×

A sequence of mappings associated with the Hermite-Hadamard inequalities and applications. (English) Zbl 1099.26016

The point of departure of the paper is the Hermite-Hadamard inequality \[ f\left (\frac {a+b}2\right ) \leq \frac 1{b-a}\int _a^ bf(t)\,\text dt \leq \frac {f(a)+f(b)}2, \] which holds for convex functions. The author studies sequences of functions defined by multiple integrals \[ H_n(t):= \frac 1{(b-a)^ n} \int _a^ b\cdots \int _a^ b f\left (t\frac {x_1+\dots +x_n}n +(1-t)\frac {a+b}2\right ) dx_1\dots dx_{n} \] and proves plenty of results about them. First, basic properties of \(H_n\) are established such as convexity, monotonicity (both in \(n\) and in \(t\)) and various lower and upper bounds. The differences \(H_n(t)-f\left (\frac {a+b}2\right )\) and \(tH_n(1)+(1-t)H_n(0)-H_n(t)\) are also studied and several bounds for them are given.

MSC:

26D15 Inequalities for sums, series and integrals
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] G. Allasia, C. Giordano, J. Pečarić: Hadamard-type inequalities for \((2r)\)-convex functions with applications. Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 133 (1999), 187-200. · Zbl 1035.26016
[2] H. Alzer: A note on Hadamard’s inequalities. C. R. Math. Rep. Acad. Sci. Canada 11 (1989), 255-258. · Zbl 0707.26012
[3] H. Alzer: On an integral inequality. Anal. Numér. Théor. Approx. 18 (1989), 101-103. · Zbl 0721.26011
[4] A. G. Azpeitia: Convex functions and the Hadamard inequality. Rev. Colombiana Mat. 28 (1994), 7-12. · Zbl 0832.26015
[5] D. Barbu, S. S. Dragomir and C. Buşe: A probabilistic argument for the convergence of some sequences associated to Hadamard’s inequality. Studia Univ. Babeş-Bolyai Math. 38 (1993), 29-33. · Zbl 0829.26009
[6] C. Buşe, S. S. Dragomir and D. Barbu: The convergence of some sequences connected to Hadamard’s inequality. Demostratio Math. 29 (1996), 53-59. · Zbl 0860.26010
[7] S. S. Dragomir: A mapping in connection to Hadamard’s inequalities. Anz. Österreich. Akad. Wiss. Math.-Natur. Kl. 128 (1991), 17-20. · Zbl 0747.26015
[8] S. S. Dragomir: A refinement of Hadamard’s inequality for isotonic linear functionals. Tamkang J. Math. 24 (1993), 101-106. · Zbl 0799.26016
[9] S. S. Dragomir: On Hadamard’s inequalities for convex functions. Mat. Balkanica (N. S.) 6 (1992), 215-222. · Zbl 0834.26010
[10] S. S. Dragomir: On Hadamard’s inequality for the convex mappings defined on a ball in the space and applications. Math. Inequal. Appl. 3 (2000), 177-187. · Zbl 0951.26010
[11] S. S. Dragomir: On Hadamard’s inequality on a disk. JIPAM. J. Inequal. Pure Appl. Math. 1 (2000), , Electronic. · Zbl 0948.26012
[12] S. S. Dragomir: Some integral inequalities for differentiable convex functions. Makedon. Akad. Nauk Umet. Oddel. Mat.-Tehn. Nauk. Prilozi 13 (1992), 13-17. · Zbl 0770.26009
[13] S. S. Dragomir: Some remarks on Hadamard’s inequalities for convex functions. Extracta Math. 9 (1994), 88-94. · Zbl 0984.26012
[14] S. S. Dragomir: Two mappings in connection to Hadamard’s inequalities. J. Math. Anal. Appl. 167 (1992), 49-56. · Zbl 0758.26014
[15] S. S. Dragomir, R. P. Agarwal: Two new mappings associated with Hadamard’s inequalities for convex functions. Appl. Math. Lett. 11 (1998), 33-38. · Zbl 0979.26013
[16] S. S. Dragomir, C. Buşe: Refinements of Hadamard’s inequality for multiple integrals. Utilitas Math. 47 (1995), 193-198. · Zbl 0838.26013
[17] S. S. Dragomir, Y. J. Cho and S. S. Kim: Inequalities of Hadamard’s type for Lipschitzian mappings and their applications. J. Math. Anal. Appl. 245 (2000), 489-501. · Zbl 0956.26015
[18] S. S. Dragomir, S. Fitzpatrick: Hadamard inequality for \(s\)-convex functions in the first sense and applications. Demonstratio Math. 31 (1998), 633-642. · Zbl 0923.26014
[19] S. S. Dragomir, S. Fitzpatrick: The Hadamard’s inequality for \(s\)-convex functions in the second sense. Demonstratio Math. 32 (1999), 687-696. · Zbl 0952.26014
[20] S. S. Dragomir, N. M. Ionescu: On some inequalities for convex-dominated functions. Anal. Numér. Théor. Approx. 19 (1990), 21-27. · Zbl 0733.26010
[21] S. S. Dragomir, D. S. Milośević and J. Sándor: On some refinements of Hadamard’s inequalities and applications. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 4 (1993), 3-10. · Zbl 0813.26005
[22] S. S. Dragomir, B. Mond: On Hadamard’s inequality for a class of functions of Godunova and Levin. Indian J. Math. 39 (1997), 1-9. · Zbl 0916.26008
[23] S. S. Dragomir, C. E. M. Pearce: Quasi-convex functions and Hadamard’s inequality. Bull. Austral. Math. Soc. 57 (1998), 377-385. · Zbl 0908.26015
[24] S. S. Dragomir, C. E. M. Pearce, and J. E. Pečarić: On Jessen’s related inequalities for isotonic sublinear functionals. Acta Sci. Math. 61 (1995), 373-382. · Zbl 0846.39012
[25] S. S. Dragomir, J. E. Pečarić, and L. E. Persson: Some inequalities of Hadamard type. Soochow J. Math. 21 (1995), 335-341. · Zbl 0834.26009
[26] S. S. Dragomir, J. E. Pečarić, and J. Sándor: A note on the Jensen-Hadamard inequality. Anal. Numér. Théor. Approx. 19 (1990), 29-34. · Zbl 0743.26016
[27] S. S. Dragomir, G. H. Toader: Some inequalities for \(m\)-convex functions. Studia Univ. Babeş-Bolyai Math. 38 (1993), 21-28. · Zbl 0829.26010
[28] A. M. Fink: A best possible Hadamard inequality. Math. Inequal. Appl. 1 (1998), 223-230. · Zbl 0907.26009
[29] A. M. Fink: Toward a theory of best possible inequalities. Nieuw Arch. Wisk. 12 (1994), 19-29. · Zbl 0827.26018
[30] A. M. Fink: Two inequalities. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 6 (1995), 48-49. · Zbl 0841.26009
[31] B. Gavrea: On Hadamard’s inequality for the convex mappings defined on a convex domain in the space. JIPAM. J. Inequal. Pure Appl. Math. 1 (2000), Article 9, , Electronic. · Zbl 0949.26008
[32] P. M. Gill, C. E. M. Pearce and J. Pečarić: Hadamard’s inequality for \(r\)-convex functions. J. Math. Anal. Appl. 215 (1997), 461-470. · Zbl 0891.26013
[33] G. H. Hardy, J. E. Littlewood, and G. Pólya: Inequalities. 2nd ed. Cambridge University Press, 1952. · Zbl 0047.05302
[34] K.-C. Lee, K.-L. Tseng: On weighted generalization of Hadamard’s inequality for \(g\) functions. Tamsui Oxf. J. Math. Sci. 16 (2000), 91-104. · Zbl 0958.26012
[35] A. Lupaş: The Jensen-Hadamard inequality for convex functions of higher order. Octogon Math. Mag. 5 (1997), 8-9.
[36] A. Lupaş: A generalization of Hadamard’s inequality for convex functions. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 544-576 (1976), 115-121.
[37] D. M. Maksimović: A short proof of generalized Hadamard’s inequalities. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 634-677 (1979), 126-128.
[38] D. S. Mitrinović, I. Lacković: Hermite and convexity. Aequationes Math. 28 (1985), 229-232. · Zbl 0572.26004
[39] D. S. Mitrinović, J. E. Pečarić, and A. M. Fink: Classical and New Inequalities in Analysis. Kluwer Academic Publishers, Dordrecht, 1993. · Zbl 0771.26009
[40] E. Neuman: Inequalities involving generalized symmetric means. J. Math. Anal. Appl. 120 (1986), 315-320. · Zbl 0601.26013
[41] E. Neuman, J. E. Pečarić: Inequalities involving multivariate convex functions. J. Math. Anal. Appl. 137 (1989), 514-549. · Zbl 0672.26010
[42] E. Neuman: Inequalities involving multivariate convex functions. II. Proc. Amer. Math. Soc. 109 (1990), 965-974. · Zbl 0699.26009
[43] C. P. Niculescu: A note on the dual Hermite-Hadamard inequality. The Math. Gazette (July 2000), .
[44] C. P. Niculescu: Convexity according to the geometric mean. Math. Inequal. Appl. 3 (2000), 155-167. · Zbl 0952.26006
[45] C. E. M. Pearce, J. Pečarić, and V. Šimić: Stolarsky means and Hadamard’s inequality. J. Math. Anal. Appl. 220 (1998), 99-109. · Zbl 0909.26011
[46] C. E. M. Pearce, A. M. Rubinov: \(P\)-functions, quasi-convex functions and Hadamard-type inequalities. J. Math. Anal. Appl. 240 (1999), 92-104. · Zbl 0939.26009
[47] J. E. Pečarić: Remarks on two interpolations of Hadamard’s inequalities. Makedon. Akad. Nauk. Umet. Oddel. Mat.-Tehn. Nauk. Prilozi 13 (1992), 9-12.
[48] J. Pečarić, S. S. Dragomir: A generalization of Hadamard’s inequality for isotonic linear functionals. Rad. Mat. 7 (1991), 103-107. · Zbl 0738.26006
[49] J. Pečarić, F. Proschan, and Y. L. Tong: Convex Functions, Partial Orderings and Statistical Applications. Academic Press, Boston, 1992. · Zbl 0749.26004
[50] J. Sándor: An application of the Jensen-Hadamard inequality. Nieuw Arch. Wisk. 8 (1990), 63-66. · Zbl 0714.26008
[51] J. Sándor: On the Jensen-Hadamard inequality. Studia Univ. Babeş-Bolyai, Math. 36 (1991), 9-15. · Zbl 0900.26039
[52] P. M. Vasić, I. B. Lacković, and D. M. Maksimović: Note on convex functions. IV. On Hadamard’s inequality for weighted arithmetic means. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 678-715 (1980), 199-205.
[53] G. S Yang, M. C. Hong: A note on Hadamard’s inequality. Tamkang J. Math. 28 (1997), 33-37. · Zbl 0880.26019
[54] G. S. Yang, K. L. Tseng: On certain integral inequalities related to Hermite-Hadamard inequalities. J. Math. Anal. Appl. 239 (1999), 180-187. · Zbl 0939.26010
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.