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In this article, we construct generalized harmonic univalent mappings and find its coefficients bounds. We present the counterexample to validate the coefficient conjecture proposed by Clunie and Sheil-Small for the class of functions…

Here is a sample of the results proved in this paper: Let $f:{\bf R}\to {\bf R}$ be a continuous function, let $\rho>0$ and let $\omega:[0,\rho[\to [0,+\infty[$ be a continuous increasing function such that $\lim_{\xi\to…

We develop further basic tools in the theory of continuous bounded cohomology of locally compact groups. We apply this tools to establish a Milnor-Wood type inequality in a very general context and to prove a global rigidity result which…

Denote by $M_n$ the set of $n\times n$ complex matrices. Let $f: M_n \rightarrow [0,\infty)$ be a continuous map such that $f(\mu UAU^*)= f(A)$ for any complex unit $\mu$, $A \in M_n$ and unitary $U \in M_n$, $f(X)=0$ if and only if $X=0$…

Let $f$ be a fixed self-contragradient Hecke-Maass form for $SL(3,\mathbb Z)$, and $u$ an even Hecke-Maass form for $SL(2,\mathbb Z)$ with Laplace eigenvalue $1/4+k^2$, $k>0$. A subconvexity bound $O\big(k^{4/3+\varepsilon}\big)$ in the…

In this paper, we prove the convexity of trace functionals $$(A,B,C)\mapsto \text{Tr}|B^{p}AC^{q}|^{s},$$ for parameters $(p,q,s)$ that are best possible, where $B$ and $C$ are any $n$-by-$n$ positive definite matrices, and $A$ is any…

The generalized Young inequality on the Lorentz spaces for commutative hypergroups is introdused and an application of it is given to the theory of fractional integrals. The boundedness on the Lorentz space and the Hardy-Littlewood-Sobolev…

In this paper we study variations of the Hopf theorem concerning continuous maps $f$ of a compact Riemannian manifold $M$ of dimension $n$ to $\mathbb{R}^n$. We investigate the case when $M$ is a closed convex $n$-dimensional surface and…

Eigenvalues inequalities involving (log) convex/concav functions and Hermitian matrices, positive unital maps are considered. Simple proofs of Bhatia-Kittaneh inequality and Naimark dilation theorem are given.

We prove the interior and global Lipschitz regularity results for a solution of fully nonlinear equations with $(p,q)$-growth. We prove that for a small gap $q-p$, a solution is locally or globally Lipschitz continuous. We also prove that a…

The purpose of this paper is to derive the Hoffman-Wielandt inequality and its generalization for quaternion matrices. Diagonalizability of the block companion matrix of certain quadratic (linear) quaternion matrix polynomials is brought…

We study multivariate entire functions and polynomials with non-negative coefficients. A class of {\bf Strongly Log-Concave} entire functions, generalizing {\it Minkowski} volume polynomials, is introduced: an entire function $f$ in $m$…

In this paper, we study the conjecture of Gardner and Zvavitch from \cite{GZ}, which suggests that the standard Gaussian measure $\gamma$ enjoys $\frac{1}{n}$-concavity with respect to the Minkowski addition of \textbf{symmetric} convex…

We prove some "power" generalizations of Marcus-Lopes-style (including McLeod and Bullen) concavity inequalities for elementary symmetric polynomials, and convexity inequalities (of McLeod and Baston) for complete homogeneous symmetric…

Lazarev and Lieb showed that finitely many integrable functions from the unit interval to $\mathbb{C}$ can be simultaneously annihilated in the $L^2$ inner product by a smooth function to the unit circle. Here we answer a question of…

In this paper, we give a new inequality for convex functions of real variables, and we apply this inequality to obtain considerable generalizations, refinements, and reverses of the Young and Heinz inequalities for positive scalars.…

We prove lifting theorems for complex representations $V$ of finite groups $G$. Let $\sigma=(\sigma_1,\dots,\sigma_n)$ be a minimal system of homogeneous basic invariants and let $d$ be their maximal degree. We prove that any continuous map…

The Brunn-Minkowski Theorem asserts that $\mu_d(A+B)^{1/d}\geq \mu_d(A)^{1/d}+\mu_d(B)^{1/d}$ for convex bodies $A,\,B\subseteq \R^d$, where $\mu_d$ denotes the $d$-dimensional Lebesgue measure. It is well-known that equality holds if and…

Given a hyperplane $H$ cutting a compact, convex body $K$ of positive Lebesgue measure through its centroid, Gr\"unbaum proved that $$\frac{|K\cap H^+|}{|K|}\geq \left(\frac{n}{n+1}\right)^n,$$ where $H^+$ is a half-space of boundary $H$.…

We prove new, general versions of Bernstein-von Mises theorem for both well-specified and misspecified models when the log-likelihood is concave in the parameter and the prior distribution is log-concave. Unlike classical versions of…