Related papers: Gr\"uss inequalities for the $\beta-$integral asso…

We establish a general operator parallelogram law concerning a characterization of inner product spaces, get an operator extension of Bohr's inequality and present several norm inequalities. More precisely, let ${\mathfrak A}$ be a…

In this paper we present a new type of fractional operator, which is a generalization of the Caputo and Caputo--Hadamard fractional derivative operators. We study some properties of the operator, namely we prove that it is the inverse…

In this paper, we study the uniform H\"older continuity of the generalized Riemann function $R_{\alpha,\beta}$ (with $\alpha>1$ and $\beta>0$) defined by \[ R_{\alpha,\beta}(x)=\sum_{n=1}^{+\infty}\frac{\sin(\pi n^\beta x)}{n^\alpha},\quad…

We obtain large sieve type inequalities for the Rayleigh quotient of the restriction of phase space representations of higher rank operators, via an operator analogue of the short-time Fourier transform (STFT). The resulting bounds are…

The generalized weighted mean operator $\mathbf{M}^{g}_{w}$ is given by $$[\mathbf{M}^{g}_{w}f](x)= g^{-1}\left(\frac{1}{W(x)}\int_{0}^{x}w(t)g(f(t))\,\mathrm{d}t\right),$$ with $$W(x)=\int_{0}^{x} w(s)\,\mathrm{d}s, \quad \textrm{for} x…

For $\alpha, \beta \in L^{\infty} (S^1),$ the singular integral operator $S_{\alpha,\beta}$ on $L^2 (S^1)$ is defined by $S_{\alpha,\beta}f:= \alpha Pf+\beta Qf$, where $P$ denotes the orthogonal projection of $L^2(S^1)$ onto the Hardy…

Iterates of quantum operations and their convergence are investigated in the context of mean ergodic theory. We discuss in detail the convergence of the iterates and show that the uniform ergodic theorem plays an essential role. Our results…

In this work, the mixed Schwarz inequality for semi-Hilbertian space operators is proved. Namely, for every positive Hilbert space operator $A$. If $f$ and $g$ are nonnegative continuous functions on $\left[0,\infty\right)$ satisfying…

In [P. Renaud, "A matrix formulation of Gr\"uss inequality", Linear Algebra Appl. 335 (2001), 95--100] it was proved an operator inequality involving the usual trace functional. In this article, we give a refinement of such result and we…

For $\a,\b>0$ and for a locally integrable function (or, more generally, a distribution) $\f$ on $(0,\be)$, we study integral ooperators ${\frak G}^{\a,\b}_\f$ on $L^2(\R_+)$ defined by $\big({\frak G}^{\a,\b}_\f…

A new Hilbert-type integral inequality in the whole plane with the non-homogeneous kernel and parameters is given. The constant factor related to the hypergeometric function and the beta function is proved to be the best possible. As…

For $\beta>0$ and $p\ge 1$, the generalized Ces\`aro operator $$ \mathcal{C}_\beta f(t):=\frac{\beta}{t^\beta}\int_0^t (t-s)^{\beta-1}f(s)ds $$ and its companion operator $\mathcal{C}_\beta^*$ defined on Sobolev spaces…

For bounded Lebesgue measurable functions $f,g,\phi$ and $\psi$ on the unit circle, $P_{+}fP_{+}+P_{-}gP_{+} +P_{+}\phi P_{-}+P_{-}\psi P_{-}$ is called a generalized singular integral operator (GSIO) on $L^{2}(\mathbb{T})$, where $P_{+}$…

In this paper, I show a close connection between renormalization and a generalization of the Dynkin operator in terms of logarithmic derivations. The geometric $\beta$ function, which describes the dependence of a Quantum Field Theory on an…

For each $ \alpha > 0 $, the $\alpha$-Bloch space is consisting of all analytic functions $f$ on the unit disk satisfying $ \sup_{|z|<1} (1-|z|^2)^\alpha |f'(z)| < + \infty.$ In this paper, we consider the following complex integral…

Let $\mu$ be a finite Borel measure on $[0,1)$. In this paper, we consider the generalized integral type Hilbert operator $$\mathcal{I}_{\mu_{\alpha+1}}(f)(z)=\int_{0}^{1}\frac{f(t)}{(1-tz)^{\alpha+1}}d\mu(t)\ \ \ (\alpha>-1).$$ The…

In this paper an analytic operator-valued generalized Feynman integral was studied on a very general Wiener space $C_{a,b}[0,T]$. The general Wiener space $C_{a,b}[0,T]$ is a function space which is induced by the generalized Brownian…

Suppose that $\Omega \in L^{\infty}(\mathbb{S} ^{n-1})$ is homogeneous of degree zero with mean value zero. Then we consider a fractional type Marcinkiewicz integral operator $$\mu_{\Omega ,\beta }f(x) = \left ( \int_{0}^{\infty } \left |…

In this article, we look for the weight functions (say $g$) that admits the following generalized Hardy-Rellich type inequality: $ \int_{\Omega} g(x) u^2 dx \leq C \int_{\Omega} |\Delta u|^2 dx, \forall u \in \mathcal{D}^{2,2}_0(\Omega), $…

A new condition is introduced by generalizing the Ritt and Kreiss operators named $(\alpha, \beta)$-RK condition. Geometrical properties of the spectrum for the case $\beta < 1$ are studied, moreover it is shown that in that case if $\alpha…