On the monotonicity of left and right Riemann sums
This paper is dedicated to proving general theorems about the monotonicity of left and right Riemann sums, a problem first raised by Fejér in 1950. We provide a much-needed review of the literature on the problem and offer several new sufficient and necessary conditions for the monotonicity of Riema...
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| Abstract | This paper is dedicated to proving general theorems about the monotonicity of left and right Riemann sums, a problem first raised by Fejér in 1950. We provide a much-needed review of the literature on the problem and offer several new sufficient and necessary conditions for the monotonicity of Riemann sums. Additionally, we present a new insightful proof of a fundamental theorem related to these sums using tools from the theory of majorization. Lastly, we delve deeper into a question posed by Borwein, almost resolving it completely. |
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| AbstractList | This paper is dedicated to proving general theorems about the monotonicity of left and right Riemann sums, a problem first raised by Fejér in 1950. We provide a much-needed review of the literature on the problem and offer several new sufficient and necessary conditions for the monotonicity of Riemann sums. Additionally, we present a new insightful proof of a fundamental theorem related to these sums using tools from the theory of majorization. Lastly, we delve deeper into a question posed by Borwein, almost resolving it completely. |
| ArticleNumber | 14 |
| Author | Bouthat, Ludovick |
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| Keywords | 26A42 Monotone sequences Riemann sums 26A48 Interpolation in approximation theory 41A05 26D15 |
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| References | JichangKSome extensions and refinements of Minc-Sathre inequalityMath. Gazette19998349612312710.2307/3618705 Bouthat, L., Mashreghi, J., Morneau-Guérin, F.: The diameter of the Birkhoff Polytope. Special Matrices (2023). Submitted Alzer, H.: On an inequality of H. Minc and L. Sathre. J. Math. Anal. Appl. 179(2), 396–402 (1993) https://doi.org/10.1006/jmaa.1993.1358 PannikovBConvergence of Riemann sums for functions which can be represented by trigonometric series with coefficients forming a monotonic sequenceMath. Notes Acad. Sci. USSR19708581081627799010.1007/BF01146937 CadilhacLMajorization, interpolation and noncommutative Khinchin inequalitiesStudia Math.20212581126421435110.4064/sm181217-30-1 AbramovichSBarićJMatićMPečarićJOn van de Lune-Alzer’s inequalityJ. Math. Inequal.200714563587240840910.7153/jmi-01-47 Gavrea, I.: Operators of Bernstein–Stancu type and the monotonicity of some sequences involving convex functions. In: inequalities and applications. Internat. Ser. Numer. Math., 157:181–192. Birkhäuser, Basel (2009). https://doi.org/10.1007/978-3-7643-8773-0_17 ChenC-PQiFCeronePDragomirSSMonotonicity of sequences involving convex and concave functionsMath. Inequal. Appl.200362229239197460310.7153/mia-06-22 BirkhoffGThree observations on linear algebraUniv. Nac. Tucumán. Revista A.1946514715120547 QiFGeneralizations of Alzer’s and Kuang’s inequalityTamkang J. Math.2000313223227177822010.5556/j.tkjm.31.2000.396 RooinJDehghanHSome monotonicity properties of convex functions with applicationsMediterr. J. Math.2015123593604337680010.1007/s00009-014-0440-z ButzerPLStensRLThe Euler-MacLaurin summation formula, the sampling theorem, and approximate integration over the real axisLinear Algebra Appl.1983525314115570934810.1016/0024-3795(83)80011-1 Lebesgue, H.L.: Leçons sur l’intégration et la recherche des fonctions primitives, professées Au Collège de France. Cambridge Library Collection. Cambridge University Press, Cambridge (2009). https://doi.org/10.1017/CBO9780511701825. Reprint of the 1904 original SzegőGTuránPOn the monotone convergence of certain Riemann sumsPubl. Math. Debrecen1961832633513781810.5486/PMD.1961.8.3-4.10 Borwein, D., Borwein, J.M., Sims, B.: Symmetry and the monotonicity of certain Riemann sums. In: From Analysis to Visualization. Springer Proc. Math. Stat., 313:7–20. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-36568-4_2 Szilárd, A.: Monotonicity of certain Riemann-type sums. The Teaching of Mathematics 15(2), 113–120 (2012). https://teaching.matf.bg.ac.rs/vol/tm1523 Bouthat, L., Mashreghi, J., Morneau-Guérin, F.: Monotonicity of certain left and right Riemann sums. In: recent developments in operator theory, mathematical physics and complex analysis. Oper. Theory Adv. Appl., 290, pp. 89–113. Birkhäuser, Cham (2023). https://doi.org/10.1007/978-3-031-21460-8_3 Kyrezi, I.: Monotonicity properties of Darboux sums. Real Anal. Exchange 35(1), 43–64 (2010) https://doi.org/10.14321/realanalexch.35.1.0043 Marshall, A.W., Olkin, I., Arnold, B.C.: Inequalities: theory of majorization and its applications, 2nd edn. Springer Series in Statistics. Springer, New York (2011). https://doi.org/10.1007/978-0-387-68276-1 PetrovichAYProperties of Riemann sums for functions representable by a trigonometric series with monotone coefficientsMatematicheskii Sbornik19751393360378380243 ButzerPLDodsonMMFerreiraPJSGHigginsJRSchmeisserGStensRLSeven pivotal theorems of Fourier analysis, signal analysis, numerical analysis and number theory: their interconnectionsBull. Math. Sci.201443481525327788310.1007/s13373-014-0057-3 QiFGuoB-NMonotonicity of sequences involving convex function and sequenceMath. Inequal. Appl.2006922472542225011 Turán, P.: On the zeros of the polynomials of Legendre. Časopis Pěst. Mat. Fys. 75, 113–122 (1950) https://doi.org/10.21136/CPMF.1950.123879 Riemann, B.: Ueber die Darstellbarkeit Einer Function Durch Eine Trigonometrische Reihe. Dieterich, Göttingen (1867). http://eudml.org/doc/203787 Minc, H., Sathre, L.: Some inequalities involving (r!)1/r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(r!)^{1/r}$$\end{document}. Proc. Edinburgh Math. Soc. 14(2), 41–46 (1964/65) https://doi.org/10.1017/S0013091500011214 Marsden, J.E., Hoffman, M.J.: Basic Complex Analysis, 2nd edn. W. H. Freeman and Company, New York (1987). https://doi.org/10.2307/3618015 BennettGJamesonGMonotonic averages of convex functionsJ. Math. Anal. Appl.20002521410430179786510.1006/jmaa.2000.7087 G Birkhoff (88_CR9) 1946; 5 88_CR23 88_CR21 PL Butzer (88_CR4) 2014; 4 K Jichang (88_CR10) 1999; 83 G Szegő (88_CR15) 1961; 8 S Abramovich (88_CR16) 2007; 1 C-P Chen (88_CR19) 2003; 6 88_CR27 88_CR24 F Qi (88_CR18) 2006; 9 B Pannikov (88_CR25) 1970; 8 L Cadilhac (88_CR5) 2021; 258 88_CR11 PL Butzer (88_CR6) 1983; 52 88_CR12 AY Petrovich (88_CR26) 1975; 139 88_CR1 88_CR2 J Rooin (88_CR22) 2015; 12 88_CR3 88_CR13 88_CR14 88_CR7 88_CR8 F Qi (88_CR17) 2000; 31 G Bennett (88_CR20) 2000; 252 |
| References_xml | – reference: BennettGJamesonGMonotonic averages of convex functionsJ. Math. Anal. Appl.20002521410430179786510.1006/jmaa.2000.7087 – reference: Turán, P.: On the zeros of the polynomials of Legendre. Časopis Pěst. Mat. Fys. 75, 113–122 (1950) https://doi.org/10.21136/CPMF.1950.123879 – reference: CadilhacLMajorization, interpolation and noncommutative Khinchin inequalitiesStudia Math.20212581126421435110.4064/sm181217-30-1 – reference: Marshall, A.W., Olkin, I., Arnold, B.C.: Inequalities: theory of majorization and its applications, 2nd edn. Springer Series in Statistics. Springer, New York (2011). https://doi.org/10.1007/978-0-387-68276-1 – reference: Marsden, J.E., Hoffman, M.J.: Basic Complex Analysis, 2nd edn. W. H. Freeman and Company, New York (1987). https://doi.org/10.2307/3618015 – reference: Kyrezi, I.: Monotonicity properties of Darboux sums. Real Anal. Exchange 35(1), 43–64 (2010) https://doi.org/10.14321/realanalexch.35.1.0043 – reference: JichangKSome extensions and refinements of Minc-Sathre inequalityMath. Gazette19998349612312710.2307/3618705 – reference: Alzer, H.: On an inequality of H. Minc and L. Sathre. J. Math. Anal. Appl. 179(2), 396–402 (1993) https://doi.org/10.1006/jmaa.1993.1358 – reference: Riemann, B.: Ueber die Darstellbarkeit Einer Function Durch Eine Trigonometrische Reihe. Dieterich, Göttingen (1867). http://eudml.org/doc/203787 – reference: Bouthat, L., Mashreghi, J., Morneau-Guérin, F.: Monotonicity of certain left and right Riemann sums. In: recent developments in operator theory, mathematical physics and complex analysis. Oper. Theory Adv. Appl., 290, pp. 89–113. Birkhäuser, Cham (2023). https://doi.org/10.1007/978-3-031-21460-8_3 – reference: SzegőGTuránPOn the monotone convergence of certain Riemann sumsPubl. Math. Debrecen1961832633513781810.5486/PMD.1961.8.3-4.10 – reference: AbramovichSBarićJMatićMPečarićJOn van de Lune-Alzer’s inequalityJ. Math. Inequal.200714563587240840910.7153/jmi-01-47 – reference: PannikovBConvergence of Riemann sums for functions which can be represented by trigonometric series with coefficients forming a monotonic sequenceMath. Notes Acad. Sci. USSR19708581081627799010.1007/BF01146937 – reference: BirkhoffGThree observations on linear algebraUniv. Nac. Tucumán. Revista A.1946514715120547 – reference: RooinJDehghanHSome monotonicity properties of convex functions with applicationsMediterr. J. Math.2015123593604337680010.1007/s00009-014-0440-z – reference: ChenC-PQiFCeronePDragomirSSMonotonicity of sequences involving convex and concave functionsMath. Inequal. Appl.200362229239197460310.7153/mia-06-22 – reference: PetrovichAYProperties of Riemann sums for functions representable by a trigonometric series with monotone coefficientsMatematicheskii Sbornik19751393360378380243 – reference: Gavrea, I.: Operators of Bernstein–Stancu type and the monotonicity of some sequences involving convex functions. In: inequalities and applications. Internat. Ser. Numer. Math., 157:181–192. Birkhäuser, Basel (2009). https://doi.org/10.1007/978-3-7643-8773-0_17 – reference: ButzerPLStensRLThe Euler-MacLaurin summation formula, the sampling theorem, and approximate integration over the real axisLinear Algebra Appl.1983525314115570934810.1016/0024-3795(83)80011-1 – reference: Bouthat, L., Mashreghi, J., Morneau-Guérin, F.: The diameter of the Birkhoff Polytope. Special Matrices (2023). Submitted – reference: Szilárd, A.: Monotonicity of certain Riemann-type sums. The Teaching of Mathematics 15(2), 113–120 (2012). https://teaching.matf.bg.ac.rs/vol/tm1523 – reference: QiFGeneralizations of Alzer’s and Kuang’s inequalityTamkang J. Math.2000313223227177822010.5556/j.tkjm.31.2000.396 – reference: Lebesgue, H.L.: Leçons sur l’intégration et la recherche des fonctions primitives, professées Au Collège de France. Cambridge Library Collection. Cambridge University Press, Cambridge (2009). https://doi.org/10.1017/CBO9780511701825. Reprint of the 1904 original – reference: Minc, H., Sathre, L.: Some inequalities involving (r!)1/r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(r!)^{1/r}$$\end{document}. Proc. Edinburgh Math. Soc. 14(2), 41–46 (1964/65) https://doi.org/10.1017/S0013091500011214 – reference: Borwein, D., Borwein, J.M., Sims, B.: Symmetry and the monotonicity of certain Riemann sums. In: From Analysis to Visualization. Springer Proc. Math. Stat., 313:7–20. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-36568-4_2 – reference: ButzerPLDodsonMMFerreiraPJSGHigginsJRSchmeisserGStensRLSeven pivotal theorems of Fourier analysis, signal analysis, numerical analysis and number theory: their interconnectionsBull. Math. Sci.201443481525327788310.1007/s13373-014-0057-3 – reference: QiFGuoB-NMonotonicity of sequences involving convex function and sequenceMath. Inequal. Appl.2006922472542225011 – volume: 31 start-page: 223 issue: 3 year: 2000 ident: 88_CR17 publication-title: Tamkang J. Math. doi: 10.5556/j.tkjm.31.2000.396 – ident: 88_CR21 doi: 10.1007/978-3-7643-8773-0_17 – volume: 8 start-page: 810 issue: 5 year: 1970 ident: 88_CR25 publication-title: Math. Notes Acad. Sci. USSR doi: 10.1007/BF01146937 – volume: 1 start-page: 563 issue: 4 year: 2007 ident: 88_CR16 publication-title: J. Math. Inequal. doi: 10.7153/jmi-01-47 – ident: 88_CR3 doi: 10.21136/CPMF.1950.123879 – ident: 88_CR14 – ident: 88_CR12 doi: 10.1017/S0013091500011214 – volume: 252 start-page: 410 issue: 1 year: 2000 ident: 88_CR20 publication-title: J. Math. Anal. Appl. doi: 10.1006/jmaa.2000.7087 – ident: 88_CR11 doi: 10.1006/jmaa.1993.1358 – ident: 88_CR7 doi: 10.1007/978-3-031-21460-8_3 – ident: 88_CR13 doi: 10.14321/realanalexch.35.1.0043 – volume: 8 start-page: 326 year: 1961 ident: 88_CR15 publication-title: Publ. Math. Debrecen doi: 10.5486/PMD.1961.8.3-4.10 – volume: 9 start-page: 247 issue: 2 year: 2006 ident: 88_CR18 publication-title: Math. Inequal. Appl. – volume: 6 start-page: 229 issue: 2 year: 2003 ident: 88_CR19 publication-title: Math. Inequal. Appl. doi: 10.7153/mia-06-22 – volume: 83 start-page: 123 issue: 496 year: 1999 ident: 88_CR10 publication-title: Math. Gazette doi: 10.2307/3618705 – volume: 12 start-page: 593 issue: 3 year: 2015 ident: 88_CR22 publication-title: Mediterr. J. Math. doi: 10.1007/s00009-014-0440-z – volume: 5 start-page: 147 year: 1946 ident: 88_CR9 publication-title: Univ. Nac. Tucumán. Revista A. – volume: 139 start-page: 360 issue: 3 year: 1975 ident: 88_CR26 publication-title: Matematicheskii Sbornik – ident: 88_CR24 doi: 10.1007/978-0-387-68276-1 – ident: 88_CR23 doi: 10.1007/978-3-030-36568-4_2 – volume: 52 start-page: 141 issue: 53 year: 1983 ident: 88_CR6 publication-title: Linear Algebra Appl. doi: 10.1016/0024-3795(83)80011-1 – ident: 88_CR1 – ident: 88_CR2 doi: 10.1017/CBO9780511701825 – volume: 4 start-page: 481 issue: 3 year: 2014 ident: 88_CR4 publication-title: Bull. Math. Sci. doi: 10.1007/s13373-014-0057-3 – volume: 258 start-page: 1 issue: 1 year: 2021 ident: 88_CR5 publication-title: Studia Math. doi: 10.4064/sm181217-30-1 – ident: 88_CR8 doi: 10.1515/spma-2023-0113 – ident: 88_CR27 doi: 10.2307/3618015 |
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| Title | On the monotonicity of left and right Riemann sums |
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