On a Multiple Hilbert's Inequality with Parameters
© Qiliang Huang. 2010
Received: 12 May 2010
Accepted: 31 August 2010
Published: 2 September 2010
By introducing multiparameters and conjugate exponents and using Hadamard's inequality and the way of real analysis, we estimate the weight coefficients and give a multiple more accurate Hilbert's inequality, which is an extension of some published results. We also prove that the constant factor in the new inequality is the best possible and consider its equivalent form.
where the constant factor is the best possible. Inequalities (1.1)–(1.4) are important in analysis and their applications (cf. ). In near one century, there are many improvements, generalizations and, applications of (1.1)–(1.4) in numerous literatures and monographs of mathematics (cf. [2–18]). Yang and Huang also considered the multiple Hilbert-type integral inequality (cf. [19, 20]). Recently, Yang summarized the methods of introducing parameters and estimating the weight coefficients to extend Hilbert-type inequalities for the past 100 years. Some representative results are as follows (cf. [21, 22]):
The constant factors in the above five inequalities are all the best possible. Inequalities (1.5) and (1.7) are generalizations of inequality (1.2), and inequality (1.9) is a multiple extension of (1.1). Inequalities (1.6) and (1.8) are the equivalent forms of (1.5) and (1.7), which are extensions of (1.4).
In this paper, by introducing multi-parameters and conjugate exponents and using Hadamard's inequality, we estimate the weight coefficients and give a multiple more accurate Hilbert 's inequality, which is an extension of inequalities (1.5), (1.7), and (1.9). We also prove that the constant factor in the new inequality is the best possible and consider its equivalent form.
2. Some Lemmas
and then (2.1) is valid.
Hence, we prove that the left-hand side of (2.3) is valid.
and then (2.11) is valid by using inequality (2.3).
Hence, (2.12) is valid.
3. Main Results
Then (3.3) is valid, which is equivalent to (3.2).
This work is supported by the Emphases Natural Science Foundation of Guangdong Institution, Higher Learning, College and University (no. 05Z026), and Guangdong Natural Science Foundation (no. 7004344).
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