General Lebesgue integral inequalities of Jensen and Ostrowski type for differentiable functions whose derivatives in absolute value are h-convex and applications

Sever Dragomir

Abstract


Some inequalities related to Jensen and Ostrowski inequalities for general Lebesgue integral of differentiable functions whose derivatives in absolute value are h-convex are obtained. Applications for f-divergence measure are provided as well.

Keywords


Ostrowski's inequality; Jensen's inequality; f-divergence measures.

Full Text:

PDF

References


Ali, S. M., Silvey, S. D., A general class of coefficients of divergence of one distribution from another, J. Roy. Statist. Soc. Sec. B 28 (1966), 131-142.

Alomari, M., Darus, M., The Hadamard's inequality for s-convex function, Int. J. Math. Anal. (Ruse) 2, No. 13-16 (2008), 639-646.

Alomari, M., Darus, M., Hadamard-type inequalities for s-convex functions, Int. Math. Forum 3, No. 37-40 (2008), 1965-1975.

Anastassiou, G. A., Univariate Ostrowski inequalities, revisited, Monatsh. Math. 135, No. 3 (2002), 175-189.

Barnett, N. S., Cerone, P., Dragomir, S. S., Pinheiro, M. R., Sofo, A., Ostrowski type inequalities for functions whose modulus of the derivatives are convex and applications, in Inequality Theory and Applications Vol. 2 (Chinju/Masan, 2001), Nova Sci. Publ., Hauppauge, NY, 2003, 19-32. Preprint: RGMIA Res. Rep. Coll. 5, No. 2 (2002), Art. 1 [Online http://rgmia.org/papers/v5n2/Paperwapp2q.pdf].

Beckenbach, E. F., Convex functions, Bull. Amer. Math. Soc. 54 (1948), 439-460.

Beth Bassat, M., f-entropies, probability of error and feature selection, Inform. Control 39 (1978), 227-242.

Bhattacharyya, A., On a measure of divergence between two statistical populations defined by their probability distributions, Bull. Calcutta Math. Soc. 35 (1943), 99-109.

Bombardelli, M., Varosanec, S., Properties of h-convex functions related to the Hermite-Hadamard-Fejer inequalities, Comput. Math. Appl. 58, No. 9 (2009), 1869-1877.

Breckner, W. W., Stetigkeitsaussagen fur eine Klasse verallgemeinerter konvexer Funktionen in topologischen linearen Raumen, Publ. Inst. Math. (Beograd) (N.S.) 23 (37) (1978), 13-20.

Breckner, W. W., Orban, G., Continuity Properties of Rationally s-Convex Mappings with Values in an Ordered Topological Linear Space, Universitatea "Babes-Bolyai", Facultatea de Matematica, Cluj-Napoca, 1978.

Burbea, I., Rao, C. R., On the convexity of some divergence measures based on entropy function, IEEE Trans. Inform. Theory 28 (3) (1982), 489-495.

Cerone, P., Dragomir, S. S., Midpoint-type rules from an inequalities point of view, in Handbook of Analytic-Computational Methods in Applied Mathematics, Anastassiou, G. A., (Ed.), CRC Press, New York, 2000, 135-200.

Cerone, P., Dragomir, S. S., New bounds for the three-point rule involving the Riemann-Stieltjes integrals, in Advances in Statistics Combinatorics and Related Areas, Gulati, C., et al. (Eds.), World Science Publishing, River Edge, N.J., 2002, 53-62.

Cerone, P., Dragomir, S. S., Pearce, C. E. M., A generalised trapezoid inequality for functions of bounded variation, Turkish J. Math. 24 (2) (2000), 147-163.

Cerone, P., Dragomir, S. S., Roumeliotis, J., Some Ostrowski type inequalities for n-time differentiable mappings and applications, Demonstratio Math. 32 (2) (1999), 697-712.

Chen, C. H., Statistical Pattern Recognition, Hoyderc Book Co., Rocelle Park, New York, 1973.

Chow, C. K., Lin, C. N., Approximating discrete probability distributions with dependence trees, IEEE Trans. Inform. Theory 14 (3) (1968), 462-467.

Cristescu, G., Hadamard type inequalities for convolution of h-convex functions, Ann. Tiberiu Popoviciu Semin. Funct. Equ. Approx. Convexity 8 (2010), 3-11.

Csiszar, I. I., Information-type measures of difference of probability distributions and indirect observations, Studia Math. Hungarica 2 (1967), 299-318.

Csiszar, I. I., On topological properties of f-divergences, Studia Math. Hungarica 2 (1967), 329-339.

Csiszar, I. I., Korner, J., Information Theory: Coding Theorem for Discrete Memoryless Systems, Academic Press, New York, 1981.

Dragomir, S. S., Ostrowski's inequality for monotonous mappings and applications, J. KSIAM 3 (1) (1999), 127-135.

Dragomir, S. S., The Ostrowski integral inequality for mappings of bounded variation, Bull. Austral. Math. Soc. 60 (1) (1999), 495-508.

Dragomir, S. S., The Ostrowski's integral inequality for Lipschitzian mappings and applications, Comp. Math. Appl. 38 (1999), 33-37.

Dragomir, S. S., A converse result for Jensen's discrete inequality via Gruss' inequality and applications in information theory, An. Univ. Oradea Fasc. Mat. 7 (1999/2000), 178-189.

Dragomir, S. S., On the midpoint quadrature formula for mappings with bounded variation and applications, Kragujevac J. Math. 22 (2000), 13-18.

Dragomir, S. S., On the Ostrowski's inequality for Riemann-Stieltjes integral, Korean J. Appl. Math. 7 (2000), 477-485.

Dragomir, S. S., On the Ostrowski's integral inequality for mappings with bounded variation and applications, Math. Inequal. Appl. 4 (1) (2001), 59-66.

Dragomir, S. S., On the Ostrowski inequality for Riemann-Stieltjes integral (int_{a}^{b}f(t) du(t)) where (f) is of Holder type and u is of bounded variation and applications, J. KSIAM 5 (1) (2001), 35-45.

Dragomir, S. S., On a reverse of Jessen's inequality for isotonic linear functionals, J. Ineq. Pure Appl. Math. 2, No. 3, (2001), Art. 36.

Dragomir, S. S., Ostrowski type inequalities for isotonic linear functionals, J. Inequal. Pure Appl. Math. 3 (5) (2002), Art. 68.

Dragomir, S. S., A refinement of Ostrowski's inequality for absolutely continuous functions whose derivatives belong to $L_{infty }$ and applications, Libertas Math. 22 (2002), 49-63.

Dragomir, S. S., An inequality improving the first Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, J. Inequal. Pure Appl. Math. 3, No. 2 (2002), Art. 31.

Dragomir, S. S., An inequality improving the second Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, J. Inequal. Pure Appl. Math. 3, No. 3 (2002), Art. 35.

Dragomir, S. S., Some companions of Ostrowski's inequality for absolutely continuous functions and applications, Preprint RGMIA Res. Rep. Coll. 5 (2002), Suppl. Art. 29. [Online http://rgmia.org/papers/v5e/COIACFApp.pdf], Bull. Korean Math. Soc. 42, No. 2 (2005), 213-230.

Dragomir, S. S., A Gruss type inequality for isotonic linear functionals and applications, Demonstratio Math. 36, No. 3 (2003), 551-562. Preprint RGMIA Res. Rep. Coll. 5 (2002), Supl. Art. 12. [Online http://rgmia.org/v5(E).php].

Dragomir, S. S., An Ostrowski like inequality for convex functions and applications, Revista Math. Complutense 16 (2) (2003), 373-382.

Dragomir, S. S., Bounds for the normalised Jensen functional, Bull. Aust. Math. Soc. 74 (2006), 471-478.

Dragomir, S. S., Bounds for the deviation of a function from the chord generated by its extremities, Bull. Aust. Math. Soc. 78, No. 2 (2008), 225-248.

Dragomir, S. S., Reverses of the Jensen inequality in terms of the first derivative and applications, Preprint RGMIA Res. Rep. Coll. 14 (2011), Art. 71 [http://rgmia.org/papers/v14/v14a71.pdf].

Dragomir, S. S., Operator Inequalities of Ostrowski and Trapezoidal Type, Springer, New York, 2012.

Dragomir, S. S., Perturbed companions of Ostrowski's inequality for absolutely continuous functions (I), Preprint RGMIA Res. Rep. Coll. 17 (2014), Art 7. [Online http://rgmia.org/papers/v17/v17a07.pdf].

Dragomir, S. S., Inequalities of Hermite-Hadamard type for (lambda)-convex functions on linear spaces, Preprint RGMIA Res. Rep. Coll. 17 (2014), Art. 13.

Dragomir, S. S., Jensen and Ostrowski type inequalities for general Lebesgue integral with applications (I), Preprint RGMIA Res. Rep. Coll. 17 (2014), Art. 25.

Dragomir, S. S., Cerone, P., Roumeliotis, J., Wang, S., A weighted version of Ostrowski inequality for mappings of Holder type and applications in numerical analysis, Bull. Math. Soc. Sci. Math. Romanie 42(90) (4) (1999), 301-314.

Dragomir, S. S., Fitzpatrick, S., The Hadamard inequalities for s-convex functions in the second sense, Demonstratio Math. 32, No. 4 (1999), 687-696.

Dragomir, S. S., Fitzpatrick, S., The Jensen inequality for s-Breckner convex functions in linear spaces, Demonstratio Math. 33, No. 1 (2000), 43-49.

Dragomir, S. S., Ionescu, N. M., Some converse of Jensen's inequality and applications, Rev. Anal. Numer. Theor. Approx. 23, No. 1 (1994), 71-78.

Dragomir, S. S., Mond, B., On Hadamard's inequality for a class of functions of Godunova and Levin, Indian J. Math. 39, No. 1 (1997), 1-9.

Dragomir, S. S., Pearce, C. E., On Jensen's inequality for a class of functions of Godunova and Levin, Period. Math. Hungar. 33, No. 2 (1996), 93-100.

Dragomir, S. S., Pearce, C. E., Quasi-convex functions and Hadamard's inequality, Bull. Aust. Math. Soc. 57 (1998), 377-385.

Dragomir, S. S., Pecaric, J., Persson, L., Some inequalities of Hadamard type, Soochow J. Math. 21, No. 3 (1995), 335-341.

Dragomir, S. S., Pecaric, J., Persson, L., Properties of some functionals related to Jensen's inequality, Acta Math. Hungarica 70 (1996), 129-143.

Dragomir, S. S., Rassias, Th. M. (Eds.), Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer Academic Publishers, Dordrecht-Boston-London, 2002.

Dragomir, S. S., Wang, S., A new inequality of Ostrowski's type in (L_{1})-norm and applications to some special means and to some numerical quadrature rules}, Tamkang J. Math. 28 (1997), 239-244.

Dragomir, S. S., Wang, S., A new inequality of Ostrowski's type in (L_{p})-norm and applications to some special means and to some numerical quadrature rules, Indian J. Math. 40 (3) (1998), 245-304.

Dragomir, S. S., Wang, S., Applications of Ostrowski's inequality to the estimation of error bounds for some special means and some numerical quadrature rules, Appl. Math. Lett. 11 (1998), 105-109.

El Farissi, A., Simple proof and refinement of Hermite-Hadamard inequality, J. Math. Ineq. 4, No. 3 (2010), 365-369.

Fink, A. M., Bounds on the deviation of a function from its averages, Czechoslovak Math. J. 42, No. 2 (1992), 298-310.

Godunova, E. K., Levin, V. I., Inequalities for functions of a broad class that contains convex, monotone and some other forms of functions, in Numerical Mathematics and Mathematical Physics, Moskov. Gos. Ped. Inst., Moscow, 1985, 138-142 (Russian).

Gokhale, D. V., Kullback, S., Information in Contingency Tables, Marcel Decker, New York, 1978.

Hudzik, H., Maligranda, L., Some remarks on s-convex functions, Aequationes Math. 48, No. 1 (1994), 100-111.

Guessab, A., Schmeisser, G., Sharp integral inequalities of the Hermite-Hadamard type, J. Approx. Theory 115 (2002), 260-288.

Havrda, J. H., Charvat, F., Quantification method classification process: concept of structural (alpha)-entropy, Kybernetika 3 (1967), 30-35.

Hellinger, E., Neue Bergruirdung du Theorie quadratisher Formerus von uneudlichvieleu Veranderlicher, J. Reine Angew. Math. 36 (1909), 210-271.

Jeffreys, H., An invariant form for the prior probability in estimating problems, Proc. Roy. Soc. London A Math. Phys. Sci. 186 (1946), 453-461.

Kadota, T. T., Shepp, L. A., On the best finite set of linear observables for discriminating two Gaussian signals, IEEE Trans. Inform. Theory 13 (1967), 288-294.

Kailath, T., The divergence and Bhattacharyya distance measures in signal selection, IEEE Trans. Comm. Technology 15 (1967), 52-60.

Kapur, J. N., A comparative assessment of various measures of directed divergence, Advances in Management Studies 3 (1984), 1-16.

Kazakos, D., Cotsidas, T., A decision theory approach to the approximation of discrete probability densities, IEEE Trans. Perform. Anal. Machine Intell. 1 (1980), 61-67.

Kikianty, E., Dragomir, S. S., Hermite-Hadamard's inequality and the p-HH-norm on the Cartesian product of two copies of a normed space, Math. Inequal. Appl. (in press).

Kirmaci, U. S., Klaricic Bakula, M., E Ozdemir, M., Pecaric, J., Hadamard-type inequalities for s-convex functions, Appl. Math. Comput. 193, No. 1 (2007), 26-35.

Kullback, S., Leibler, R. A., On information and sufficiency, Annals Math. Statist. 22 (1951), 79-86.

Lin, J., Divergence measures based on the Shannon entropy, IEEE Trans. Inform. Theory 37 (1) (1991), 145-151.

Latif, M. A., On some inequalities for h-convex functions, Int. J. Math. Anal. (Ruse) 4, No. 29--32 (2010), 1473-1482.

Mei, M., The theory of genetic distance and evaluation of human races, Japan J. Human Genetics 23 (1978), 341-369.

Mitrinovic, D. S., Lackovic, I. B., Hermite and convexity, Aequationes Math. 28 (1985), 229-232.

Mitrinovic, D. S., Pecaric, J. E., Note on a class of functions of Godunova and Levin, C. R. Math. Rep. Acad. Sci. Canada 12, No. 1 (1990), 33-36.

Ostrowski, A., Uber die Absolutabweichung einer differentienbaren Funktionen von ihren Integralmittelwert, Comment. Math. Helv. 10 (1938), 226-227.

Pearce, C. E. M., Rubinov, A. M., P-functions, quasi-convex functions, and Hadamard-type inequalities, J. Math. Anal. Appl. 240, No. 1 (1999), 92-104.

Pecaric, J. E., Dragomir, S. S., On an inequality of Godunova-Levin and some refinements of Jensen integral inequality, "Babes-Bolyai" University, Research Seminars, Preprint No. 6, Cluj-Napoca, 1989.

Pecaric, J., Dragomir, S. S., A generalization of Hadamard's inequality for isotonic linear functionals, Radovi Mat. (Sarajevo) 7 (1991), 103-107.

Pielou, E. C., Ecological Diversity, Wiley, New York, 1975.

Radulescu, M., Radulescu, S., Alexandrescu, P., On the Godunova-Levin-Schur class of functions, Math. Inequal. Appl. 12, No. 4 (2009), 853-862.

Rao, C. R., Diversity and dissimilarity coefficients: a unified approach, Theoretic Population Biology 21 (1982), 24-43.

Renyi, A., On measures of entropy and information, in Proc. Fourth Berkeley Symp. Math. Stat. and Prob., Vol. 1, University of California Press, 1961, 547-561.

Roberts, A. W., Varberg, D. E., Convex Functions, Academic Press, New York, 1973.

Sarikaya, M. Z., Saglam, A., Yildirim, H., On some Hadamard-type inequalities for h-convex functions, J. Math. Inequal. 2, No. 3 (2008), 335-341.

Sarikaya, M. Z., Set, E., Ozdemir, M. E., On some new inequalities of Hadamard type involving h-convex functions, Acta Math. Univ. Comenian. (N.S.) 79, No. 2 (2010), 265-272.

Set, E., Ozdemir, M. E., Sarikaya, M. Z., New inequalities of Ostrowski's type for s-convex functions in the second sense with applications, Facta Univ. Ser. Math. Inform. 27, No. 1 (2012), 67-82.

Sharma, B. D., Mittal, D. P., New non-additive measures of relative information, J. Comb. Inf. Syst. Sci. 2 (4) (1977), 122-132.

Sen, A., On Economic Inequality, Oxford University Press, London, 1973.

Shioya, H., Da-Te, T., A generalisation of Lin divergence and the derivative of a new information divergence, Electronics and Communications in Japan 78 (7) (1995), 37-40.

Taneja, I. J., Generalised Information Measures and Their Applications [http://www.mtm.ufsc.br/symbol{126}taneja/bhtml/bhtml.html].

Theil, H., Economics and Information Theory, North-Holland, Amsterdam, 1967.

Theil, H., Statistical Decomposition Analysis, North-Holland, Amsterdam, 1972.

Topsoe, F., Some inequalities for information divergence and related measures of discrimination, Preprint RGMIA Res. Rep. Coll. 2 (1) (1999), 85-98.

Tunc, M., Ostrowski-type inequalities via h-convex functions with applications to special means, J. Inequal. Appl. 2013, 2013:326.

Vajda, I., Theory of Statistical Inference and Information, Kluwer Academic Publishers, Dordrecht-Boston, 1989.

Varosanec, S., On h-convexity, J. Math. Anal. Appl. 326, No. 1 (2007), 303-311.




DOI: http://dx.doi.org/10.17951/a.2015.69.2.17-45
Date of publication: 2015-12-30 22:51:59
Date of submission: 2015-12-29 22:54:26


Statistics


Total abstract view - 990
Downloads (from 2020-06-17) - PDF - 646

Indicators



Refbacks

  • There are currently no refbacks.


Copyright (c) 2015 Sever Dragomir