Related papers: Some sharp inequalities for the Toader-Qi mean

We establish sharp inequalities involving the incomplete Beta and Gamma functions. These inequalities arise in the approximation of generalized Bernstein functions by higher order Thorin-Bernstein functions. Furthermore, new properties of a…

In this paper, our aim is to show some mean value inequalities for the Fox-Wright functions, such as Tur\'an--type inequalities, Lazarevi\'c and Wilker--type inequalities. As applications we derive some new type inequalities for…

The modified q-Bessel functions and the q-Bessel-Macdonald functions of the first and second kind are introduced. Their definition is based on representations as power series. Recurrence relations, the q-Wronskians, asymptotic…

Using the properties of geometric mean, we shall show for any $0\le \alpha ,\beta \le 1$, \[f\left( A{{\nabla }_{\alpha }}B \right)\le f\left( \left( A{{\nabla }_{\alpha }}B \right){{\nabla }_{\beta }}A \right){{\sharp}_{\alpha }}f\left(…

Let $q$ be a positive integer, $\chi$ a nontrivial character mod $q$, $\mathcal{I}$ an interval of length not exceeding $q.$ In this paper we shall study the character sum analogue of the well-known Kloosterman…

We introduce the class of functions positively associated with a linear operator. We describe these classes for several integral operators including the $q$-cosine transform and the spherical Radon transform. We show that positively…

We study the weighted norm inequality of $(1,q)$-type, \[ \Vert \mathbf{G}\nu \Vert_{L^q(\Omega, d\sigma)} \le C \Vert \nu \Vert, \quad \text{ for all } \nu \in \mathcal{M}^+(\Omega), \] along with its weak-type analogue, for $0 < q < 1$,…

We prove $q$-analogues of identities that are equivalent to the functional equation of the arithmetic-geometric mean. We also present $q$-analogues of $F(\sqrt{k},\frac{\pi}{2})$, the complete elliptical integral of the first kind, and its…

Some mathematical inequalities among various weighted means are studied. Inequalities on weighted logarithmic mean are given. Besides, the gap in Jensen's inequality is studied as a convex function approach. Consequently, some non-trivial…

In this paper, we study Bessel operators and Bessel Laplace equations studied by Weinstein, Huber, and related the harmonic function theory introduced by Muckenhoupt--Stein. We establish the Moser type inequality for these harmonic…

We study $q$-variation inequality for bilinear averaging operators over convex bodies $(G_t)_{t>0}$ defined by \begin{align*} \mathbf{A}_t^G(f_1,f_2)(x) & =\frac{1}{|G_t|}\int_{G_t} f_1(x+y_1)f_2(x+y_2)\, dy_1\, dy_2, \quad x\in \Bbb R^d.…

For a real-valued measurable function $f$ and a nonnegative, nondecreasing function $\phi$, we first obtain a Chebyshev type inequality which provides an upper bound for $\displaystyle \phi(\lambda_{1}) \mu(\{x \in \Omega : f(x) \geq…

There exists a positive function $\psi(t)${on}$t\geq0${, with fast decay at infinity, such that for every measurable set}$\Omega${in the Euclidean space and}$R>0${, there exist entire functions}$A(x) ${and}$B(x) ${of exponential type}$R${,…

Let $A, B$ and $X$ be $n\times n$ matrices such that $A, B$ are positive semidefinite. We present some refinements of the matrix Cauchy-Schwarz inequality by using some integration techniques and various refinements of the Hermite--Hadamard…

The celebrated Hardy inequality can be written in the form $$\int_0^\infty \mathcal{P}_p \big(f|_{[0,x]}\big)dx \le (1-p)^{-1/p} \int_0^\infty f(x)\:dx \qquad \text{ for }p\in(0,1)\text{ and }f \in L^1\text{ with }f\ge0,$$ where…

We present here several versions of the Grothendieck inequality over the skew field of quaternions: The first one is the standard Grothendieck inequality for rectangular matrices, and two additional inequalities for self-adjoint matrices,…

Quantum functional inequalities (e.g. the logarithmic Sobolev- and Poincar\'e inequalities) have found widespread application in the study of the behavior of primitive quantum Markov semigroups. The classical counterparts of these…

This note is concerned with an extension, at second order, of an inequality on the discrete cube $C_n=\{-1,1\}$ (equipped with the uniform measure) due to Talagrand (\cite{TalL1L2}). As an application, the main result of this note is a…

In this paper, we prove a trace inequality $\text{Tr}[ f(A) A^s B^s ] \leq \text{Tr}[ f(A) (A^{1/2} B A^{1/2} )^s ]$ for any positive and monotone increasing function $f$, $s\in[0,1]$, and positive semi-definite matrices $A$ and $B$. On the…

Let $I_0$ and $K_0$ be modified Bessel functions of the zeroth order. We use Vanhove's differential operators for Feynman integrals to derive upper bounds for dimensions of the $\mathbb Q$-vector space spanned by certain sequences of Bessel…