A New Hilbert-type Integral Inequality and the Equivalent Form

We give a new Hilbert-type integral inequality with the best constant factor by estimating the weight function. And the equivalent form is considered.


Introduction
If f ,g are real functions such that 0 < ∞ 0 f 2 (x)dx < ∞ and 0 < ∞ 0 g 2 (x)dx < ∞, then we have (see [1]) x + y dx dy < π where the constant factor π is the best possible.Inequality (1.1) is the well-known Hilbert's inequality.And inequality (1.1) had been generalized by Hardy in 1925 as follows.
In this paper, we give a new type of Hilbert's integral inequality as follows:

Main results
Lemma 2.1.
For 0 2) is not the best possible, then there exists a positive number K with K < c, such that (2.5) is valid by changing c to K. We have (2.15) Setting y = tx, by (2.1), we find 2) ≤ K, then c ≤ K, which contradicts the hypothesis.Hence the constant factor c in (2.5) is the best possible.
6 A new Hilbert-type integral inequality If the constant c 2 = 2(π − 2arctan √ 2) 2 in (2.18) is not the best possible, by (2.22), we may get a contradiction that the constant factor c in (2.5) is not the best possible.Thus we complete the proof of the theorem.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: .5) where the constant factor c = √ 2(π − 2arctan √ 2) = 1.7408... is the best possible.