical point of view the theory of octonionic slice regular functions. We first prove a boundary Schwarz lemma for slice regular self-mappings of the open unit ball of the…
We study the structure of the space of coarse Lipschitz maps between Banach spaces. In particular we introduce the notion of norm attaining coarse Lipschitz maps. We extend to the case of norm attaining coarse Lipschitz equivalences, a…
In this paper, we establish some Schwarz type lemmas for mappings $\Phi$ satisfying the inhomogeneous biharmonic Dirichlet problem $ \Delta (\Delta(\Phi)) = g$ in $\mathbb{D}$, $\Phi=f$ on $\mathbb{T}$ and $\partial_n \Phi=h$ on…
We prove several sharp distortion and monotonicity theorems for spherically convex functions defined on the unit disk involving geometric quantities such as spherical length, spherical area and total spherical curvature. These results can…
We prove a Schwarz-type lemma for noncompact manifolds with possibly noncompact boundary. The result is a consequence of a suitable form of the weak maximum principle of independent interest. The paper is enriched with applications to…
We prove a version of the Schwarz lemma for holomorphic mappings from the unit disk into the symmetric product of a Riemann surface. Our proof is function-theoretic and self-contained. The main novelty in our proof is the use of the…
We prove a Hopf bifurcation theorem in general Banach spaces, which improves a classical result by Crandall and Rabinowitz. Actually, our theorem does not need any compactness conditions, which leads to wider applications. In particular,…
The aim of this paper is twofold. First, we obtain a Schwarz-Pick type lemma for the $\alpha$-harmonic mapping $u=P_{\alpha}[\phi]$, where $\phi\in L^{p}(\mathbb{S}^{n-1},\mathbb{R} )$ and $p\in[1,\infty]$. We get an explicit form of the…
For a family of systems of linear elasticity with rapidly oscillating periodic coefficients, we establish sharp boundary estimates with either Dirichlet or Neumann conditions, uniform down to the microscopic scale, without smoothness…
In this paper we give sharp extension results for convoluted solutions of abstract Cauchy problems in Banach spaces. The main technique is the use of algebraic structure (for usual convolution product $\ast$) of these solutions which are…
These notes concern the nonlinear geometry of Banach spaces, asymptotic uniform smoothness and several Banach-Saks-like properties. We study the existence of certain concentration inequalities in asymptotically uniformly smooth Banach…
Existing concentration bounds for bounded vector-valued random variables include extensions of the scalar Hoeffding and Bernstein inequalities. While the latter is typically tighter, it requires knowing a bound on the variance of the random…
In this paper, we give a general boundary Schwarz lemma for holomorphic mappings between unit balls in any dimensions. It is proved that if the mapping $f\in C^{1+\alpha}$ at $z_0\in \partial \mathbb B^n$ with $f(z_0)=w_0\in \partial…
The question of extension of locally defined maps to the entire space arises in many problems of analysis (e.g., local linearization of functional equations). A known classical method of extension of smooth local maps on Banach spaces uses…
This is an attempt to build Banach space valued theory for certain singular integrals on Hamming cube. Of course all estimates below are dimension independent, and we tried to find ultimate sharp assumptions on the Banach space for a…
In this paper we lay the foundations for the Morse theoretical study of strongly indefinite functionals on Banach manifolds by developing the local theory for a specific model class that captures several key analytical features also arising…
The numerical range of holomorphic mappings arises in many aspects of nonlinear analysis, finite and infinite dimensional holomorphy, and complex dynamical systems. In particular, this notion plays a crucial role in establishing exponential…
The Schwarz lemma as one of the most influential results in complex analysis and it has a great impact to the development of several research fields, such as geometric function theory, hyperbolic geometry, complex dynamical systems, and…