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Let $\Omega \subset \mathbb{R}^n$ be a convex domain and let $f:\Omega \rightarrow \mathbb{R}$ be a subharmonic function, $\Delta f \geq 0$, which satisfies $f \geq 0$ on the boundary $\partial \Omega$. Then $$ \int_{\Omega}{f ~dx} \leq…

Let $r\geq3$ and $G$ be an $r$-uniform hypergraph with vertex set $\left\{ 1,\ldots,n\right\} $ and edge set $E$. Let \[ \mu\left( G\right) :=\max {\textstyle\sum\limits_{\left\{ i_{1},\ldots,i_{r}\right\} \in E}} x_{i_{1}}\cdots x_{i_{r}},…

We prove existence of solutions for a class of singular elliptic problems with a general measure as source term whose model is $$\begin{cases} -\Delta u = \frac{f(x)}{u^{\gamma}} +\mu & \text{in}\ \Omega, u=0 &\text{on}\ \partial\Omega, u>0…

We prove the continuity of Sobolev functions $\varphi \in W^{1,n}_{\mathrm{loc}}(\Omega)$, $\Omega \subset \mathbb{R}^n$, that satisfy \[ \lvert\nabla \varphi(x)\rvert^n \le K(x)\bigl(\langle \nabla \varphi(x), \xi(x)\rangle + A(x)\bigr),…

Let $\Omega$ be a domain in a smooth complete Finsler manifold, and let $G$ be the largest open subset of $\Omega$ such that for every $x$ in $G$ there is a unique closest point from $\partial \Omega$ to $x$ (measured in the Finsler…

Let $\mathcal{S}$ denote the class of functions analytic and univalent (i.e. one-to-one) in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:\, |z|<1\}$ normalized by $f(0)=0=f'(0)-1$. The logarithmic coefficients $\gamma_n$ of $f\in\mathcal{S}$…

This paper is concerned with the Dirichlet problem for an equation involving the 1--Laplacian operator $\Delta_1 u$ and having a singular term of the type $\frac{f(x)}{u^\gamma}$. Here $f\in L^N(\Omega)$ is nonnegative, $0<\gamma\le1$ and…

A normalized univalent function is uniformly convex if it maps every circular arc contained in the open unit disk with center in it into a convex curve. This article surveys recent results on the class of uniformly convex functions and on…

For $\Gamma$ a cocompact or cofinite Fuchsian group, we study the lattice point problem on the Riemann surface $\Gamma\backslash\mathbb{H}$. The main asymptotic for the counting of the orbit $\Gamma z$ inside a circle of radius $r$ centered…

We consider three classes of functions defined using the class $\mathcal{P}$ of all analytic functions $p(z)=1+cz+\dotsb$ on the open unit disk having positive real part and study several radius problems for these classes. The first class…

Let $f:[0,+\infty) \to \mathbb{R}$ be a (locally) Lipschitz function and $\Omega \subset \mathbb{R}^2$ a $C^{1,\alpha}$ domain whose boundary is unbounded and connected. If there exists a positive bounded solution to the overdetermined…

Let $\Omega $ be a bounded domain in $\mathbb{R} ^N $, and let $u\in C^1 (\overline{\Omega }) $ be a weak solution of the following overdetermined BVP: $-\nabla (g(|\nabla u|)|\nabla u|^{-1} \nabla u )=f(|x|,u)$, $ u>0 $ in $\Omega $ and…

For the standard Ma-Minda class $\mathcal{S}^{*}(\psi)$ of univalent starlike functions, we derive $\mathcal{S}^{*}(\psi)$-radii for some well-known special functions. In addition, we obtain the set of extremal functions for the classical…

We consider certain inequalities among the Apollonian metric, the Apollonian inner metric, the $j$ metric and the quasihyperbolic metric. We verify that whether these inequalities can occur in simply connected planar domains and in proper…

We study the elliptic inclusion given in the following divergence form \begin{align*} & -\mathrm{div}\, A(x,\nabla u) \ni f\quad \mathrm{in}\quad \Omega, & u=0\quad \mathrm{on}\quad \partial \Omega. \end{align*} As we assume that $f\in…

Let $\mathcal{A}$ be a unital $\mathbf{C}^*$-algebra with unit $e$. We develop several inequalities for a positive linear functional $f$ on $\mathcal{A}$ and obtain several bounds for the numerical radius $v(a)$ of an element $a\in…

For a real constant $b,$ we give sharp estimates of $\log|f(z)/z|+b\arg[f(z)/z]$ for subclasses of normalized univalent functions $f$ on the unit disk.

In the context of global optimization of mixed-integer nonlinear optimization formulations, we consider smoothing univariate functions $f$ that satisfy $f(0)=0$, $f$ is increasing and concave on $[0,+\infty)$, $f$ is twice differentiable on…

Let $\|\cdot\|_{\mathbf A}$ be a norm on $\mathbb C^m$ given by the formula $\|(z_1,\ldots,z_m)\|_{\mathbf A}=\|z_1A_1+\cdots+z_mA_m\|_{\rm op}$ for some choice of an $m$-tuple of $n\times n$ linearly independent matrices $\mathbf A=(A_1,…

We prove that for every $n \ge 2$, there exists a pseudoconvex domain $\Omega \subset \mathbb{C}^n$ such that $\mathfrak{c}^0(\Omega) \subsetneq \mathfrak{c}^1(\Omega)$, where $\mathfrak{c}^k(\Omega)$ denotes the core of $\Omega$ with…