Characterizations of weight functions are given for which integral inequalities of monotone and concave functions are satisfied. The constants in these inequalities are sharp and in the case of concave functions, constitute weighted forms of Favard-Berwald inequalities on finite and infinite intervals. Related inequalities, some of Hardy type, are also given.
@article{bwmeta1.element.bwnjournal-article-smv116i2p133bwm,
author = {Hans Heinig and Lech Maligranda},
title = {Weighted inequalities for monotone and concave functions},
journal = {Studia Mathematica},
volume = {113},
year = {1995},
pages = {133-165},
zbl = {0851.26012},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv116i2p133bwm}
}
Heinig, Hans; Maligranda, Lech. Weighted inequalities for monotone and concave functions. Studia Mathematica, Tome 113 (1995) pp. 133-165. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv116i2p133bwm/
[00000] [1] K. F. Andersen, Weighted generalized Hardy inequalities for nonincreasing functions, Canad. J. Math. 43 (1991), 1121-1135. | Zbl 0757.26018
[00001] [2] M. Ariño and B. Muckenhoupt, Maximal functions on classical spaces and Hardy's inequality with weights for nonincreasing functions, Trans. Amer. Math. Soc. 320 (1990), 727-735. | Zbl 0716.42016
[00002] [3] R. W. Barnard and J. Wells, Weighted inverse Hölder inequalities, J. Math. Anal. Appl. 147 (1990), 198-213.
[00003] [4] D. C. Barnes, Some complements of Hölder's inequality, ibid. 26 (1969), 82-87. | Zbl 0186.37302
[00004] [5] E. F. Beckenbach and R. Bellman, Inequalities, Springer, New York, 1983.
[00005] [6] R. Bellman, Converses of Schwarz's inequality, Duke Math. J. 23 (1956), 429-434. | Zbl 0070.28503
[00006] [7] J. Bergh, V. Burenkov and L. E. Persson, Best constants in reversed Hardy's inequalities for quasimonotone functions, Acta Sci. Math. (Szeged) 59 (1994), 221-239. | Zbl 0805.26008
[00007] [8] L. Berwald, Verallgemeinerung eines Mittelwertsatzes von J. Favard, für positive konkave Funktionen, Acta Math. 79 (1947), 17-37. | Zbl 0029.11704
[00008] [9] C. Borell, Inverse Hölder inequalities in one and several dimensions, J. Math. Anal. Appl. 41 (1973), 300-312. | Zbl 0249.26015
[00009] [10] J. Bradley, Hardy inequalities with mixed norms, Canad. Math. Bull. 21 (1978), 405-408. | Zbl 0402.26006
[00010] [11] H. Bückner, Untere Schranken für skalare Produkte von Vektoren und für analoge Integralausdrücke, Ann. Mat. Pura Appl. 28 (1949), 237-261. | Zbl 0036.17101
[00011] [12] V. I. Burenkov, On the best constant in Hardy's inequality for 0 < p < 1, Trudy Mat. Inst. Steklov. 194 (1992), 58-62 (in Russian).
[00012] [13] R. J. Bushell and W. Okrasiński, Nonlinear Volterra equations with convolution kernel, J. London Math. Soc. 41 (1990), 503-510. | Zbl 0714.45006
[00013] [14] A. P. Calderón and R. Scott, Sobolev type inequalities for p> 0, Studia Math. 62 (1978), 75-92. | Zbl 0399.46031
[00014] [15] M. J. Carro and J. Soria, Weighted Lorentz spaces and the Hardy operator, J. Funct. Anal. 112 (1993), 480-494. | Zbl 0784.46022
[00015] [16] M. J. Carro and J. Soria, Boundedness of some linear operators, Canad. J. Math. 45 (1993), 1155-1166. | Zbl 0798.42010
[00016] [17] A. Clausing, Disconjugacy and integral inequalities, Trans. Amer. Math. Soc. 260 (1980), 293-307. | Zbl 0453.26009
[00017] [18] J. Favard, Sur les valeurs moyennes, Bull. Sci. Math. 57 (1933), 54-64.
[00018] [19] P. Frank und G. Pick, Distanzschätzungen im Funktionenraum. I, Math. Ann. 76 (1915), 354-375. | Zbl 45.0456.01
[00019] [20] A. García del Amo, On reverse Hardy's inequality, Collect. Math. 44 (1993), 115-123. | Zbl 0816.26006
[00020] [21] G. Grűss, Über das Maximum des absoluten Betrages von , Math. Z. 39 (1935), 215-226. | Zbl 60.0189.02
[00021] [22] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Univ. Press, 1952.
[00022] [23] H. Heinig and L. Maligranda, Chebyshev inequality in function spaces, Real Anal. Exchange 17 (1991-92), 211-247. | Zbl 0748.26011
[00023] [24] H. Heinig and V. D. Stepanov, Weighted Hardy inequalities for increasing functions, Canad. J. Math. 45 (1993), 104-116. | Zbl 0796.26008
[00024] [25] R. A. Hunt, On L(p, q) spaces, Enseign. Math. 12 (1966), 249-276. | Zbl 0181.40301
[00025] [26] S. Karlin and W. J. Studden, Tchebycheff Systems: with Applications in Analysis and Statistics, Pure Appl. Math. 15, Wiley, New York, 1966. | Zbl 0153.38902
[00026] [27] S. Lai, Weighted norm inequalities for general operators on monotone functions, Trans. Amer. Math. Soc. 340 (1993), 811-836. | Zbl 0819.47044
[00027] [28] G. G. Lorentz, Some new functional spaces, Ann. of Math. 51 (1950), 37-55. | Zbl 0035.35602
[00028] [29] L. Maligranda, J. Pečarić and L. E. Persson, Weighted Favard and Berwald inequalities, J. Math. Anal. Appl. 190 (1995), 248-262. | Zbl 0834.26012
[00029] [30] V. M. Manakov, On the best constant in weighted inequalities for Riemann-Liouville integrals, Bull. London Math. Soc. 24 (1991), 442-448. | Zbl 0763.26006
[00030] [31] W. G. Mazja [V. G. Maz'ya], Einbettungssätze für Sobolewsche Räume, Teil 1, Teubner, Leipzig, 1979. | Zbl 0429.46018
[00031] [32] B. Muckenhoupt, Hardy's inequality with weights, Studia Math. 44 (1972), 31-38. | Zbl 0236.26015
[00032] [33] E. Myasnikov, L. E. Persson and V. Stepanov, On the best constants in certain integral inequalities for monotone functions, Acta Sci. Math. (Szeged) 59 (1994), 613-624. | Zbl 0814.26011
[00033] [34] C. J. Neugebauer, Weighted norm inequalities for averaging operators of monotone functions, Publ. Mat. 35 (1991), 429-447. | Zbl 0746.42014
[00034] [35] R. O'Neil, Convolution operators and L(p,q) spaces, Duke Math. J. 30 (1963), 129-142.
[00035] [36] B. Opic and A. Kufner, Hardy-Type Inequalities, Pitman Res. Notes in Math. Ser. 219, Longman, 1990.
[00036] [37] M. Petschke, Extremalstrahlen konvexer Kegel und komplementäre Ungleichungen, Ph.D. Dissertation, Darmstadt, 1989, 106 pp.
[00037] [38] E. Sawyer, Weighted Lebesgue and Lorentz norm inequalities, Trans. Amer. Math. Soc. 281 (1984), 329-337. | Zbl 0538.47020
[00038] [39] E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math. 96 (1990), 145-158. | Zbl 0705.42014
[00039] [40] G. Sinnamon, Weighted Hardy and Opial-type inequalities, J. Math. Anal. Appl. 160 (1991), 434-445. | Zbl 0756.26011
[00040] [41] E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N.J., 1971. | Zbl 0232.42007
[00041] [42] V. D. Stepanov, On boundedness of linear integral operators on the class of monotone functions, Sibirsk. Mat. Zh. 32 (1991), 222-224 (in Russian).
[00042] [43] V. D. Stepanov, Integral operators on the cone of monotone functions, J. London Math. Soc. 48 (1994), 465-487. | Zbl 0837.26011
[00043] [44] V. D. Stepanov, Weighted Hardy's inequality for nonincreasing functions, Trans. Amer. Math. Soc. 338 (1993), 173-186. | Zbl 0786.26015
[00044] [45] W. Walter and V. Weckesser, An integral inequality of convolution type, Aequationes Math. 46 (1993), 212-219. | Zbl 0787.26015
[00045] [46] C.-L. Wang, An extension of a Bellman inequality, Utilitas Math. 8 (1975), 251-256. | Zbl 0324.49025
[00046] [47] H.-T. Wang and S.-Y. Chen, Inverse Hölder inequalities with weight , J. Math. Anal. Appl. 176 (1993), 92-107.