A Hilbert-type fractal integral inequality and its applications

By using thefractal theory and the methods of weight function, a Hilbert-type fractal integral inequality and its equivalent form are given. Their constant factors are proved being the best possible, and their applications are discussed briefly.


Introduction
If f , g ≥ , satisfying  < ∞  f  (x) dx < ∞,  < ∞  g  (y) dy < ∞, then there is the following basic Hilbert-type integral inequality and its equivalent form where the constants are optimal. Inequalities () and () are important in the analysis and partial differential equations [, ]. In  and , respectively, () and () were generalized and improved by introducing an independent parameter λ and two parameters λ  , In recent years, the fractal theory has been developed rapidly, and it has been widely used in the fields of science and engineering. Some researchers have used the fractal theory to discuss and generalize some classical inequalities on fractal sets [, ], but the research into the Hilbert-type integral inequality on the fractal set is still not involved. In this paper, by using the fractal theory and the method of weight function to make a meaningful attempt, a Hilbert-type integral inequality and its equivalent form on a fractal set are established.

Preliminaries
. Then the following differentiation rules are valid: . Then the local fractional integral is defined by , we have the following equations:

fractal surface, then we have (i) Hölder's inequality on the fractal set
(ii) Hölder's weighted inequality on the fractal set The inequality keeps the form of equality, then there exist constants A and B such that they are not all zero and AF p (x, y) = BG q (x, y) a.e. on S (β) . Then Similarly, we obtain ω(α, q, y) = η(α)y α  (q-) .
Further, let y x = t, and by Lemma ., we have

Main results and applications
Introducing the mark: where the constant factor η(α) defined in () is the best possible.
Proof By Hölder's weighted inequality on the fractal set and Lemma ., we obtain Now assume that equality holds in (), there exist two nonzero constants A and B such is not optimal, then there exists positive K < η(α) such that inequality () is still valid if we replace η(α) by K . Hence by () and (), we have Letting ε →  + , we get K ≥ η(α), which contradicts the fact that K < η(α), therefore η(α) in () is the best possible.

Conclusions
In the paper, based on the local fractional calculus theory, a Hilbert-type fractional integral inequality and its equivalent form are tentatively researched. The results show that some methods and skills of the Hilbert-type integral inequality can be transplanted to the research of Hilbert-type fractional integral inequality, which provides a new direction and field to research Hardy-Hilbert's integral inequalities.