Further improvement of an extension of Hölder-type inequality. (English) Zbl 1435.26022
Summary: In [Bull. Aust. Math. Soc. 51, No. 3, 453–458 (1995; Zbl 0835.26014)], C. E. M. Pearce and the second author proved an extension of Hölder’s inequality. In this paper we extend their result in a measure theoretic sense and further improve it using log-convexity of related linear functionals. Moreover, we study the action of related linear functionals on families of exponentially convex functions.
MSC:
| 26D15 | Inequalities for sums, series and integrals |
Keywords:
Hölder inequality; Steffensen inequality; measure theory; exponential convexity; log-convexityCitations:
Zbl 0835.26014References:
| [1] | N. I. AKHIEZER,The Classical Moment Problem and Some Related Questions in Analysis, Oliver and Boyd, Edinburgh, 1965. · Zbl 0135.33803 |
| [2] | S. N. BERNSTEIN,Sur les fonctions absolument monotones, Acta Math.52(1929), 1-66. · JFM 55.0142.07 |
| [3] | J. C. EVARD, H. GAUCHMAN,Steffensen type inequalites over general measure spaces, Analysis17, 2-3 (1997), 301-322. · Zbl 0887.26013 |
| [4] | H. GAUCHMAN,A Steffensen type inequality, J. Inequal. Pure Appl. Math.1, 1 (2000), Article 3. · Zbl 0952.26012 |
| [5] | H. GAUCHMAN,On further generalization of Steffensen’s inequality, J. Inequal. Appl.5, 5 (2000), 505-513. · Zbl 0968.26014 |
| [6] | J. JAKSETIˇC´, J. PECARIˇC´,Exponential convexity method, J. Convex Anal.20, 1 (2013), 181-197. · Zbl 1275.26038 |
| [7] | J. JAKˇSETIC´, J. PECARIˇC´,Steffensen’s inequality for positive measures, Math. Inequal. Appl.18, 3 (2015), 1159-1170. · Zbl 1321.26049 |
| [8] | J. JAKSETIˇC´, J. PECARIˇC´, K. SMOLJAKKALAMIR,Some measure theoretic aspects of Steffensen’s and reversed Steffensen’s inequality, J. Math. Inequal.10, 2 (2016), 459-469. · Zbl 1334.26048 |
| [9] | J. JAKˇSETIC´, J. PECARIˇC´, K. SMOLJAKKALAMIR,Exponential convexity induced by Steffensen’s inequality and positive measures, Results Math.73:136, 4 (2018), 18 pages. · Zbl 1404.26014 |
| [10] | C. E. M. PEARCE, J. PECARIˇC´,On an extension of H¨older’s inequality, Bull. Austral. Math. Soc.51, 3 (1995), 453-458. · Zbl 0835.26014 |
| [11] | J. PECARIˇC´, J. PERIC´,Improvements of the Giaccardi and the Petrovi´c inequality and related results, An. Univ. Craiova Ser. Mat. Inform.39, 1 (2012), 65-75. · Zbl 1274.26069 |
| [12] | J. E. PECARIˇC´, F. PROSCHAN, Y. L. TONG,Convex functions, partial orderings, and statistical applications, Mathematics in science and engineering187, Academic Press, 1992. |
| [13] | J. PECARIˇC´, K. SMOLJAK,Improvement of an extension of H¨older-type inequality, Anal. Math.38, 2 (2012), 135-146. · Zbl 1265.26082 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
