Further inequalities for log-convex functions related to Hermite-Hadamard result.

Authors

Keywords:

Convex functions, Integral inequalities, Log-convex functions

Abstract

Some unweighted and weighted inequalities of Hermite-Hadamard type for log-convex functions defined on real intervals are given.

Author Biography

S. S. Dragomir, Victoria University.

College of Engineering and Science.

References

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

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

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

G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen, Generalized convexity and inequalities, J. Math. Anal. Appl. 335, pp. 1294—1308, (2007).

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

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

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

W. W. Breckner, Stetigkeitsaussagen für eine Klasse verallgemeinerter konvexer Funktionen in topologischen linearen Räumen. (German) Publ. Inst. Math. (Beograd) (N.S.) 23 (37), pp. 13—20, (1978).

W. W. Breckner and G.Orbán, Continuity properties of rationally s-convex mappings with values in an ordered topological linear space. Universitatea Babeş-Bolyai, Facultatea de Matematica, Cluj-Napoca, viii+92, (1978).

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

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

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

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

S. S. Dragomir, Some remarks on Hadamard’s inequalities for convex functions, Extracta Math., 9 (2), pp. 88-94, (1994).

S. S. Dragomir, Refinements of the Hermite-Hadamard integral inequality for log-convex functions, Austral. Math. Soc. Gaz. 28, No. 3, pp. 129—134, (2001).

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

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

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

S. S. Dragomir, On the Ostrowski’s inequality for mappings of bounded variation and applications, Math. Ineq. & Appl., 4 (1), pp. 33-40, (2001).

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

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

S. S. Dragomir, 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, Article 31, 8, (2002).

S. S. Dragomir, An inequality improving the first Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, J. Inequal. Pure and Appl. Math., 3 (2), Art. 31, (2002).

S. S. Dragomir, An inequality improving the second HermiteHadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, J. Inequal. Pure and Appl. Math., 3 (3), Art. 35, (2002).

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

S. S. Dragomir, Operator Inequalities of Ostrowski and Trapezoidal Type. Springer Briefs in Mathematics. Springer, New York, (2012). x+112 pp. ISBN: 978-1-4614-1778-1

S. S. Dragomir, Some new inequalities of Hermite-Hadamard type for GA-convex functions, Preprint RGMIA Res. Rep. Coll. 18 (2015), Art 33. [http://rgmia.org/papers/v18/v18a33.pdf].

S. S. Dragomir, New inequalities of Hermite-Hadamard type for logconvex functions, Preprint RGMIA Res. Rep. Coll. 18, (2015), Art 42. [http://rgmia.org/papers/v18/v18a42.pdf].

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

S. S. Dragomir and S. Fitzpatrick, The Hadamard inequalities for sconvex functions in the second sense. Demonstratio Math. 32, No. 4, pp. 687—696, (1999).

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

S. S. Dragomir and B. Mond, Integral inequalities of Hadamard’s type for log-convex functions, Demonstratio Math., 31 (2), pp. 354-364, (1998).

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

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

S. S. Dragomir and C. E. M. Pearce, Selected Topics on HermiteHadamard Inequalities and Applications, RGMIA Monographs, Victoria University, (2000).

S. S. Dragomir, J. Pečarić and L. Persson, Some inequalities of Hadamard type. Soochow J. Math. 21, No. 3, pp. 335—341, (1995).

S. S. Dragomir and Th. M. Rassias (Eds), Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer Academic Publisher, (2002).

S. S. Dragomir and S. Wang, A new inequality of Ostrowski’s type in L1−norm and applications to some special means and to some numerical quadrature rules, Tamkang J. of Math., 28, pp. 239-244, (1997).

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

S. S. Dragomir and S. Wang, 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. of Math., 40 (3), pp. 245-304, (1998).

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

L. Fejér, Über die Fourierreihen, II, Math. Naturwiss. Anz Ungar. Akad. Wiss. 24, pp. 369-390, (1906). (In Hungarian).

P. M. Gill, C. E. M. Pearce and J. Pečarić, Hadamard’s inequality for rconvex functions, Journal of Mathematical Analysis and Applications. 215, No. 2, pp. 461-470, (1997).

E. K. Godunova and V. I. Levin, Inequalities for functions of a broad class that contains convex, monotone and some other forms of functions. (Russian) Numerical mathematics and mathematical physics (Russian), 138—142, 166, Moskov. Gos. Ped. Inst., Moscow, (1985).

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

E. Kikianty and S. S. Dragomir, Hermite-Hadamard’s inequality and the p-HH-norm on the Cartesian product of two copies of a normed space, Math. Inequal. Appl. Mathematical Inequalities Applications, Volume 13, Number 1, pp. 1-32, (2010).

U. S. Kirmaci, M. Klaričić Bakula, M. E Özdemir and J. Pečarić, Hadamard-type inequalities for s-convex functions. Appl. Math. Comput. 193, No. 1, pp. 26—35, (2007).

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

D. S. Lacković and I. B. Lacković, Hermite and convexity, Aequationes Math. 28, pp. 229—232, (1985).

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

M. A. Noor, K. I. Noor and M. U. Awan, Some inequalities for geometrically-arithmetically h-convex functions, Creat. Math. Inform. 23, No. 1, 91-98, (2014).

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

J. E. Pečarić and S. S. Dragomir, On an inequality of Godunova-Levin and some refinements of Jensen integral inequality. Itinerant Seminar on Functional Equations, Approximation and Convexity (Cluj-Napoca, 263—268, Preprint, 89-6, Univ. Babeş-Bolyai, Cluj-Napoca, (1989).

J. Pečarić and S. S. Dragomir, A generalization of Hadamard’s inequality for isotonic linear functionals, Radovi Mat. (Sarajevo) 7, pp. 103—107, (1991).

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

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

E. Set, M. E. Özdemir and M. Z. Sarıkaya, New inequalities of Ostrowski’s type for s-convex functions in the second sense with applications. Facta Univ. Ser. Math. Inform. 27, No. 1, pp. 67—82, (2012).

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

W. T. Sulaiman, Refinements to Hadamard’s inequality for log-convex functions. Applied Mathematics, 2, pp. 899-903, (2011).

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

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

X.-M. Zhang, Y.-M. Chu and X.-H. Zhang, The Hermite-Hadamard type inequality of GA-convex functions and its application, Journal of Inequalities and Applications, Volume, Article ID 507560, 11 pages, (2010).

Published

2019-05-30

How to Cite

[1]
S. S. Dragomir, “Further inequalities for log-convex functions related to Hermite-Hadamard result.”, Proyecciones (Antofagasta, On line), vol. 38, no. 2, pp. 267-293, May 2019.

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