Companions of Hermite-Hadamard Inequality for Convex Functions (II)

S. S. Dragomir

I. Gomm

Victoria University

Australia

ABSTRACT

Companions of Hermite-Hadamard inequalities for convex functions defined on the positive axis in the case when the integral has either the weight ψ or ¹ ,t > 0 are given. Applications for special means are provided as well.

Subjclass : 26D15; 25D10.

Keywords : Convex functions, Hermite-Hadamard inequality, Special means

REFERENCES

[1] G. ALLASIA, C. GIORDANO, J. PECARIC, Hadamard-type ln-equalltles for (2r)-convex functlons wlth appllcatlons, Atti Acad. Sci. Torino-Cl. Sc. Fis., 133, pp. 1-14, (1999).

[2] H. ALZER, A note on Hadamard's lnequalltles, C.R. Math.Rep.Acad. Sci. Canada, 11, pp. 255-258, (1989).

[3] H. ALZER, On an lntegral lnequallty, Math.Rev. Anal.Numer.Theor. Approx., 18, pp. 101-103, (1989).

[4] A. G. AZPEITIA, Convex functlons and the Hadamard lnequallty, Rev.-Colombiana-Mat., 28 (1), pp. 7-12, (1994).

[5] D.BARBU,S. S. DRAGOMIR and C. BU§E, A probablllstlc argument for the convergence of some sequences assoclated to Hadamard's lnequallty, Studia Univ. Babe§-Bolyai, Math., 38 (1), pp. 29-33, (1993).

[6] C. BU§E,S.S.DRAGOMIRand D. BARBU, Theconvergence of some sequences connected to Hadamard's lnequallty, Demostratio Math.,29 (1), pp. 53-59, (1996).

[7] S. S. DRAGOMIR, A mapplng ln connectlon to Hadamard's lnequal-ltles, An. Oster. Akad. Wiss. Math.-Natur., (Wlen), 128, pp. 17-20. MR 934:26032. ZBL No. 747:26015, (1991).

[8] S. S. DRAGOMIR, A refinement of Hadamard's lnequallty for lsotonlc llnear functlonals, Tamkang J. of Math. (Talwan), 24 , pp. 101-106. MR 94a: 26043. 2BL No. 799: 26016, (1993).

[9] S. S. DRAGOMIR, On Hadamard's lnequalltles for convex functlons, Mat. Balkanica, 6, pp. 215-222. MR: 934, (1992). 26033.

[10] S. S. DRAGOMIR, On Hadamard's lnequallty for the convex mapplngs defined on a ball ln the space and appllcatlons, Math. Ineq. & Appl., 3 (2), pp. 177-187, (2000).

[11] S. S. DRAGOMIR, On Hadamard's lnequallty on a dlsk, Journal of Ineq. in Pure & Appl. Math., 1, No. 1, Artlcle 2, http://jlpam.vu.edu.au/, (2000).

[12] S. S. DRAGOMIR, Some lntegral lnequalltles for dlfferentlable convex functlons, Contributions, Macedonian Acad. of Sci. and Arts,13(1), pp. 13-17, (1992).

[13] S. S. DRAGOMIR, Some remarks on Hadamard's lnequalltles for convex functlons, Extracta Math., 9 (2), pp. 88-94, (1994).

[14] S.S. DRAGOMIR, Two mapplngs ln connectlon to Hadamard's lnequalltles, J. Math. Anal. Appl., 167, pp. 49-56. MR:934:26038, ZBL No. 758:26014, (1992).

[15] S. S. DRAGOMIR, , An lnequallty lmprovlng the first Hermlte-Hadamard lnequallty for convex functlons defined on llnear spaces and appllcatlons for seml-lnner products, J. Inequal. Pure Appl. Math. 3, No. 2, Artlcle 31, (2002).

[16] S. S. DRAGOMIR, An lnequallty lmprovlng the second Hermlte-Hadamard lnequallty for convex functlons defined on llnear spaces and appllcatlons for seml-lnner products, J. Inequal. Pure Appl. Math. 3, No. 3, Artlcle 35, (2002).

[17] S.S . DRAGOMIR and R. P. AGARWAL, Two new mapplngs assocl-ated wlth Hadamard's lnequalltles for convex functlons, Appl. Math. Lett., 11, No. 3, pp. 33-38, (1998).

[18] S. S. DRAGOMIR and C. BU§E, Refinements of Hadamard's lnequal-lty for multlple lntegrals, Utilitas Math (Canada), 47, pp. 193-195, (1995).

[19&093; S. S. DRAGOMIR, Y.J.CHO and S.S. KIM, Inequalltlesof Hadamard's type for Llpschltzlan mapplngs and thelr appllcatlons, J. ofMath. Anal. Appl., 245 (2), pp. 489-501, (2000).

[20] S. S. DRAGOMIR and I. GOMM, Bounds for two mapplngs assoclated to the Hermlte-Hadamard lnequallty, Aust. J. Math. Anal. Appl.,8, Art. 5, 9 pages, (2011).

[21] S. S. DRAGOMIR and I. GOMM, Some new bounds for two map-plngs related to the Hermlte-Hadamard lnequallty for convex func-tlons, Num. Alg. Cont. & Opt. 2, No. 2, pp. 271-278, (2012).

[22] S. S. DRAGOMIR and I. GOMM, Companlons of Hermlte-Hadamard Inequallty for Convex Functlons (I), Preprlnt RGMIA Res. Rep. Coll., 17 (2014), Art.

[23] S. S. DRAGOMIR and S. FITZPATRICK, The Hadamard's lnequallty for s-convex functlons ln the first sense, Demonstratio Math., 31 (3), pp. 633-642, (1998).

[24] S. S. DRAGOMIR and S. FITZPATRICK, The Hadamard's lnequallty for s-convex functlons ln the second sense, Demonstratio Math., 32 (4), pp. 687-696, (1999).

[25] S. S. DRAGOMIR and N. M. IONESCU, On some lnequalltles for convex-domlnated functlons, Anal. Num. Theor. Approx., 19, pp. 2128. MR 936: 26014 ZBL No. 733 : 26010, (1990).

[26] S.S. DRAGOMIR, D. S. MILOSEVIC and J. SANDOR, On some refinements of Hadamard's lnequalltles and appllcatlons, Univ. Belgrad, Publ. Elek. Fak. Sci. Math., 4, pp. 21-24, (1993).

[27] S.S. DRAGOMIR and B. MOND, On Hadamard's lnequallty for a class of functlons of Godunova and Levln, Indian J. Math., 39, No. 1, pp. 1-9, (1997).

[28] S. S. DRAGOMIR and C. Ε. M. PEARCE, Quasl-convex functlons and Hadamard's lnequallty, Bull. Austral. Math. Soc., 57 , pp. 377385, (1998).

[29] S.S. DRAGOMIR and C. E.M.PEARCE, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, (2000). [;Onllne http://rgmla.org/monographs/hermlteJ1adamard.html] .

[30] S. S. DRAGOMIR,C.E. M.PEARCEand J. Ε. PECARIC, On Jessen's and related lnequalltles for lsotonlc subllnear functlonals, Acta Math. Sci. (Szeged), 61, pp. 373-382, (1995).

[31] S. S. DRAGOMIR, J. Ε. PECARIC and L. Ε. PERSSON, Some ln-equalltles of Hadamard type, SoochowJ.ofMath. (Talwan), 21, pp. 335-341, (1995).

[32] S. S. DRAGOMIR,J.E.PECAriC and J. SANDOR, A note on the Jensen-Hadamard lnequallty, Anal. Num. Theor. Approx., 19, pp. 2128. MR 93b : 260 14.ZBL No. 733 : 26010, (1990).

[33] S. S. DRAGOMIR and G. H. TOADER, Some lnequalltles for m-convex functlons, Studia Univ. Babes-Bolyai, Math., 38 (1), pp. 21-28, (1993).

[34] A. M. FINK, A best posslble Hadamard lnequallty, Math. Ineq. & Appl., 2 , pp. 223-230, (1998).

[35] A. M. FINK, Toward a theory of best posslble lnequalltles, Nieuw Archief von Wiskunde, 12 , pp. 19-29, (1994).

[36] A. M. FINK, Two lnequalltles, Univ.BeogradPubl.Elektrotehn.Fak. Ser. Mat., 6, pp. 48-49, (1995).

[37] B. GAVREA, On Hadamard's lnequallty for the convex mapplngs defined on a convex domaln ln the space, Journal of Ineq. in Pure & Appl. Math., 1 (2000), No. 1, Artlcle 9, http://jlpam.vu.edu.au/

[38] P. M. GILL, C.E.M.PEARCEand J. PECARlC, Hadamard's ln-equallty for r-convex functlons, J. of Math. Anal. and Appl., 215 , pp. 461-470, (1997).

[39] G. H. HARDY, J. Ε. LITTLEWOOD and G. pOlYA, Inequalities, 2nd Ed., Cambrldge Unlverslty Press, (1952).

[40] K.-C. LEE and K.-L. TSENG, On a welghted generallsatlon of Hadamard's lnequallty for G-convex functlons, Tamsui Oxford Journal of Math. Sci., 16 (1), pp. 91-104, (2000).

[41] A. LUPA§, The Jensen-Hadamard lnequallty for convex functlons of hlgher order, Octogon Math. Mag., 5, No. 2, pp. 8-9, (1997).

[42] A. LUPA§, A generallsatlon of Hadamard's lnequallty for convex func-tlons, Univ. Beograd. Publ. Elek. Fak. Ser. Mat. Fiz., No. 544-576, pp. 115-121, (1976).

[43] D. M. MAKI IMOVIC^S, A short proof of generallzed Hadamard's lnequalltles, Univ.Beograd. Publ.Elektrotehn.Fak.Ser.Mat.Fiz.,No. 634-677, pp. 126-128, (1979). [;44] D. . MITRINOVIC^S andI. LACKOVIC^S , Hermlte and convexlty, Aequat. Math., 28, pp. 229-232, (1985).

[45] D. S. MITRINOV1C, J. Ε. PECARIC and A. M. FINK, Classical and New Inequalities in Analysis, Kluwer Academlc Publlshers, Dordrecht/Boston/London.

[46] Ε. NEUMAN, lnequalltles lnvolvlng generallsed symmetrlc means, J. Math. Anal. Appl., 120, pp. 315-320, (1986).

[47] Ε. NEUMAN and J. Ε. PECARlC, lnequalltles lnvolvlng multlvarlate convex functlons, J. Math. Anal. Appl., 137, pp. 514-549, (1989).

[48] Ε. NEUMAN, lnequalltles lnvolvlng multlvarlate convex functlons II, Proc.Amer.Math.Soc., 109, pp. 965-974, (1990).

[49] C. P. NICULESCU, A note on the dual Hermlte-Hadamard lnequallty, The Math. Gazette, July (2000).

[50] C. P. NICULESCU, Convexlty accordlng to the geometrlc mean, Math. Ineq. & Appl., 3 (2), pp. 155-167, (2000).

[51] C. Ε. M. PEARCE,J.PECARlC and V. SIM1C, Stolarsky means and Hadamard's lnequallty, J. Math. Anal. Appl., 220, pp. 99-109, (1998).

[52] C. Ε. M. PEARCE and A. M. RUBINOV, P-functlons, quasl-convex functlons and Hadamard-type lnequalltles, J. Math. Anal. Appl., 240, (1), pp. 92-104, (1999).

[53] J. Ε. ΡΕ(]ARlC, Remarks on two lnterpolatlons of Hadamard's ln-equalltles, Contributions, Macedonian Acad. ofSci. and Arts, Sect. of Math. and Technical Sciences, (Scopje), 13, pp. 9-12, (1992).

[54] J. PEC ARlC and S. S. DRAGOMIR, A generallzatlon of Hadamard's lntegral lnequallty for lsotonlc llnear functlonals, Rudovi Mat. (Sarajevo), 7 (1991), 103-107. MR 924: 26026. 2BL No. 738: 26006.

[55] J. PECARIC, F. PROSCHAN and Y. L. TONG, Convex Functions, Partial Orderings and Statistical Applications, Academlc Press, Inc., (1992).

[56] J. SA^S NDOR, A note on the Jensen-Hadamard lnequallty, Anal. Nu-mer. Theor. Approx., 19, No. 1, pp. 29-34, (1990).

[57] J. SA^S NDOR, An appllcatlon of the Jensen-Hadamard lnequallty, Nieuw-Arch.-Wisk., 8, No. 1, pp. 63-66, (1990).

[58] J. SA^S NDOR, On the Jensen-Hadamard lnequallty, Studia Univ. Babes-Bolyai, Math., 36, No. 1, pp. 9-15, (1991).

[59] Ρ. M. VASlC, I. B. LACKOV1C and D. M. MAKSIMOV1C, Note on convex functlons IV: OnHadamard's lnequallty for welghted arlthmetlc means, Univ. Beograd Publ. Elek. Fak., Ser. Mat. Fiz., No. 678-715, pp. 199-205, (1980).

[60] G. S. YANG and M. C. HONG, A note on Hadamard's lnequallty, Tamkang J. Math., 28 (1), pp. 33-37, (1997).

[61] G. S. YANG and K. L. TSENG, On certaln lntegral lnequalltles related to Hermlte-Hadamard lnequalltles, J. Math. Anal. Appl., 239, pp.180-187, (1999).

 

S. S. Dragomir

Mathematlcs

College of Englneerlng & Sclence Vlctorla Unlverslty Ρ. O. Box 14428 Melbourne Clty

MC 8001

Australla

e-mall : sever.dragomlr@vu.edu.au

Companions of Hermite-Hadamard Inequality (II) 367

and

School of Computatlonal & Applled Mathematlcs Unlverslty of the Wltwatersrand Prlvate Bag 3, Johannesburg 2050 South Afrlca

and

I. Gomm

Mathematlcs

College of Englneerlng & Sclence Vlctorla Unlverslty Ρ. O. Box 14428 Melbourne Clty

MC 8001

Australla

e-mall : Ian.Gomm@vu.edu.au

Received : March 2014. Accepted : July 2014