Half-discrete Hilbert-type inequalities involving differential operators
© Adiyasuren et al.; licensee Springer. 2014
Received: 4 December 2013
Accepted: 10 February 2014
Published: 19 February 2014
Motivated by some recent results, in this paper we derive several half-discrete Hilbert-type inequalities with a homogeneous kernel involving some differential operators. The main results are proved for the case of non-conjugate parameters. After reduction to the conjugate case, we show that the constants appearing on the right-hand sides of these inequalities are the best possible.
MSC:26D10, 26D15, 33B15.
The Hilbert inequality is one of the most interesting inequalities in mathematical analysis and its applications. Although classical, it is still of interest to numerous authors.
Recently, Azar , obtained a new form of the Hilbert inequality including a differential operator. In order to state this result and to summarize our further discussion, we start by giving some notation. We denote by , , a differential operator defined by , where stands for the m th derivative of a function . In addition, throughout this article, is the set of non-negative measurable functions such that exists a.e. on , , a.e. on , and , .
holds for all , , and for all non-negative sequences , , provided that the integral and the series on the right-hand side converge. Moreover, the constant C is the best possible in the sense that it cannot be replaced with a smaller constant so that (1) still holds for all and for all non-negative sequences . The above inequality may be regarded as an extension of a classical Hilbert inequality since for , , and , we obtain the non-weighted inequality with the previously known sharp constant (for more details, see ). For a comprehensive study of an initial development of the Hilbert inequality, the reader is referred to a classical monograph .
It should be noticed here that the inequality (1) is usually referred to as a half-discrete Hilbert-type inequality since it includes both integral and a sum.
If nothing else is explicitly stated, we assume that the integral converges for considered values of η.
On the other hand, the recent paper  provides a unified treatment of half-discrete Hilbert-type inequalities with a homogeneous kernel and in the setting with non-conjugate exponents.
and observe that holds for all p and q as in (2). In particular, equality holds in (3) if and only if , that is, only if p and q are mutually conjugate. Otherwise, we have , and such parameters p and q will be referred to as non-conjugate exponents.
holds for any non-negative measurable function and a non-negative sequence , where , and , are real parameters such that the function is decreasing on for any . Clearly, in the above inequalities all integrals and sums are assumed to be convergent, and the function and the sequence are not equal to zero. Inequalities (4) and (5) will be an important tool in our extension of (1) to a general homogeneous case. For some particular half-discrete inequalities and related results, the reader is referred to recent papers [4–8] and references therein. Moreover, for comprehensive accounts on Hilbert inequality including history, different proofs, refinements and diverse applications, we refer to recent monograph .
The paper is divided into four sections as follows: After this Introduction, in Section 2, we cite several auxiliary results needed for our study. Further, in Section 3 we give several extensions of inequality (1) to a general homogeneous case and in the setting with non-conjugate parameters. It should be noticed here that our methods of proving differ from the techniques presented in . After reduction to conjugate case, in Section 4, we establish conditions for which derived inequalities include the best constants on their right-hand sides.
2 Notation and auxiliary results
holds for and , provided that (for more details, see ).
3 Half-discrete inequalities in the non-conjugate case
Our first intention is to give an extension of inequality (1) to the case of non-conjugate exponents and a general homogeneous function. Having in mind relations (4) and (5), our results will be given in two equivalent forms. It is interesting that the constants appearing in our extended inequalities are also expressed in terms of the Gamma function.
where , hold for a non-negative function and a non-negative sequence , provided that the integral and series on their right-hand sides converge to positive numbers.
Finally, the inequality (8) holds due to (4), (10), and (11). In the same way the inequality (9) follows by virtue of (5) and (11), which completes the proof. □
The previous theorem is derived by virtue of the Hardy inequality and covers the case when , where m is a fixed non-negative integer. Our next result is in some way complementary to Theorem 1 since it covers the case when . The crucial step in proving the corresponding relations will be the dual Hardy inequality (7).
In order to state this result, we define a differential operator by , where m is a non-negative integer. Moreover, the following theorem holds for all non-negative functions such that the m th derivative exists a.e. on , , , a.e. on , and for . This set of functions will be denoted by .
hold for any non-negative function and a non-negative sequence , provided that the integral and series on their right-hand sides converge to positive numbers.
Now, the relations (4), (14), and (15) entail the desired inequality (12). Similarly, the inequality (13) follows by virtue of (5) and (15). □
Remark 1 It should be noticed here that Theorem 1 and Theorem 2 coincide in the case of . Therefore, presented results may be regarded as the differential extensions of inequalities (4) and (5).
We have already mention that the problem of the best constants is one of the most interesting questions in connection with Hilbert-type inequalities. Unfortunately, there is still no evidence that the constants appearing on the right-hand sides of relations (8), (9), (12), and (13) are the best possible. This problem seems to be very hard in the non-conjugate case and remains still open. Luckily, we can solve the mentioned problem for some particular settings in the conjugate case.
4 Reduction to conjugate case and the best constants
Now, our goal is to determine conditions under which the constants appearing on the right-hand sides of inequalities (8), (9), (12), and (13) are the best possible.
Therefore, in this section we deal with non-negative conjugate exponents p, q, that is, with parameters p and q such that , . In this case , , and .
since in this case relation holds. With this assumption, the constant L appearing in Theorem 1 and Theorem 2 reduces to .
Remark 2 Let , , and , . In this case the constant appearing in inequalities (17), (18), (19), and (20) is expressed in terms of the Beta function, that is, . Then, utilizing the relationship between the Beta and the Gamma functions, we have , that is, the relation (17) becomes the inequality (1) from the Introduction, with a weaker condition . Thus, the dual form of (1) includes the constant which reduces to , after applying the Euler reflection formula .
Now, our aim is to show that the constants appearing in (17), (18), (19), and (20) are the best possible. The corresponding proofs are the substance of the following two theorems.
Theorem 3 Let be conjugate parameters and be a non-negative measurable homogeneous function of degree −s, . Further, let and be real parameters fulfilling condition (16) and , , where m is a fixed non-negative integer. If the function is decreasing on for any fixed , then the constant is the best possible in (17) and (18).
holds for any non-negative function and a non-negative sequence , provided that the integral and series on its right-hand side converge.
since the function is decreasing on for any fixed . Here, φ stands for the function .
Therefore, by the Fatou lemma, as , it follows that , which is in contrast to our assumption. Hence, is the best constant in (17).
which leads to the result that is not the best possible constant in (17). With this contradiction, the proof is completed. □
Theorem 4 Let be conjugate parameters and be a non-negative measurable homogeneous function of degree −s, . Further, let and be real parameters fulfilling condition (16) and . If the function is decreasing on for any fixed , then is the best constant in (19) and (20).
and consequently, , after letting . This means that the constant is the best possible in (19).
which is impossible since is the best constant in (19). With this contradiction, the proof is completed. □
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