In this paper we prove new continuous refinements of some Jensen type inequalities in both direct and reversed forms. As applications we also derive some continuous refinements of Hermite–Hadamard, Hölder, and Popoviciu type inequalities. As particular cases we point out the corresponding results for sums and integrals showing that our results contain both several well-known but also some new ...

In this paper, we explore a generalised solution of the Cauchy problems for the q-heat and q-wave equations which are generated by Jackson’s and the q-Sturm-Liouville operators with respect to t and x, respectively. For this, we use a new method, where a crucial tool is used to represent functions in the Fourier series expansions in a Hilbert space on quantum calculus. We show that these solutions ...

This paper is devoted to investigating the sequence of some linear functionals in the space BV of finite variation functions. We prove that under certain conditions this sequence is bounded. We also prove that this result is sharp. In particular, the obtained results can be used to study convergence of some general Fourier series. Moreover, the obtained conditions seem to be new and useful also for ...

Some new refined versions of the Jensen, Minkowski, and Hardy inequalities are stated and proved. In particular, these results both generalize and unify several results of this type. Some results are also new for the classical situation.

We consider the generalized Lorentz space L_ψ,q defined via a continuous and concave function ψ and the Fourier series of a function with respect to an unbounded orthonormal system. Some new Fourier and Jackson-Nikol’skii type inequalities in this frame are stated, proved and discussed. In particular, the derived results generalize and unify several well-known results but also some new applications ...

We prove and discuss some new refined Hölder inequalities for any p>1 and also a reversed version for 0<p<1. The key is to use the concepts of superquadraticity, strong convexity, and to first prove the corresponding refinements of the Young and reversed Young inequalities. Refinements of the Minkowski and reversed Minkowski inequalities are also given.

In this paper, we introduce some new weighted maximal operators of the partial sums of the Walsh–Fourier series. We prove that for some “optimal” weights these new operators indeed are bounded from the martingale Hardy space H_p(G) to the Lebesgue space weak−L_p(G), for 0<p<1. Moreover, we also prove sharpness of this result. As a consequence we obtain some new and well-known ...

The concept of Vilenkin–Lebesgue points was introduced in [12], where the almost everywhere convergence of Fejer means of Vilenkin–Fourier series was proved. In this paper, we present a different (and simpler) approach to prove a similar result, which can be used to prove that the corresponding result holds also in a more general context, namely for regular Norlund and T-means.

In this paper, we investigate difference-differential operators of parabolic and hyperbolic types. Namely, we considern on-homogenous heat and wave equations for Rubin’s difference operator. Wellposedness results are obtained in appropriate Sobolev type spaces. In particular, we prove that the heat and wave equations generated by Rubin’s difference operator have unique solutions. We even show that ...