- Open Access
On a more accurate multidimensional Mulholland-type inequality
© Chen and Yang; licensee Springer 2014
Received: 17 March 2014
Accepted: 29 July 2014
Published: 22 August 2014
In this paper, by using the way of weight coefficients and technique of real analysis, a more accurate multidimensional discrete Mulholland-type inequality with the best possible constant factor is given, which is an extension of the Mulholland inequality. The equivalent form, the operator expression with the norm as well as a few particular cases are also considered.
where the constant factor is still the best possible.
In this paper, by using the way of weight coefficients and technique of real analysis, a more accurate multidimensional discrete Mulholland-type inequality with the best possible constant factor is given, which is an extension of (3). The equivalent form, the operator expression with the norm as well as a few particular cases are also considered.
2 Some lemmas
Then we have (7). □
Hence we prove that (12) is valid for . Therefore, we have (11). □
Then by (13), we obtain (14). □
where and .
- (i)we have(17)
- (ii)for , , setting , , we have(20)
Hence, we have (17). In the same way, we have (18).
Hence, we have (20) and (21). □
3 Main results and operator expressions
we have the following.
is the best possible ( is indicated by (15)).
Then by (17) and (18), we have (23).
and then . Hence, is the best possible constant factor of (23). □
which is equivalent to (23).
namely, , and then (27) follows.
Then by (27), we have (23). Hence (27) and (23) are equivalent.
By the equivalency, the constant factor in (27) is the best possible. Otherwise, we would reach a contradiction by (28) that the constant factor in (23) is not the best possible. □
With the assumptions of Theorem 2, in view of , we have the following definition.
Since the constant factor in (32) is the best possible, we have:
- (ii)Putting , , () and (), in (32), we have the following new inequality:(37)
In particular, for , , in (37), we can deduce (4). Hence, (23) is an extension of (4).
This work is supported by the National Natural Science Foundation of China (No. 61370186), and 2013 Knowledge Construction Special Foundation Item of Guangdong Institution of Higher Learning College and University (No. 2013KJCX0140).
- Hardy GH, Littlewood JE, Pólya G: Inequalities. Cambridge University Press, Cambridge; 1934.MATHGoogle Scholar
- Mitrinović DS, Pečarić JE, Fink AM: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Academic, Boston; 1991.View ArticleMATHGoogle Scholar
- Yang BC: Hilbert-Type Integral Inequalities. Bentham Science Publishers, Sharjah; 2009.Google Scholar
- Yang BC: Discrete Hilbert-Type Inequalities. Bentham Science Publishers, Sharjah; 2011.Google Scholar
- Yang BC: The Norm of Operator and Hilbert-Type Inequalities. Science Press, Beijing; 2009.Google Scholar
- Yang BC: Two Types of Multiple Half-Discrete Hilbert-Type Inequalities. Lambert Academic Publishing, Saarbrücken; 2012.Google Scholar
- Yang BC: On Hilbert’s integral inequality. J. Math. Anal. Appl. 1998, 220: 778–785. 10.1006/jmaa.1997.5877MathSciNetView ArticleMATHGoogle Scholar
- Yang BC, Brnetić I, Krnić M, Pečarić JE: Generalization of Hilbert and Hardy-Hilbert integral inequalities. Math. Inequal. Appl. 2005,8(2):259–272.MathSciNetMATHGoogle Scholar
- Krnić M, Pečarić JE: Hilbert’s inequalities and their reverses. Publ. Math. (Debr.) 2005,67(3–4):315–331.MATHGoogle Scholar
- Yang BC, Rassias TM: On the way of weight coefficient and research for Hilbert-type inequalities. Math. Inequal. Appl. 2003,6(4):625–658.MathSciNetMATHGoogle Scholar
- Yang BC, Rassias TM: On a Hilbert-type integral inequality in the subinterval and its operator expression. Banach J. Math. Anal. 2010,4(2):100–110. 10.15352/bjma/1297117244MathSciNetView ArticleMATHGoogle Scholar
- Azar L: On some extensions of Hardy-Hilbert’s inequality and applications. J. Inequal. Appl. 2009., 2009: Article ID 546829Google Scholar
- Arpad B, Choonghong O: Best constant for certain multilinear integral operator. J. Inequal. Appl. 2006., 2006: Article ID 28582Google Scholar
- Kuang JC, Debnath L: On Hilbert’s type inequalities on the weighted Orlicz spaces. Pac. J. Appl. Math. 2007,1(1):95–103.MathSciNetMATHGoogle Scholar
- Zhong WY: The Hilbert-type integral inequality with a homogeneous kernel of − λ -degree. J. Inequal. Appl. 2008., 2008: Article ID 917392Google Scholar
- Hong Y: On Hardy-Hilbert integral inequalities with some parameters. J. Inequal. Pure Appl. Math. 2005.,6(4): Article ID 92Google Scholar
- Zhong WY, Yang BC: On a multiple Hilbert-type integral inequality with the symmetric kernel. J. Inequal. Appl. 2007., 2007: Article ID 27962 10.1155/2007/27962Google Scholar
- Yang BC, Krnić M: On the norm of a multi-dimensional Hilbert-type operator. Sarajevo J. Math. 2011,7(20):223–243.MathSciNetMATHGoogle Scholar
- Krnić M, Pečarić JE, Vuković P: On some higher-dimensional Hilbert’s and Hardy-Hilbert’s type integral inequalities with parameters. Math. Inequal. Appl. 2008, 11: 701–716.MathSciNetMATHGoogle Scholar
- Krnić M, Vuković P: On a multidimensional version of the Hilbert-type inequality. Anal. Math. 2012, 38: 291–303. 10.1007/s10476-012-0402-2MathSciNetView ArticleMATHGoogle Scholar
- Adiyasuren V, Batbold T: Some new inequalities similar to Hilbert-type integral inequality with a homogeneous kernel. J. Math. Inequal. 2012,6(2):183–193.MathSciNetView ArticleMATHGoogle Scholar
- Adiyasuren V, Batbold T, Krnić M: On several new Hilbert-type inequalities involving means operators. Acta Math. Sin. Engl. Ser. 2013,29(8):1493–1514. 10.1007/s10114-013-2545-xMathSciNetView ArticleMATHGoogle Scholar
- Rassias MT, Yang BC: A multidimensional half-discrete Hilbert-type inequality and the Riemann zeta function. Appl. Math. Comput. 2013, 225: 263–277.MathSciNetView ArticleGoogle Scholar
- Li YJ, He B: On inequalities of Hilbert’s type. Bull. Aust. Math. Soc. 2007,76(1):1–13. 10.1017/S0004972700039423View ArticleMATHGoogle Scholar
- Yang BC: A mixed Hilbert-type inequality with a best constant factor. Int. J. Pure Appl. Math. 2005,20(3):319–328.MathSciNetMATHGoogle Scholar
- Yang BC: A half-discrete Hilbert-type inequality. J. Guangdong Univ. Educ. 2011,31(3):1–7.Google Scholar
- Zhong WY: A mixed Hilbert-type inequality and its equivalent forms. J. Guangdong Univ. Educ. 2011,31(5):18–22.MATHGoogle Scholar
- Zhong WY: A half discrete Hilbert-type inequality and its equivalent forms. J. Guangdong Univ. Educ. 2012,32(5):8–12.MATHGoogle Scholar
- Zhong JH, Yang BC: On an extension of a more accurate Hilbert-type inequality. J. Zhejiang Univ. Sci. Ed. 2008,35(2):121–124.MathSciNetMATHGoogle Scholar
- Zhong JH: Two classes of half-discrete reverse Hilbert-type inequalities with a non-homogeneous kernel. J. Guangdong Univ. Educ. 2012,32(5):11–20.Google Scholar
- Zhong WY, Yang BC: A best extension of Hilbert inequality involving several parameters. J. Jinan Univ., Nat. Sci. 2007,28(1):20–23.Google Scholar
- Zhong WY, Yang BC: A reverse Hilbert’s type integral inequality with some parameters and the equivalent forms. Pure Appl. Math. 2008,24(2):401–407.MathSciNetGoogle Scholar
- Rassias MT, Yang BC: On half-discrete Hilbert’s inequality. Appl. Math. Comput. 2013, 220: 75–93.MathSciNetView ArticleMATHGoogle Scholar
- Yang BC, Chen Q: A half-discrete Hilbert-type inequality with a homogeneous kernel and an extension. J. Inequal. Appl. 2011., 2011: Article ID 124 10.1186/1029-242X-2011-124Google Scholar
- Yang BC: A half-discrete Hilbert-type inequality with a non-homogeneous kernel and two variables. Mediterr. J. Math. 2012,10(2):677–692. 10.1007/s00009-012-0213-5View ArticleMathSciNetMATHGoogle Scholar
- Yang BC: Hilbert-type integral operators: norms and inequalities. In Nonlinear Analysis: Stability, Approximation, and Inequalities. Edited by: Pardalos PM, Georgiev PG, Srivastava HM. Springer, New York; 2012:771–859. Chapter 42View ArticleGoogle Scholar
- Kuang JC: Applied Inequalities. Shangdong Science Technic Press, Jinan; 2004.Google Scholar
- Pan YL, Wang HT, Wang FT: On Complex Functions. Science Press, Beijing; 2006.Google Scholar
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