Related papers: Improving Beckner's bound via Hermite functions

We consider degenerate fully nonlinear parabolic equations, which generalize the p-parabolic equation with $p>2$ to nondivergence form operators. We prove an intrinsic Harnack inequality for nonnegative solutions and a weak Harnack…

In this note we are concerned with interior regularity properties of the $p$-Poisson problem $\Delta_p(u)=f$ with $p>2$. For all $0<\lambda\leq 1$ we constuct right-hand sides $f$ of differentiability $-1+\lambda$ such that the (Besov-)…

In this paper, we obtain new bounds for the inequalities of Simpson and Hermite-Hadamard type for functions whose second derivatives absolute values are P-convex. These bounds can be much better than some obtained bounds. Some applications…

We consider weak solutions to a class of Dirichlet boundary value problems invloving the $p$-Laplace operator, and prove that the second weak derivatives are in $L^{q}$ with $q$ as large as it is desirable, provided $p$ is sufficiently…

We improve an $L^2\times L^2\to L^2$ estimate for a certain bilinear operator in the finite field of size $p$, where $p$ is a prime sufficiently large. Our method carefully picks the variables to apply the Cauchy-Schwarz inequality. As a…

In this paper, we prove a self-improvement result for $(\theta,p)$-fractional Hardy inequalities, in both the exponent $1<p<\infty$ and the regularity parameter $0<\theta<1$, for bounded domains in doubling metric measure spaces. The key…

A classical result of Hensley provides a sharp lower bound for the functional $\int_\mathbb{R} t^2f$, where $f$ is a non-negative, even log-concave function. In the context of studying the minimal slabs of the unit cube, Barthe and…

We consider second-order elliptic equations in a half space with leading coefficients measurable in a tangential direction. We prove the $W^2_p$-estimate and solvability for the Dirichlet problem when $p\in (1,2]$, and for the Neumann…

In this paper, we establish some upper bounds for Berezin number inequalities including of $2\times 2$ operator matrices and their off-diagonal parts. Among other inequalities, it is shown that if $T=\left[\begin{array}{cc} 0&X, Y&0…

New proofs of the classical Hermite-Hadamard inequality are presented and several applications are given, including Hadamard-type inequalities for the functions, whose derivatives have inflection points or whose derivatives are convex.…

Let $f\in \ell^2(\mathbb Z)$. Define the average of $ f$ over the square integers by $ A_N f(x):=\frac{1}{N}\sum_{k=1}^N f(x+k^2) $. We show that $ A_N$ satisfies a local scale-free $ \ell ^{p}$-improving estimate, for $ 3/2 < p \leq 2$:…

A function $f:[a,b] \rightarrow \mathbb{R}$ is called $(p,a,b)$-convex if $f$ is $p$ times continuously differentiable, $f^{(p)}$ is convex and increasing, and $f^{(k)}(a)=0$ for all $k=1,\ldots,p$ where $f^{(j)}$ is the $j$th derivative of…

We show that an almost trivial inequality for the first and second mean of a random variable can be used to give non-trivial improvements on deep results. As applications we improve on results on lower bounds for the Riemann zeta-function…

The main result of this paper are dimension-free $L^p$ inequalities, $1<p<\infty$, for low degree scalar-valued functions on the Hamming cube. More precisely, for any $p>2,$ $\varepsilon>0,$ and $\theta=\theta(\varepsilon,p)\in (0,1)$…

We obtain the inequality $$\int_{\Omega}|\nabla u(x)|^ph(u(x))dx\leq C(n,p)\int_{\Omega} \left( \sqrt{ |\nabla^{(2)} u(x)||{\cal T}_{h,C}(u(x))|}\right)^{p}h(u(x))dx,$$ where $\Omega\subseteq {\bf R}^n$ and $n\ge 2$, $u:\Omega\rightarrow…

In this paper, we study the $L^p(\mathbb{R}^2)$-improving bounds, i.e., $L^p(\mathbb{R}^2)\rightarrow L^q(\mathbb{R}^2)$ estimates, of the maximal function $M_{\gamma}$ along a plane curve $(t,\gamma(t))$, where…

In this paper, we prove the existence of a weak solution for the Dirichlet boundary value problem related to the $p(x)-$Laplacian $$ -\mbox{div}(|\nabla u|^{p(x)-2}\nabla u)+u\in -[\underline{g}(x,u),\overline{g}(x,u)], $$ by using the…

Let $P(z)$ be a polynomial of degree $n,$ then it is known that for $\alpha\in\mathbb{C}$ with $|\alpha|\leq \frac{n}{2},$ \begin{align*} \underset{|z|=1}{\max}|\left|zP^{\prime}(z)-\alpha P(z)\right|\leq…

We prove fractional boundary Hardy's inequality in dimension one for the critical case $sp =1$. Optimality of the inequality is obtained for any $p$. The extra logarithmic correction term appears in usual fashion. We also provide a concrete…

For $f\in \mathcal{S}$, the class univalent functions in the unit disk $\mathbb{D}$ and given by $f(z)=z+\sum_{n=2}^{\infty} a_n z^n$ for $z\in \mathbb{D}$, we improve previous bounds for the second and third Hankel determinants in case…