On Hardy-type integral inequalities in the whole plane related to the extended Hurwitz-zeta function

Using weight functions, we establish a few equivalent statements of two kinds of Hardy-type integral inequalities with nonhomogeneous kernel in the whole plane. The constant factors related to the extended Hurwitz-zeta function are proved to be the best possible. In the form of applications, we deduce some special cases involving homogeneous kernel. We additionally consider some particular inequalities and operator expressions.

In the present paper, using weight functions, we establish a few equivalent statements of two kinds of Hardy-type integral inequalities with nonhomogeneous kernel and multiparameters in the whole plane. The constant factors related to the extended Hurwitz-zeta function are proved to be the best possible. In the form of applications, we deduce a few equivalent statements of two kinds of Hardy-type integral inequalities with homogeneous kernel in the whole plane. As corollaries, we also consider some particular cases and operator expressions. For β > 0, σ > -α -1, it follows that
In the sequel, we assume that p > 1, 1 Lemma 1 If σ 1 ∈ R, there exists a constant M 1 such that, for any nonnegative measurable functions f (x) and g(y) in R, the following inequality holds true, then we have σ 1 = σ > -α -1 and β > 0.

Lemma 2
If σ 1 ∈ R and there exists a constant M 2 such that, for any nonnegative measurable functions f (x) and g(y) in R, the following inequality holds true, then we have σ 1 = σ , μ > -α -1, and β > 0.

Main results and some corollaries
Theorem 1 If σ 1 ∈ R, then the following statements (i), (ii), and (iii) are equivalent: we have the following Hardy-type integral inequality of the first kind with the nonhomogeneous kernel: we have the following inequality: (iii) σ 1 = σ > -α -1 and β > 0.
(iii) ⇒ (i). We obtain the following weight function: For y = 0, By the weighted Hölder inequality and (17), we obtain 1 |y| If (18) takes the form of equality for some y ∈ R \ {0}, then (cf. [37]) there exist constants A and B such that they are not both zero and Let us assume that A = 0 (otherwise B = A = 0). It follows that which contradicts the fact that Hence, (18) takes the form of strict inequality.
This completes the proof of the theorem.

Corollary 1
The following statements (i), (ii), and (iii) are equivalent: (i) There exists a constant M 1 such that, for any f (x) ≥ 0 satisfying the following inequality is satisfied: (ii) There exists a constant M 1 such that, for any f (x), g(y) ≥ 0 satisfying we have the following inequality: (iii) α > -1 p -1 and β > 0. If statement (iii) holds true, then the constant M 1 = K (1) ( 1 p ) (∈ R + ) in (19) and (20) is the best possible.
in Theorem 1, and then replacing Y by y, we obtain the following corollary.

Corollary 3
If μ 1 ∈ R, then the following statements (i), (ii), and (iii) are equivalent: we have the following Hardy-type integral inequality of the first kind with homogeneous kernel: (ii) There exists a constant M 1 such that, for any f (x), g(y) ≥ 0 satisfying we have the following inequality: (iii) μ 1 = μ < λ + α + 1 and β > 0.
In particular, for σ = σ 1 = 1 p in Theorem 2, we obtain the following corollary.

Corollary 5
The following statements (i), (ii), and (iii) are equivalent: (i) There exists a constant M 2 such that, for any f (x) ≥ 0 satisfying we have the following inequality: (ii) There exists a constant M 2 such that, for any f (x), g(y) ≥ 0 satisfying we have the following inequality: (iii) α > -λ -1 q and β > 0. If statement (iii) holds true, then the constant M 2 = K (2) ( 1 p ) (∈ R + ) in (32) and (33) is the best possible.
in Theorem 2, and then replacing Y by y, we deduce the following corollary.

Corollary 8
The following statements (i), (ii), and (iii) are equivalent: (i) There exists a constant M 2 such that, for any f (x) ≥ 0 satisfying we have the following inequality: (ii) There exists a constant M 2 such that, for any f (x), g(y) ≥ 0 satisfying we have the following inequality: (iii) α > -1 p -1 and β > 0. If statement (iii) holds true, then the constant M 2 = K (2) ( 1 q ) (∈ R + ) in (38) and (39) is the best possible.

Conclusions
In the present paper, using weight functions we obtain in Theorems 1, 2 a few equivalent statements of two kinds of Hardy-type integral inequalities with nonhomogeneous kernel and multi-parameters in the whole plane. The constant factors related to the extended Hurwitz-zeta function are proved to be the best possible. In the form of applications, a few equivalent statements of two kinds of Hardy-type integral inequalities with the homogeneous kernel in the whole plane are also deduced in Corollaries 3, 7. We also consider some particular cases in Corollaries 1, 4, 5, 8 and in Remarks 2, 3. We additionally consider operator expressions in Theorems 3, 4 and Corollaries 9, 10. The lemmas and theorems within the present work provide an extensive account of this type of inequalities.