On the best Hardy constant for quasi-arithmetic means and homogeneous deviation means. (English) Zbl 1384.26058
Summary: The aim of this paper is to characterize the so-called Hardy means, i.e., those means \(M : \bigcup^\infty_{n=1}\mathbb{R}^n_+ \to \mathbb{R}_+\) that satisfy the inequality
\[
\sum^\infty_{n=1} M (x_1,\ldots,x_n) \leqslant C \sum^\infty_{n=1} X_n
\]
for all positive sequences \((X_n)\) with some finite positive constant \(C\). The smallest constant \(C\) satisfying this property is called the Hardy constant of the mean \(M\).
In this paper we determine the Hardy constant in the cases when the mean \(M\) is either a concave quasi-arithmetic or a concave and homogeneous deviation mean.
In this paper we determine the Hardy constant in the cases when the mean \(M\) is either a concave quasi-arithmetic or a concave and homogeneous deviation mean.
Keywords:
mean; Hardy mean; Hardy constant; Hardy inequality; quasi-arithmetic mean; deviation mean; Gini meanReferences:
| [1] | J. A CZ´EL, {\it On mean values}, Bull. Amer. Math. Soc. 54 (1948), 392-400. · Zbl 0030.02702 |
| [2] | K. J. A RROW, {\it Aspects of the Theory of Risk-Bearing}, Yrj¨o Jahnsson Foundation, Helsinki, 1965. |
| [3] | F. B ERNSTEIN AND G. D OETSCH, {\it Zur Theorie der konvexen Funktionen}, Math. Ann. 76 (1915), 514-526. · JFM 45.0627.02 |
| [4] | T. C ARLEMAN, {\it Sur les fonctions quasi-analitiques}, Conf´erences faites au cinqui‘eme congr‘es des math´ematiciens scandinaves, Helsinki (1932), 181-196. |
| [5] | J. C HUDZIAK, D. GŁAZOWSKA, J. J ARCZYK,AND W. J ARCZYK, {\it On weighted quasi-arithmetic} {\it means which are convex}, manuscript. |
| [6] | Z. D AR OCZY´, {\it A general inequality for means}. Aequationes Math. 7 (1971), 16-21. · Zbl 0242.26015 |
| [7] | Z. D AR OCZY´, {\it ¨}{\it Uber eine Klasse von Mittelwerten}, Publ. Math. Debrecen 19 (1972), 211-217. · Zbl 0265.26010 |
| [8] | Z. D AR OCZY AND´L. L OSONCZI, {\it ¨}{\it Uber den Vergleich von Mittelwerten}, Publ. Math. Debrecen 17 (1970), 289-297. · Zbl 0227.26010 |
| [9] | Z. D AR OCZY AND´Z S. P ´ALES, {\it On comparison of mean values}, Publ. Math. Debrecen 29 (1982), 107-115.. · Zbl 0508.26010 |
| [10] | Z. D AR OCZY AND´Z S. P ´ALES, {\it Multiplicative mean values and entropies}, In Functions, series, oper ators, vol. I, II (Budapest, 1980), North-Holland, Amsterdam 1983, 343-359. · Zbl 0558.94002 |
| [11] | B.DE F INETTI, {\it Sul concetto di media}, Giornale dell’ Instituto, Italiano degli Attuarii 2 (1931), 369- 396. · Zbl 0002.27802 |
| [12] | J. D UNCAN AND C. M. M C G REGOR, {\it Carleman’s inequality}, Amer. Math. Monthly 110 (2003), 424-431. · Zbl 1187.26011 |
| [13] | C. G INI, {\it Di una formula compressiva delle medie}, Metron 13 (1938), 3-22. · Zbl 0018.41404 |
| [14] | G. H. H ARDY, {\it Note on a theorem of Hilbert}, Math. Z. 6 (1920), 314-317. · JFM 47.0207.01 |
| [15] | K. K NOPP, {\it ¨}{\it Uber Reihen mit positiven Gliedern}, J. London Math. Soc. 3 (1928), 205-211. · JFM 54.0225.01 |
| [16] | A. N. K OLMOGOROV, {\it Sur la notion de la moyenne}, Rend. Accad. dei Lincei (6), 12 (1930), 388-391. · JFM 56.0198.02 |
| [17] | A. K UFNER, L. M ALIGRANDA,AND L. E. P ERSSON, {\it The Hardy Inequality: About Its History and} {\it Some Related Results}, Vydavatelsk‘y servis, 2007. · Zbl 1213.42001 |
| [18] | E. L ANDAU, {\it A note on a theorem concerning series of positive terms}, J. London Math. Soc. 1 (1921), 38-39. · JFM 52.0207.01 |
| [19] | L. L OSONCZI, {\it ¨}{\it Uber den Vergleich von Mittelwerten die mit Gewichtsfunktionen gebildet sind}, Publ. Math. Debrecen 17 (1970), 203-208. · Zbl 0229.26013 |
| [20] | L. L OSONCZI, {\it Subadditive Mittelwerte}, Arch. Math. (Basel) 22 (1971), 168-174. · Zbl 0226.26023 |
| [21] | L. L OSONCZI, {\it Subhomogene Mittelwerte}, Acta Math. Acad. Sci. Hungar. 22 (1971), 187-195. · Zbl 0226.26022 |
| [22] | L. L OSONCZI, {\it ¨}{\it Uber eine neue Klasse von Mittelwerten}, Acta Sci. Math. (Szeged) 32 (1971), 71-81. · Zbl 0217.37503 |
| [23] | L. L OSONCZI, {\it General inequalities for nonsymmetric means}, Aequationes Math. 9 (1973), 221-235. · Zbl 0266.26018 |
| [24] | L. L OSONCZI, {\it Inequalities for integral mean values}, J. Math. Anal. Appl. 61 (1977), 586-606. · Zbl 0378.26008 |
| [25] | J. G. M IKUSI NSKI´, Sur les moyennes de la formeψ{\it −}1[∑{\it q}ψ({\it x})]. Studia Mathematica 10 (1948), 90-96. · Zbl 0036.31901 |
| [26] | P. M ULHOLLAND, {\it On the generalization of Hardy’s inequality}, J. London Math. Soc. 7 (1932), 208- 214. · JFM 58.0212.04 |
| [27] | M. N AGUMO, {\it ¨}{\it Uber eine Klasse der Mittelwerte}, Japanese J. Math. 7 (1930), 71-79. · JFM 56.0198.03 |
| [28] | Z S. P ´ALES, {\it Characterization of quasideviation means}, Acta Math. Acad. Sci. Hungar. 40 (1982), 243-260. · Zbl 0541.26006 |
| [29] | Z S. P ´ALES, {\it On complementary inequalities}, Publ. Math. Debrecen 30 (1983), 75-88. · Zbl 0544.26009 |
| [30] | Z S. P ´ALES, {\it Inequalities for comparison of means}, In General Inequalities 4 (Oberwolfach, 1983) (W. Walter, ed.), International Series of Numerical Mathematics, vol. 71, Birkh¨auser, Basel, 1984, 59-73. · Zbl 0584.26013 |
| [31] | Z S. P ´ALES, {\it Ingham Jessen’s inequality for deviation means}, Acta Sci. Math. (Szeged) 49 (1985), 131-142. · Zbl 0596.26013 |
| [32] | Z S. P ´ALES, {\it General inequalities for quasideviation means}, Aequationes Math. 36 (1988), 32-56. · Zbl 0652.26023 |
| [33] | Z S. P ´ALES, {\it On a Pexider-type functional equation for quasideviation means}, Acta Math. Hungar. 51 (1988), 205-224. · Zbl 0641.39005 |
| [34] | Z S. P ´ALES, {\it On homogeneous quasideviation means}, Aequationes Math. 36 (1988), 132-152. · Zbl 0664.39005 |
| [35] | Z S. P ´ALES AND P. P ASTECZKA, {\it Characterization of the Hardy property of means and the best Hardy} {\it constants}, Math. Inequal. Appl. 19, 4 (2016), 1141-1158. · Zbl 1353.26030 |
| [36] | P. P ASTECZKA, {\it When is a family of generalized means a scale?}, Real Anal. Exchange 38, 1 (2012/13), 193-209. |
| [37] | J. E. P E CARIˇC AND´K. B. S TOLARSKY, {\it Carleman’s inequality: history and new generalizations}, Aequationes Math. 61, 1-2 (2001), 49-62. · Zbl 0990.26016 |
| [38] | J. W. P RATT, {\it Risk aversion in the small and in the large}, Econometrica 32 (1964), 122-136. · Zbl 0132.13906 |
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