Hilbert-Type Inequalities Revisited

unless f(x) ≡ 0 or g(x) ≡ 0, where p>1, q = p/(p-1). Inequality (1), would be invalid for some f(x), g(x) if the constant π cosec (π/p) were replaced by a smaller number see (Hardy et al., 1934). Inequality (1) with its modifications have played an important role in the raise of many mathematical and physical branches see for instance (Xingdong and Bicheng, 2010; Jichang and Debnath, 2000). In this study we are concerned with the case when p = q = 2, i.e., we focus on the inequality:


Introduction
We establish more general variants of the integral Hilbert-type inequality (Hardy et al., 1934): unless f(x) ≡ 0 or g(x) ≡ 0, where p>1, q = p/(p-1). Inequality (1), would be invalid for some f(x), g(x) if the constant π cosec (π/p) were replaced by a smaller number see (Hardy et al., 1934). Inequality (1) with its modifications have played an important role in the raise of many mathematical and physical branches see for instance (Xingdong and Bicheng, 2010;Jichang and Debnath, 2000). In this study we are concerned with the case when p = q = 2, i.e., we focus on the inequality: Many mathematicians have worked on generalizing inequality (2) in different ways. Some of them developed half discrete analogues of (2) see for instance (Xin and Yang, 2012;Zhenxiao and Yang, 2013), while others worked on developing different variants of the denominator of the left hand side see for example (Bicheng, 1998;Bicheng and Qiang, 2015;Bing et al., 2015;Jichang and Debnath, 2000). For example in (Bicheng, 1998) the following inequality can be found: for 0 < a < b and 0 < λ ≤ 1, f(x), g(x)∈ L 2 [0, ∞) we have: The objective of this paper is to derive more general form of Hilbert's inequality (2) by introducing some parameters. In particular we generalize inequality (3) focusing on developing the denominator of the left hand side. In this study β(p, q) is the β-function.

Main Results and Discussion
This section states and discusses the main theorem which will be proved in the fourth section. For different parameters t, λ∈ (0, 1] we have the following theorem.

Remark 2.3
If a = c and b = d, then inequality (5) reduces to inequality (3), which in turn leads to the original Hilbert's inequality (2) if λ = 1 and a → 0 and b → ∞.
To prove Theorem 2.1, we prove first two lemmas introduced in the following section.

Lemmas
In this section we present and prove two needed lemmas.

Proof
Put u t = v in ψ t,λ to obtain: However, it is known that the Beta function is given by (see for instance (Greene and Krantz, 2006)): Hence, from (7) and (8)
We use Lemmas 3.1 and 3.2 to prove our main result.

Proof of Theorem 2.1
By Cauchy's inequality, we can estimate the left hand side of (4) as follows: where: ( ) ( )