In this paper, Jensen’s inequality and Fubini’s Theorem are extended for the function of several variables via diamond integrals time scale calculus. These extensions used to generalize Hardy-type inequalities with general kernels variables. Some Hardy Hilbert Polya Knop type also discussed as special cases. Classical new deduced from main results using particular scales.

We give a necessary and sufficient condition on a radially symmetric potential V on a bounded domain Omega of (n) that makes it an admissible candidate for an improved Hardy inequality of the following type. For every element in H(1)(0)(Omega) integral(Omega) |vector differential u|2 dx - ((n - 2)/2)2 integral(Omega) |u|2/|x|2 dx > or = c integral(Omega) V(x)|u|2 dx. A characterization of the b...

In this paper a slightly stronger version of the Second Hardy-Littlewood Conjecture (see [1]), namely that inequality π(x)+π(y) > π(x+y) is examined, where π(x) denotes the number of primes not exceeding x. It is shown that the inequality holds for all su ciently large x and y. It has also been shown that for a given value of y ≥ 55 the inequality π(x)+π(y) > π(x+y) holds for all su ciently lar...

The Hardy-Littlewood-P\'{o}lya inequality of majorization is extended to the framework ordered Banach spaces. Several applications illustrating our main results are also included.

In this paper, we investigate a stochastic Hardy–Littlewood–Sobolev inequality. Due to the non-homogenous nature of potential in inequality, show that constant proportional length interval appears on right-hand-side. As direct application, derive local Strichartz estimates for randomly modulated dispersions and solve Cauchy problem critical nonlinear Schrödinger equation.

In this note, we obtain a reverse version of the integral Hardy inequality on metric measure spaces. Moreover, give necessary and sufficient conditions for weighted to be true. The main tool in our proof is continuous Minkowski inequality. addition, present some consequences obtained homogeneous groups, hyperbolic spaces Cartan-Hadamard manifolds.

By using the methods of real analysis and mid-value theorem, we introduce some lemmas obtain a new reverse extended Hardy–Hilbert’s inequality with two partial sums multi-parameters. We also give few equivalent conditions best possible constant factor related to several parameters in inequality. Some particular inequalities are deduced.

Abstract In this paper, by virtue of the symmetry principle, applying techniques real analysis and Euler–Maclaurin summation formula, we construct proper weight coefficients use them to establish a reverse extended Hardy–Hilbert’s inequality with multi-parameters. Then, obtain equivalent forms some statements best possible constant factor related several parameters. Finally, illustrate how obta...

We prove new Hardy–Copson-type (γ,a)-nabla fractional dynamic inequalities on time scales. Our results are proven by using Keller’s chain rule, the integration parts formula, and Hölder inequality When γ=1, then we obtain some well-known time-scale due to Hardy. As special cases, continuous discrete inequalities. Symmetry plays an essential role in determining correct methods solve