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 and 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, Edinburgh And London, 1965.

[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), 171-189. doi: 10.7153/jmi-04-17.

[3]

M. Anwar and J. Pečarić, Cauchy means for signed measures, Bull. Malays. Math. Sci. Soc., 34 (2011), 31-38.

[4]

N. Elezović, K. Krulić and J. Pečarić, Bounds for Hardy type differences, Acta Math. Sin. (Engl. Ser.), 27 (2011), 671-684. doi: 10.1007/s10114-011-9707-5.

[5]

G. B. Folland, "Real Analysis, Modern Techniques and Their Applications," A Wiley-Interscience publication, John Wiley & Sons, INC., New York / Chichester / Weinheim / Brisbane / Singapore / Toronto, 1984.

[6]

G. H. Hardy, J. E. Littlewooda and G. Pólya, "Inequalities," 2nd edition, Cambridge University Press, Cambridge, 1952.

[7]

J. Jakšetić and J. Pečarić, Means involving linear functionals and $n$-convex functions, Math. Inequal. Appl., 14 (2011), 657-675.

[8]

D. S. Mitrinović, J. E. Pečarić and A. M. Fink, "Classical and New Inequalities in Analysis," Kluwer Academic Publishers, Dordrecht/Boston/London, 1993.

[9]

D. S. Mitrinović, J. E. Pečarić and A. M. Fink, "Inequalities Involving Functions and Their Integrals and Derivatives," Kluwer Academic Publishers, Dordrecht/Boston/London, 1991.

[10]

L. Olsen, A new proof of Darboux's theorem, Amer. Math. Monthly, 111 (2004), 713-715. doi: 10.2307/4145046.

[11]

J. E. Pečarić, F. Proschan and Y. L. Tong, "Convex Functions, Partial Orderings, and Statistical Applications," Academic Press, San Diego, 1992.

show all references

References:
[1]

N. I. Akhiezer, "The Classical Moment Problem and Some Related Questions in Analysis," Oliver & Boyd Ltd, Edinburgh And London, 1965.

[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), 171-189. doi: 10.7153/jmi-04-17.

[3]

M. Anwar and J. Pečarić, Cauchy means for signed measures, Bull. Malays. Math. Sci. Soc., 34 (2011), 31-38.

[4]

N. Elezović, K. Krulić and J. Pečarić, Bounds for Hardy type differences, Acta Math. Sin. (Engl. Ser.), 27 (2011), 671-684. doi: 10.1007/s10114-011-9707-5.

[5]

G. B. Folland, "Real Analysis, Modern Techniques and Their Applications," A Wiley-Interscience publication, John Wiley & Sons, INC., New York / Chichester / Weinheim / Brisbane / Singapore / Toronto, 1984.

[6]

G. H. Hardy, J. E. Littlewooda and G. Pólya, "Inequalities," 2nd edition, Cambridge University Press, Cambridge, 1952.

[7]

J. Jakšetić and J. Pečarić, Means involving linear functionals and $n$-convex functions, Math. Inequal. Appl., 14 (2011), 657-675.

[8]

D. S. Mitrinović, J. E. Pečarić and A. M. Fink, "Classical and New Inequalities in Analysis," Kluwer Academic Publishers, Dordrecht/Boston/London, 1993.

[9]

D. S. Mitrinović, J. E. Pečarić and A. M. Fink, "Inequalities Involving Functions and Their Integrals and Derivatives," Kluwer Academic Publishers, Dordrecht/Boston/London, 1991.

[10]

L. Olsen, A new proof of Darboux's theorem, Amer. Math. Monthly, 111 (2004), 713-715. doi: 10.2307/4145046.

[11]

J. E. Pečarić, F. Proschan and Y. L. Tong, "Convex Functions, Partial Orderings, and Statistical Applications," Academic Press, San Diego, 1992.

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