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On a Hilbert-type inequality with a homogeneous kernel in ℝ2 and its equivalent form
© He; licensee Springer. 2012
Received: 19 August 2011
Accepted: 20 April 2012
Published: 20 April 2012
By using the way of weight functions and the technique of real analysis, a new integral inequality with a homogeneous kernel and the best constant factor in ℝ2 is given. The equivalent form and the reverses are considered.
Mathematics Subject Classification (2000): 26D15.
The inequality (1.1) may be classified into several types (discrete and integral etc.), which is of great importance in analysis and its applications [1, 2]. Ever since the advent of inequality (1.1), all kinds of improvements and extensions can be seen in [3–12]. Note that the kernel of (1.1) is homogeneous of degree -1. In 2009,  reviews the negative degree homogeneous kernel of the parameterized Hilbert-type inequalities.
where the constant factor is the best possible.
Motivated by (1.2) and the technique of real analysis, we establish a new inequality in ℝ2 with a homogeneous kernel of 0-degree. Furthermore, the equivalent form and the corresponding reverse inequalities are also considered.
In what follows, α1, α2 will be real numbers such that 0 < α1 < α2 < π.
Similarly, ϖ(x) = k for x ∈ (0, ∞). Hence (2.2) is valid for x ∈ (-∞, 0) ∪ (0, ∞). □
and ϖ(x) = 2 ln (2 sin α) + (π - 2α) cot α.
Proof. It can be completed similarly by following the proof of Lemma 2.2 as long as applying the reverse Hölder's inequality , hence we omit the details. Since q < 0, thus (2.7) takes the positive inequality. □
3. Main results and applications
where the constant factors and k p are both the best possible.
i.e., A|x| p f p (x) = B|y| q a.e. in (-∞,∞) × (-∞, ∞). We conform that A ≠ 0 (otherwise B = A = 0). Then a.e. in(-∞,∞), which contradicts the fact that . Hence (2.5) takes a strict inequality and the same as (2.4), thus (3.2) is valid.
Hence we obtain (3.2). Thus (3.2) and (3.1) are equivalent.
Hence k is the best value of (3.1). We conform that k p is also the best value of (3.2). Otherwise, we can get a contradiction by (3.3) that (3.1) is not the best possible. □
where the constant factors , both k p and k q are the best possible.
Hence we obtain (3.10). Thus (3.10) and (3.9) are equivalent.
By (3.11), we obtain (3.9), and it is equivalent between (3.11) and (3.9). Thus (3.9), (3.10), and (3.11) are equivalent.
By (3.14), if follows that k ≥ M for ε → 0+. Hence k is the best value of (3.9). Furthermore, the constant factors in (3.10) and (3.11) are both the best value too. Otherwise, by (3.3) or (3.13), we may get a contradiction that the constant factor in (3.9) is not the best possible. □
By Note (ii), Theorems 3.1 and 3.2, it follows that
The study was partially supported by the Emphases Natural Science Foundation of Guangdong Institution of Higher Learning, College and University (No. 05Z026).
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