General Opial type inequality
In this paper our goal is to give a general Opial type inequality. We consider two functions, convex and concave and prove a new general inequality on a measure space (Ω,Σ, μ ). The obtained inequalities are not direct generalizations of the Opial inequality but are of Opial type because the integra...
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| Published in | Aequationes mathematicae Vol. 89; no. 3; pp. 641 - 655 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Basel
Springer Basel
01.06.2015
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0001-9054 1420-8903 |
| DOI | 10.1007/s00010-013-0252-4 |
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| Abstract | In this paper our goal is to give a general Opial type inequality. We consider two functions, convex and concave and prove a new general inequality on a measure space (Ω,Σ,
μ
). The obtained inequalities are not direct generalizations of the Opial inequality but are of Opial type because the integrals contain a function and its integral representation. We apply our result to numerous symmetric functions and obtain new results that involve Green’s functions, Lidstone series and the Hermite’s interpolating polynomials. |
|---|---|
| AbstractList | In this paper our goal is to give a general Opial type inequality. We consider two functions, convex and concave and prove a new general inequality on a measure space (Ω,Σ,
μ
). The obtained inequalities are not direct generalizations of the Opial inequality but are of Opial type because the integrals contain a function and its integral representation. We apply our result to numerous symmetric functions and obtain new results that involve Green’s functions, Lidstone series and the Hermite’s interpolating polynomials. In this paper our goal is to give a General Opial type inequality. We consider two functions, convex and concave and prove a new general inequality on a measure space ([Omega],[Sigma],[mu]). The obtained inequalities are not direct generalizations of the Opial inequality but are of Opial type because the integrals contain a function and its integral representation. We apply our result to numerous symmetric functions and obtain new results that involve Green's functions, Lidstone series and the Hermite's interpolating polynomials. |
| Author | Barbir, Ana Pečarić, Josip Himmelreich, Kristina Krulić |
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| Keywords | jensen inequality green function Opial inequality hermite interpolating polynomial Secondary 26A51 Primary 26D15 lidstone polynomial |
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| References | Agarwal, Pang (CR3) 1995 Widder (CR8) 1942; 51 Krulić, Pečarić, Persson (CR4) 2009; 12 Opial (CR6) 1960; 8 Krulić Himmelreich, Pečarić, Pokaz (CR5) 2013 Andrić, Pečarić, Perić (CR1) 2013; 7 Agarwal, Wong (CR2) 1993 Whittaker (CR7) 1933; 36 M. Andrić (252_CR1) 2013; 7 R.P. Agarwal (252_CR2) 1993 K. Krulić (252_CR4) 2009; 12 D.V. Widder (252_CR8) 1942; 51 K. Krulić Himmelreich (252_CR5) 2013 R.P. Agarwal (252_CR3) 1995 Z. Opial (252_CR6) 1960; 8 252_CR7 |
| References_xml | – volume: 51 start-page: 387 year: 1942 end-page: 398 ident: CR8 article-title: Completly convex function and Lidstone series publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-1942-0006356-4 – year: 1993 ident: CR2 publication-title: Error Inequalities in Polynomial Interpolation and Their Applications doi: 10.1007/978-94-011-2026-5 – year: 1995 ident: CR3 publication-title: Opial Inequalities with Applications in Differential and Difference Equations doi: 10.1007/978-94-015-8426-5 – year: 2013 ident: CR5 publication-title: Inequalities of Hardy and Jensen – volume: 12 start-page: 473 issue: 3 year: 2009 end-page: 485 ident: CR4 article-title: A new Hardy–type inequality with kernel publication-title: Math. Inequal. Appl. – volume: 7 start-page: 139 issue: 1 year: 2013 end-page: 150 ident: CR1 article-title: A multiple Opial type inequality for the Riemann–Liouville fractional derivatives publication-title: J. Math. Inequal. doi: 10.7153/jmi-07-13 – volume: 8 start-page: 29 year: 1960 end-page: 32 ident: CR6 article-title: Sur une inéqalité publication-title: Ann. Polon. Math. – volume: 36 start-page: 451 year: 1933 end-page: 469 ident: CR7 article-title: On Lidstone series and two-point expansions of analytic functions publication-title: Proc. Lond. Math. Soc. – volume-title: Opial Inequalities with Applications in Differential and Difference Equations year: 1995 ident: 252_CR3 doi: 10.1007/978-94-015-8426-5 – volume: 8 start-page: 29 year: 1960 ident: 252_CR6 publication-title: Ann. Polon. Math. doi: 10.4064/ap-8-1-29-32 – volume-title: Inequalities of Hardy and Jensen year: 2013 ident: 252_CR5 – ident: 252_CR7 doi: 10.1112/plms/s2-36.1.451 – volume-title: Error Inequalities in Polynomial Interpolation and Their Applications year: 1993 ident: 252_CR2 doi: 10.1007/978-94-011-2026-5 – volume: 51 start-page: 387 year: 1942 ident: 252_CR8 publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-1942-0006356-4 – volume: 12 start-page: 473 issue: 3 year: 2009 ident: 252_CR4 publication-title: Math. Inequal. Appl. – volume: 7 start-page: 139 issue: 1 year: 2013 ident: 252_CR1 publication-title: J. Math. Inequal. doi: 10.7153/jmi-07-13 |
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| Snippet | In this paper our goal is to give a general Opial type inequality. We consider two functions, convex and concave and prove a new general inequality on a... In this paper our goal is to give a General Opial type inequality. We consider two functions, convex and concave and prove a new general inequality on a... |
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| SubjectTerms | Analysis Combinatorics Mathematical functions Mathematics Mathematics and Statistics Polynomials Symmetry |
| Title | General Opial type inequality |
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