PL EN

Preferencje
Język
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo

## Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

2015 | 69 | 2 |
Tytuł artykułu

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

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
EN
Rocznik
Tom
Numer
Opis fizyczny
Daty
wydano
2015
online
2015-12-30
Twórcy
autor
Bibliografia
• 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.
Typ dokumentu
Bibliografia
Identyfikatory