zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On strengthened Hardy and Pólya-Knopp’s inequalities. (English) Zbl 1034.26008
Summary: In this paper we prove a strengthened general inequality of the Hardy-Knopp type and also derive its dual inequality. Furthermore, we apply the obtained results to unify the strengthened classical Hardy and Pólya-Knopp’s inequalities deriving them as special cases of the obtained general relations. We discuss Pólya-Knopp’s inequality, compare it with Levin-Cochran-Lee’s inequalities and point out that these results are mutually equivalent. Finally, we also point out a reversed Pólya-Knopp type inequality.

26D15Inequalities for sums, series and integrals of real functions
Full Text: DOI
[1] T. Carleman, Sur les fonctions quasi-analytiques, Comptes rendus du Ve Congres des Mathematiciens Scandinaves, Helsingfors, 1922, pp. 181--196. · Zbl 48.1227.02
[2] Čižmešija, A.; Pečarić, J.: Mixed means and Hardy’s inequality. Math. inequal. Appl. 1, No. 4, 491-506 (1998) · Zbl 0921.26015
[3] Čižmešija, A.; Pečarić, J.: Classical Hardy’s and Carleman’s inequalities and mixed means. Survey on classical inequalities, 27-65 (2000) · Zbl 1024.26011
[4] Čižmešija, A.; Pečarić, J.: Some new generalisations of inequalities of Hardy and levin--cochran--Lee. Bull. austral. Math. soc. 63, No. 1, 105-113 (2001) · Zbl 0982.26018
[5] Čižmešija, A.; Pečarić, J.: On bicheng--debnath’s generalizations of Hardy’s integral inequality. Int. J. Math. math. Sci. 27, No. 4, 237-250 (2001) · Zbl 0995.26015
[6] Cochran, J. A.; Lee, C. -S.: Inequalities related to Hardy’s and heinig’s. Math. proc. Cambridge philos. Soc. 96, 1-7 (1984) · Zbl 0556.26012
[7] Hardy, G. H.: Note on a theorem of Hilbert. Math. Z. 6, 314-317 (1920) · Zbl 47.0207.01
[8] Hardy, G. H.: Notes on some points in the integral calculus (60). Messenger math. 54, 150-156 (1925)
[9] Hardy, G. H.; Littlewood, J. E.; Pólya, G.: Inequalities. (1967) · Zbl 0010.10703
[10] Kaijser, S.; Persson, L. -E.; Öberg, A.: On Carleman and Knopp’s inequalities. J. approx. Theory 117, 140-151 (2002) · Zbl 1049.26014
[11] Knopp, K.: Über reihen mit positiven gliedern. J. London math. Soc. 3, 205-211 (1928) · Zbl 54.0225.01
[12] A. Kufner, L.-E. Persson, The Hardy inequality--about its history and current status, Research Report 6, Dept. of Math., Luleå University of Technology, 2002 (ISSN 1400-4003).
[13] Kufner, A.; Persson, L. -E.: Weighted inequalities of Hardy type. (2003) · Zbl 1065.26018
[14] Love, E. R.: Inequalities related to those of Hardy and of cochran and Lee. Math. proc. Cambridge philos. Soc. 99, 395-408 (1986) · Zbl 0606.26013
[15] Mitrinović, D. S.; Pečarić, J. E.; Fink, A. M.: Inequalities involving functions and their integrals and derivatives. (1991) · Zbl 0744.26011
[16] B. Opic, A. Kufner, Hardy-type inequalities, Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, 1990. · Zbl 0698.26007
[17] Sinnamon, G.: One-dimensional Hardy-type inequalities in many dimensions. Proc. roy. Soc. Edinburgh sect. A 128, No. 4, 833-848 (1998) · Zbl 0910.26009
[18] Yang, B.; Debnath, L.: Generalizations of Hardy’s integral inequalities. Internat. J. Math. math. Sci. 22, No. 3, 535-542 (1999) · Zbl 0971.26012
[19] Yang, B.; Zeng, Z.; Debnath, L.: On new generalizations of Hardy’s integral inequality. J. math. Anal. appl. 217, 321-327 (1998) · Zbl 0893.26008