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On some improvements of the Jensen inequality with some applications. (English) Zbl 1175.26041

Summary: An improvement of the Jensen inequality for convex and monotone function is given as well as various applications for means. Similar results for related inequalities of the Jensen type are also obtained. Also some applications of the Cauchy mean and the Jensen inequality are discussed.

MSC:

26D15 Inequalities for sums, series and integrals
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[1] Anwar M, Pečarić J: On logarithmic convexity for differences of power means and related results.Mathematical Inequalities & Applications 2009,12(1):81-90. · Zbl 1179.26039 · doi:10.7153/mia-12-07
[2] Hussain S, Pečarić J: An improvement of Jensen’s inequality with some applications.Asian-European Journal of Mathematics 2009,2(1):85-94. 10.1142/S179355710900008X · Zbl 1180.26013 · doi:10.1142/S179355710900008X
[3] Simić, S., On logarithmic convexity for differences of power means, No. 2007, 8 (2007) · Zbl 1133.26007
[4] Anwar M, Pečarić J: New means of Cauchy’s type.Journal of Inequalities and Applications 2008, 2008: 10. · Zbl 1148.26035
[5] Dragomir SS, McAndrew A: Refinements of the Hermite-Hadamard inequality for convex functions.Journal of Inequalities in Pure and Applied Mathematics 2005,6(2, article 140):-6. · Zbl 1085.26012
[6] Alzer H: The inequality of Ky Fan and related results.Acta Applicandae Mathematicae 1995,38(3):305-354. 10.1007/BF00996150 · Zbl 0834.26013 · doi:10.1007/BF00996150
[7] Beckenbach EF, Bellman R: Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F.. Volume 30. Springer, Berlin, Germany; 1961:xii+198.
[8] Csiszár, I., Information measures: a critical survey, 73-86 (1978) · Zbl 0401.94010
[9] Pardo MC, Vajda I: On asymptotic properties of information-theoretic divergences.IEEE Transactions on Information Theory 2003,49(7):1860-1868. 10.1109/TIT.2003.813509 · Zbl 1301.94049 · doi:10.1109/TIT.2003.813509
[10] Kullback S, Leibler RA: On information and sufficiency.Annals of Mathematical Statistics 1951, 22: 79-86. 10.1214/aoms/1177729694 · Zbl 0042.38403 · doi:10.1214/aoms/1177729694
[11] Cressie P, Read TRC: Multinomial goodness-of-fit tests.Journal of the Royal Statistical Society. Series B 1984,46(3):440-464. · Zbl 0571.62017
[12] Kafka P, Österreicher F, Vincze I: On powers of -divergences defining a distance.Studia Scientiarum Mathematicarum Hungarica 1991,26(4):415-422. · Zbl 0771.94004
[13] Liese F, Vajda I: Convex Statistical Distances, Teubner Texts in Mathematics. Volume 95. BSB B. G. Teubner Verlagsgesellschaft, Leipzig, Germany; 1987:224.
[14] Österreicher F, Vajda I: A new class of metric divergences on probability spaces and its applicability in statistics.Annals of the Institute of Statistical Mathematics 2003,55(3):639-653. 10.1007/BF02517812 · Zbl 1052.62002 · doi:10.1007/BF02517812
[15] Csiszár I, Körner J: Information Theory: Coding Theorems for Discrete Memoryless System, Probability and Mathematical Statistics. Academic Press, New York, NY, USA; 1981:xi+452. · Zbl 0568.94012
[16] Anwar M, Hussain S, Pečarić J: Some inequalities for Csiszár-divergence measures.International Journal of Mathematical Analysis 2009,3(26):1295-1304. · Zbl 1204.62004
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