Our work in this paper is based on the reverse Hölder-type dynamic inequalities illustrated by El-Deeb 2018 and Hilbert-type Rezk 2021 2022. With help of Specht’s ratio, concept supermultiplicative functions, chain rule, Jensen’s inequality time scales, we can establish some comprehensive generalize a number classical to general scale space. In calculus, results are unified extended. At same ti...

In this paper, two pairs of new inequalities are given, which decompose two Hilbert-type inequalities.

The Hardy inequality ∫ Ω |u(x)|pd(x)−p dx c ∫ Ω |∇u(x)|p dx with d(x) = dist(x, ∂Ω) holds for u ∈ C∞ 0 (Ω) if Ω ⊂ n is an open set with a sufficiently smooth boundary and if 1 < p < ∞. P.Haj lasz proved the pointwise counterpart to this inequality involving a maximal function of Hardy-Littlewood type on the right hand side and, as a consequence, obtained the integral Hardy inequality. We extend...

It is nowadays well-known that Hardy’s inequality (like many other inequalities) follows directly from Jensen’s inequality. Most of the development of Hardy type inequalities has not used this simple fact, which obviously was unknown by Hardy himself and many others. Here we report on some results obtained in this way mostly after 2002 by mainly using this fundamental idea.

where the coefficients are entire functions. In [8], equations of the form (1) with coefficients in weighted Bergman or Hardy spaces are studied. The direct problem is proved, that is, if the coefficients aj(z), j = 0, ..., k − 1 of (1) belong to the weighted Bergman space, then all solutions are of finite order of growth and belong to weighted Bergman space. The inverse problem is also investi...