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A Hardy–Littlewood integral inequality on finite intervals with a concave weight

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Abstract

We prove: For all concave functions \( w: [a,b] \rightarrow [0,\infty )\) and for all functions \(f \in C^2[a,b]\) with \(f(a)=f(b)=0\) we have

$$\begin{aligned} \left( \int _{a}^{b} w(x) f^{\prime }(x)^2 \,dx \right) ^2 \le \left( \int _{a}^{b} w(x) f(x)^2 \,dx \right) \left( \int _{a}^{b} w(x)f^{\prime \prime }(x)^2 \,dx\right) . \end{aligned}$$

Moreover, we determine all cases of equality.

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References

  1. V.F. Babenko, N.P. Korneichuk, V.A. Kofanov, S.A. Pichugov, Inequalities for Derivatives and their Applications (Naukova Dumka, Kiev, 2003)

    Google Scholar 

  2. V.I. Berdyshev, The best approximation in \( L(0,\infty ) \) to the differentiation operator. Mat. Zametki 5, 477–481 (1971)

    Google Scholar 

  3. H.J. Bremermann, Complex convexity. Trans. Am. Math. Soc. 82, 17–51 (1956)

    Article  MATH  MathSciNet  Google Scholar 

  4. W.D. Evans, W.N. Everitt, HELP inequalities for limit-circle and regular problems. Proc. R. Soc. London Ser. A 432, 367–390 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  5. W.N. Everitt, On an extension to an integro-differential inequality of Hardy, Littlewood and Pólya. Proc. R. Soc. Edinb., 69, 295–333 (1971/1972)

  6. G.H. Hardy, J.E. Littlewood, Some integral inequalities connected with the calculus of variations. Q. J. Math, Oxford Ser. 2 3, 241–252 (1932)

    Article  Google Scholar 

  7. G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities (Cambridge University Press, Cambridge, 1934)

    Google Scholar 

  8. T. Kato, On an inequality of Hardy, Littlewood and Pólya. Adv. Math. 7, 217–218 (1971)

    Article  MATH  Google Scholar 

  9. M.K. Kwong, A. Zettl, An extension of the Hardy–Littlewood inequality. Proc. Am. Math. Soc. 77, 117–118 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  10. M.K. Kwong, A. Zettl, Remarks on best constants for norm inequalities among powers of an operator. J. Approx. Theory 26(3), 249–258 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  11. M.K. Kwong, A. Zettl, Norm inequalities of product form in weighted \(L^p\) spaces. Proc. R. Soc. Edinb. 89A, 293–307 (1981)

    Article  MathSciNet  Google Scholar 

  12. M.K. Kwong, A. Zettl, Norm Inequalities for Derivatives and Differences, Lecture Notes in Mathematics 1536. (Springer, New York, 1992)

  13. M.K. Kwong, A. Zettl, An alternate proof of Kato’s inequality. Evolution Equations. Lecture Notes in Pure and Applied Mathematics, vol. 234 (Dekker, New York, 2003), pp. 275–279

  14. C. Niculescu, L.-E. Persson, Convex Functions and Their Applications, CMS Books in Mathematics, vol 23 (Springer, New York, 2006)

  15. A.W. Roberts, D.E. Varberg, Convex Functions (Academic Press, New York, 1973)

    MATH  Google Scholar 

  16. H.L. Royden, P.M. Fitzpatrick, Real Analysis, 4th edn. (Prentice Hall, Boston, 2010)

    MATH  Google Scholar 

  17. E.C. Titchmarsh, The Theory of Functions (Oxford University Press, London, 1939)

    MATH  Google Scholar 

Download references

Acknowledgments

We thank the referee for careful reading of the manuscript and for helpful comments. The research of this author is fully supported by the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. PolyU 5003/12P) and The Hong Kong Polytechnic University Research Grant G-UC22.

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Authors and Affiliations

  1. Morsbacher Str. 10, 51545, Waldbröl, Germany

    Horst Alzer

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hunghom, Hong Kong

    Man Kam Kwong

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  1. Horst Alzer
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  2. Man Kam Kwong
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Correspondence to Horst Alzer.

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Alzer, H., Kwong, M.K. A Hardy–Littlewood integral inequality on finite intervals with a concave weight. Period Math Hung 71, 184–192 (2015). https://doi.org/10.1007/s10998-015-0096-x

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  • DOI: https://doi.org/10.1007/s10998-015-0096-x

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