# American Institute of Mathematical Sciences

2012, 2(2): 223-231. doi: 10.3934/naco.2012.2.223

## On a family of means generated by the Hardy-Littlewood maximal inequality

 1 Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia 2 Faculty of Food Technology and Biotechnology, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia 3 Faculty of Textile Technology, University of Zagreb, Prilaz baruna Filipovića 28a, 10000 Zagreb, Croatia 4 Faculty of Civil Engineering, University of Zagreb, Fra Andrije Kačića Miošića 26, 10000 Zagreb, Croatia

Received  October 2011 Revised  February 2012 Published  May 2012

The functional defined as the difference between the right-hand and the left-hand side of the Hardy-Littlewood maximal inequality is studied and its properties, such as exponential and logarithmic convexity, are explored. Furthermore, related analogues of the Lagrange and Cauchy mean value theorems are derived. Finally, using this functional, a new family of the Cauchy-type means is generated. These means are shown to be monotone.
Citation: Aleksandra Čižmešija, Iva Franjić, Josip Pečarić, Dora Pokaz. On a family of means generated by the Hardy-Littlewood maximal inequality. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 223-231. doi: 10.3934/naco.2012.2.223
##### References:
 [1] N. I. Akhiezer, "The Classical Moment Problem and Some Related Questions in Analysis,", Oliver & Boyd Ltd, (1965).   Google Scholar [2] M. Anwar, J. Jakšetić, J. Pečarić and Atiq Ur Rehman, Exponential convexity, positive semi-definite matrices and fundamental inequalities,, J. Math. Inequal., 4 (2010), 171.  doi: 10.7153/jmi-04-17.  Google Scholar [3] M. Anwar and J. Pečarić, Cauchy means for signed measures,, Bull. Malays. Math. Sci. Soc., 34 (2011), 31.   Google Scholar [4] N. Elezović, K. Krulić and J. Pečarić, Bounds for Hardy type differences,, Acta Math. Sin. (Engl. Ser.), 27 (2011), 671.  doi: 10.1007/s10114-011-9707-5.  Google Scholar [5] G. B. Folland, "Real Analysis, Modern Techniques and Their Applications,", A Wiley-Interscience publication, (1984).   Google Scholar [6] G. H. Hardy, J. E. Littlewooda and G. Pólya, "Inequalities,", 2nd edition, (1952).   Google Scholar [7] J. Jakšetić and J. Pečarić, Means involving linear functionals and $n$-convex functions,, Math. Inequal. Appl., 14 (2011), 657.   Google Scholar [8] D. S. Mitrinović, J. E. Pečarić and A. M. Fink, "Classical and New Inequalities in Analysis,", Kluwer Academic Publishers, (1993).   Google Scholar [9] D. S. Mitrinović, J. E. Pečarić and A. M. Fink, "Inequalities Involving Functions and Their Integrals and Derivatives,", Kluwer Academic Publishers, (1991).   Google Scholar [10] L. Olsen, A new proof of Darboux's theorem,, Amer. Math. Monthly, 111 (2004), 713.  doi: 10.2307/4145046.  Google Scholar [11] J. E. Pečarić, F. Proschan and Y. L. Tong, "Convex Functions, Partial Orderings, and Statistical Applications,", Academic Press, (1992).   Google Scholar

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##### References:
 [1] N. I. Akhiezer, "The Classical Moment Problem and Some Related Questions in Analysis,", Oliver & Boyd Ltd, (1965).   Google Scholar [2] M. Anwar, J. Jakšetić, J. Pečarić and Atiq Ur Rehman, Exponential convexity, positive semi-definite matrices and fundamental inequalities,, J. Math. Inequal., 4 (2010), 171.  doi: 10.7153/jmi-04-17.  Google Scholar [3] M. Anwar and J. Pečarić, Cauchy means for signed measures,, Bull. Malays. Math. Sci. Soc., 34 (2011), 31.   Google Scholar [4] N. Elezović, K. Krulić and J. Pečarić, Bounds for Hardy type differences,, Acta Math. Sin. (Engl. Ser.), 27 (2011), 671.  doi: 10.1007/s10114-011-9707-5.  Google Scholar [5] G. B. Folland, "Real Analysis, Modern Techniques and Their Applications,", A Wiley-Interscience publication, (1984).   Google Scholar [6] G. H. Hardy, J. E. Littlewooda and G. Pólya, "Inequalities,", 2nd edition, (1952).   Google Scholar [7] J. Jakšetić and J. Pečarić, Means involving linear functionals and $n$-convex functions,, Math. Inequal. Appl., 14 (2011), 657.   Google Scholar [8] D. S. Mitrinović, J. E. Pečarić and A. M. Fink, "Classical and New Inequalities in Analysis,", Kluwer Academic Publishers, (1993).   Google Scholar [9] D. S. Mitrinović, J. E. Pečarić and A. M. Fink, "Inequalities Involving Functions and Their Integrals and Derivatives,", Kluwer Academic Publishers, (1991).   Google Scholar [10] L. Olsen, A new proof of Darboux's theorem,, Amer. Math. Monthly, 111 (2004), 713.  doi: 10.2307/4145046.  Google Scholar [11] J. E. Pečarić, F. Proschan and Y. L. Tong, "Convex Functions, Partial Orderings, and Statistical Applications,", Academic Press, (1992).   Google Scholar
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