On Hardy-type integral inequalities with the gamma function

By means of real analysis and weight functions, we obtain a few equivalent conditions of two kinds of Hardy-type integral inequalities with the non-homogeneous kernel and parameters. The constant factors related to the gamma function are proved to be the best possible. We also consider the operator expressions and some cases of homogeneous kernel.

In , Hardy et al. gave an extension of () as follows: If k  (x, y) is a non-negative homogeneous function of degree -, where the constant factor k p is the best possible (cf. [], Theorem ). Additionally, a Hilbert-type integral inequality with the non-homogeneous kernel is proved as follows: where the constant factor φ(  p ) is still the best possible (cf. [], Theorem ). In , by introducing an independent parameter λ > , Yang gave a best extension of () with the kernel  (x+y) λ (cf. [, ]). In , by introducing another pair conjugate exponents (r, s), Yang [] gave an extension of () as follows: where the constant factor π λ sin(π /r) is the best possible. For λ = , r = q, s = p, () reduces to (); For λ = , r = p, s = q, () reduces to the dual form of () as follows: x + y dx dy (  ) For p = q = , both () and () reduce to (). In , in [] one also gave an extension of () and () with the kernel  (x+y) λ . Krnić et al. [-] provided some extensions and particular cases of (), () and () with parameters.
then () reduces to the following Hardy-type integral inequality with the non-homogeneous kernel: then () reduces to the following another kind of Hardy-type integral inequality with the non-homogeneous kernel: (   ) In this paper, by real analysis and the weight functions, we obtain a few equivalent conditions of two kinds of Hardy-type integral inequalities with the non-homogeneous kernel and parameters as (min{xy,}) α | ln xy| β (max{xy,}) λ+α . The constant factors related to the gamma function are proved to be the best possible. We also consider the operator expressions and some cases of homogeneous kernel.
Proof If σ  > σ , then, for n ≥  σ  -σ (n ∈ N), we set the following functions: and find Setting u = xy, we obtain Then by (), we have which is a contradiction. If σ  < σ , then, for n ≥  σ -σ  (n ∈ N), we set the following functions: and find Setting u = xy, we obtaiñ Then by the Fubini theorem (cf. []) and (), we have Hence, we conclude that σ  = σ . For σ  = σ , we reduce () as follows: is non-negative and increasing in (, ], by Levi theorem (cf. []), we find The lemma is proved.
there exists a constant M  , such that, for any non-negative measurable functions f (x) and g(y) in (, ∞), the following inequality: holds true, then we have σ  = σ , and then M  ≥ k  (μ).
Proof If σ  < σ , then, for n ≥  σ -σ  (n ∈ N), we set two functionsf n (x) andg n (y) as in Lemma , and find Setting u = xy, we obtaiñ and then by (), we obtain we have ∞ < ∞, which is a contradiction.
If σ  > σ , then, for n ≥  σ  -σ (n ∈ N), we set two functions f n (x) and g n (y) as in Lemma , and we find Setting u = xy, we obtain and then by Fubini theorem (cf. []) and (), we have we have ∞ < ∞, which is a contradiction. Hence, we conclude that σ  = σ . For σ  = σ , we reduce () as follows: is non-negative and increasing in [, ∞), still by the Levi theorem (cf. []), we have The lemma is proved.

Main results and corollaries
Theorem  If p > ,  p +  q = , σ  ∈ R, β > -, σ > -α, then the following conditions are equivalent: (i) There exists a constant M  , such that, for any f (x) ≥ , satisfying we have the following Hardy-type integral inequality of the first kind with the non-homogeneous kernel: we have the following inequality: Then by (), we have ().
(ii)⇒(iii). By Lemma , we have σ  = σ . (iii)⇒(i). Setting u = xy, we obtain the following weight function: If () obtains the form of equality for a y ∈ (, ∞), then (cf. []), there exist constants A and B, such that they are not all zero, and We suppose that A =  (otherwise B = A = ). It follows that Ax a.e. in R + , which contradicts the fact that  < Hence, () assumes the form of strict inequality. Hence, for σ  = σ , by () and by the Fubini theorem (cf. []), we obtain Setting M  ≥ k  (σ ), () follows. Therefore, conditions (i), (ii) and (iii) are equivalent. When condition (iii) follows, if there exists a constant factor M  ≥ k  (σ ), such that () is valid, then by Lemma , we have M  ≥ k  (σ ). Hence, the constant factor M  = k  (σ ) in () is the best possible. The constant factor M  = k  (σ ) in () is still the best possible. Otherwise, by () (for σ  = σ ), we can conclude that the constant factor M  = k  (σ ) in () is not the best possible.
, μ  = λσ  in Theorem , then replacing Y by y and G(Y ) by g(y), we have then the following conditions are equivalent: (i) There exists a constant M  , such that, for any f (x) ≥ , satisfying we have the following Hardy-type inequality of the first kind with the homogeneous kernel: we have the following inequality: If condition (iii) holds true, then we have M  ≥ k  (σ ), and the constant M  = k  (σ ) in () and () is the best possible. Similarly, we obtain the following weight function:

Remark  On the other hand, setting
and then in view of Lemma  and in the same way, we have Theorem  If p > ,  p +  q = , σ  ∈ R, β > -, μ = λσ > -α, then the following conditions are equivalent: we have the following Hardy-type inequality of the second kind with the non-homogeneous kernel: we have the following inequality: If condition (iii) holds true, then we have M  ≥ k  (μ) and the constant factor M  = k  (μ) = (β + ) (μ + α) β+ in () and () is the best possible.
, μ  = λσ  in Theorem , then replacing Y by y and G(Y ) by g(y), we have we have the following Hardy-type inequality of the second kind with the homogeneous kernel: There exists a constant M  , such that, for any f (x), g(y) ≥ , satisfying  < ∞  x p(-σ )- f p (x) dx < ∞, and  < ∞  y q(-μ  )- g q (y) dy < ∞, we have the following inequality: If condition (iii) holds true, then we have M  ≥ k  (μ), and the constant M  = k  (μ) in () and () is the best possible.
Remark  Theorem  and Corollary  are still equivalent.

Conclusions
In this paper, by means of real analysis and weight functions a few equivalent conditions of two kinds of Hardy-type integral inequalities with the non-homogeneous kernel and parameters are obtained by Theorem , . The constant factors related to the gamma function are proved to be the best possible. We also consider the operator expressions in Theorem , . The dependent cases of homogeneous kernel are assumed by Corollary -. The method of weight functions is very important, it is the key to help us proving the main inequalities with the best possible constant factor. The lemmas provide an extensive account of this type of inequalities.