Periodica Mathematica Hungarica

, Volume 71, Issue 2, pp 184–192 | Cite as

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

  • Horst Alzer
  • Man Kam Kwong


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.


Integral inequality HELP-type inequality Concave weight function 

Mathematics Subject Classification

26D10 26D15 



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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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2015

Authors and Affiliations

  1. 1.WaldbrölGermany
  2. 2.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityHunghomHong Kong

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