46 S. S. DRAGOMIR Le. If f is ;convex nd n< then f is n;convex. Proof. If x y [ ]ndt[ ] then f (tx + n ( ; t) y) =f tx + ( ; t) n y n tf (x)+( ; t) f

Transcription

1 TAMKANG JOURNAL OF MATHEMATICS Volue 33, Nuer, Spring ON SOME NEW INEQUALITIES OF HERMITE-HADAMARD TYPE FOR { CONVEX FUNCTIONS S. S. DRAGOMIR Astrct. Soe new inequlities for ;convex functions re otined.. Introduction In [7], G.H. Toder dened the ;convexity, n interedite etween the usul convexity nd strshped property. In the rst prt of this section we shll present properties of ;convex functions in siilr nner to convex functions. The following concept hs een introduced in [7](see lso [34]). Denition. The function f :[ ]! R is sid to e -convex, where [ ] if for every x y [ ] nd t [ ] we hve: f (tx + ( ; t) y) tf (x)+ ( ; t) f (y) : (.) Denote y K () the set of the ;convex functions on [ ] for which f () : Rerk. For = we recpture the concept of convex functions dened on [ ] nd for =we get the concept of strshped functions on [ ] : We recll tht f :[ ]! R is strshped if The following les hold [7]. f (tx) tf (x) for ll t [ ] nd x [ ] : (.) Le. If f is in the clss K () then it is strshped. Proof. For ny x [ ]ndt [ ] we hve: f (tx) =f (tx + ( ; t) ) tf (x)+ ( ; t) f () tf (x) : Received Mrch 5, revised My 9,. Mthetics Suject Clssiction. Priry 6D5, 6D Secondry 6D99. Key words nd phrses. Herite-Hdrd Inequlity, ;Convex functions. 45

2 46 S. S. DRAGOMIR Le. If f is ;convex nd n< then f is n;convex. Proof. If x y [ ]ndt[ ] then f (tx + n ( ; t) y) =f tx + ( ; t) n y n tf (x)+( ; t) f tf (x)+ ( ; t) n f (y) = tf (x)+n ( ; t) f (y) y nd the le is proved. As in pper [48] due to V. G. Mihesn, for pping f K () consider the function f (x) ; f () p (x) := x ; dened for x [ ] nfg for xed [ ] nd x x x 3 f (x ) f (x ) f (x 3 ) r (x x x 3 ):= x x x 3 x x x 3 where x x x 3 [ ] (x ; x )(x 3 ; x ) > x 6= x 3 : The following theore holds [48]. Theore. The following ssertions re equivlent: : f K () : p is incresing on the intervls [ ) ( ] for ll [ ] 3 : r (x x x 3 ) : Proof. ) : Let x y [ ] : If <x<y then there exists t ( ) such tht We thus hve x = ty + ( ; t) : (.3) f (x) ; f () p (x) = x ; f (ty + ( ; t) ) ; f () = ty + ( ; t) ; tf (y)+( ; t) f () ; f () t (y ; ) f (y) ; f () = y ; = p (y) :

3 NEW INEQUALITIES OF HERMITE-HADAMARD TYPE 47 If y < x < there lso exists t ( ) for which (:3) holds. Then we hve: f (x) ; f () p (x) = x ; f () ; f (ty + ( ; t) ) = ; ty ; ( ; t) f () ; tf (y)+( ; t) f () t ( ; y) f (y) ; f () = y ; = p (y) : )3 : A siple clcultion shows tht r (x x x 3 )= p x (x 3 ) ; p x (x ) x 3 ; x : Since p x is incresing on the intervls [ x ) (x ] one otins r (x x x 3 ) : 3 ) : Let x x 3 [ ] nd let x = tx 3 + ( ; t) x t ( ) : Oviously x <x <x 3 or x 3 <x <x hence r (x x x 3 )= tf (x 3)+ ( ; t) f (x ) ; f (tx 3 + ( ; t) x ) t ( ; t)(x 3 ; x ) fro where we otin (.), i.e., f K () : The following corollry holds for strshped functions. Corollry.Let f :[ ]! R: The following stteents re equivlent (i) f is strshped (ii) The pping p (x) := f(x) x is incresing on ( ]: The following le is lso interesting in itself. Le 3. If f is dierentile on [ ], then f K () if nd only if: 8 >< f (x) f(x);f(y) x;y for x>y y ( ] >: f (x) f(x);f(y) x;y for x<y y ( ]: (.4) Proof. The pping p y is incresing on (y ]ip y (x) which isequivlent with the condition (.4). Corollry. If f is dierentile in [ ] then f is strshped if (x) f(x) ll x ( ] : x for

4 48 S. S. DRAGOMIR The following inequlities of Herite-Hdrd type for ;convex functions hold [34]. Theore. Let f : [ )! R e ;convex function with ( ] : If << nd f L [ ] then one hs the inequlity: Z ( f ()+f ; f (x) dx in ; f ()+f ; ) : (.5) Proof. Since f is ;convex, we hve which gives: f (tx + ( ; t) y) tf (x)+ ( ; t) f (y) for ll x y f (t +(; t) ) tf ()+ ( ; t) f nd f (t +(; t) ) tf ()+( ; t) f for ll t [ ] : Integrting on [ ] we otin f (t +(; t) ) dt f ()+f ; nd ; f ()+f f (t +(; t) ) dt : However, f (t +(; t) ) dt = f (t +(; t) ) dt = Z f (x) dx ; nd the inequlity (.5) is otined. Another result of this type which holds for dierentile functions is eodied in the following theore [34]. Theore 3. Let f : [ )! R e ;convex function with ( ] : If << nd f is dierentile on ( ) then one hs the inequlity: f () ; ; f () Z f (x) dx (.6) ; ( ; ) f () ; ( ; ) f () : ( ; )

5 NEW INEQUALITIES OF HERMITE-HADAMARD TYPE 49 Proof. Using Le 3, we hve for ll x y withx y tht (x ; y) f (x) f (x) ; f (y) : (.7) Choosing in the ove inequlity x = nd y then x y nd Integrting over y on [ ] we get thus proving the rst inequlity in (.6). Putting in (.7) y = we hve ( ; y) f () f () ; f (y) : Z ( ; ) f () ( ; ) f () ; f (y) dy (x ; ) f (x) f (x) ; f () x : Integrting over x on [ ] we otin the second inequlity in(.6). Rerk. The second inequlity fro (.6) is lso vlid for = : Tht is, if f :[ )! R is dierentile strshped function, then for ll << one hs: Z f () ; f () f (x) dx ; ( ; ) which lso holds fro Corollry.. The New Results We will now point out soe new results of the Herite-Hdrd type. Theore 4. Let f : [ )! R e ;convex function with ( ] nd <:If f L [ ] then one hs the inequlities f + Z ; + 4 f (x)+f ; x " f ()+f () + f ; dx (.) ; + f # : Proof. By the ;convexity of f we hve tht for ll x y [ ) : x + y f h y i f (x)+f

6 5 S. S. DRAGOMIR If we choose x = t +(; t) y =(; t) + t we deduce + f f (t +(; t) )+f for ll t [ ] : Integrting over t [ ] we get + f f (t +(; t) ) dt + f Tking into ccount tht ( ; t) + t ( ; t) + t dt : (.) nd f f (t +(; t) ) dt = Z f (x) dx ; t +(; t) dt = Z ; f (x) dx = Z x f dx ; we deduce fro (.) the rst prt of (.). By the ;convexity off welsohve f (t +(; t) )+f ( ; t) + t tf ()+( ; t) f + ( ; t) f + tf (.3) for ll t [ ] : Integrting the inequlity (.3) over t on [ ] we deduce ; Z f (x)+f ; " x dx By siilr rguent we cn stte: 8 ; Z f (x)+f ; x f f ()+f ()+ nd the proof is copleted. f ()+f ; + f ; + f ; # : (.4) dx (.5) + f + f + f Rerk 3. For = we cn drop the ssuption f L [ ] nd (.) exctly ecoes the Herite-Hdrd inequlity.

7 NEW INEQUALITIES OF HERMITE-HADAMARD TYPE 5 The following result lso holds. Theore 5. Let f : [ )! R e ;convex function with ( ] : If f L [ ] where < then one hs the inequlity: " Z f (x) dx + ; Z # f (x) dx ( ; ) + ; f ()+f () : (.6) nd Proof. By the ;convexity of f we cn write: f (t + ( ; t) ) tf ()+ ( ; t) f () for ll t [ ] nd s ove. If we dd the ove inequlities we get f (( ; t) + t) ( ; t) f ()+tf () f (t +(; t) ) tf ()+ ( ; t) f () f (( ; t) + t) ( ; t) f ()+tf () f (t + ( ; t) )+f (( ; t) + t) +f (t +(; t) )+f (( ; t) + t) f ()+f ()+ (f ()+f ())=( +)(f ()+f ()) : Integrting over t [ ] we otin + As it is esy to see tht f (t + ( ; t) ) dt + f (t + ( ; t) ) dt + ( +)(f ()+f ()) : f (( ; t) + t) dt (.7) f (( ; t) + t) dt nd f (t + ( ; t) ) dt = f (( ; t) + t) dt = ; Z f (x) dx f (t + ( ; t) ) dt = f (( ; t) + t) dt = ; fro (.7) we deduce the desired result, nely, the inequlity (.6). Z f (x) dx

8 5 S. S. DRAGOMIR Rerk 4. For n extensive literture on Herite-Hdrd type inequlities, see the references enclosed. Acknowledgeents The uthor would like to thnk the nonyous referee for soe vlule suggestions on iproving the pper. References [] G. Allsi, C. Giordno, J. Pecric, Hdrd-type inequlities for (r)-convex functions with pplictions, Atti Acd. Sci. Torino-Cl. Sc. Fis., 33 (999), -4. [] H. Alzer, A note on Hdrd's inequlities, C.R. Mth. Rep. Acd. Sci. Cnd, (989), [3] H. Alzer, On n integrl inequlity, Mth. Rev. Anl. Nuer. Theor. Approx., 8(989), -3. [4] A. G. Azpeiti, Convex functions nd the Hdrd inequlity, Rev.-Coloin-Mt., 8(994), 7-. [5] D. Bru, S. S. Drgoir nd C. Buse, A proilistic rguent for the convergence of soe sequences ssocited to Hdrd's inequlity, Studi Univ. Bes-Bolyi, Mth., 38 (993), [6] E. F. Beckench, Convex functions, Bull. Aer. Mth. Soc., 54(948), [7] C. Borell, Integrl inequlities for generlised concve nd convex functions, J. Mth. Anl. Appl., 43(973), [8] C. Buse, S. S. Drgoir nd D. Bru, The convergence of soe sequences connected to Hdrd's inequlity, Deostrtio Mth., 9 (996), [9] L. J. Dedic, C. E. M. Perce nd J. Pecric, The Euler forule nd convex functions, Mth. Ineq. & Appl., (), -. [] L. J. Dedic, C. E. M. Perce nd J. Pecric, Hdrd nd Drgoir-Argrwl inequlities, high-order convexity nd the Euler Forul, suitted. [] S. S. Drgoir, A pping in connection to Hdrd's inequlities, An. Oster. Akd. Wiss. Mth.-Ntur., (Wien), 8(99), 7-. MR 934:63. ZBL No. 747:65. [] S. S. Drgoir, A reneent of Hdrd's inequlity for isotonic liner functionls, Tkng J. of Mth. (Tiwn), 4(993), -6. MR 94: 643. BL No. 799: 66. [3] S. S. Drgoir, On Hdrd's inequlities for convex functions, Mt. Blknic, 6(99), 5-. MR: 934: 633. [4] S. S. Drgoir, On Hdrd's inequlity for the convex ppings dened on ll in the spce nd pplictions, Mth. Ineq. & Appl., 3 (), [5] S. S. Drgoir, On Hdrd's inequlity on disk, Journl of Ineq. in Pure & Appl. Mth., (), No., Article, [6] S. S. Drgoir, On soe integrl inequlities for convex functions, Z.-Rd. (Krgujevc),(996), No. 8, -5. [7] S. S. Drgoir, Soe integrl inequlities for dierentile convex functions, Contriutions, Mcedonin Acd. of Sci. nd Arts, 3(99), 3-7.

9 NEW INEQUALITIES OF HERMITE-HADAMARD TYPE 53 [8] S. S. Drgoir, Soe rerks on Hdrd's inequlities for convex functions, Extrct Mth., 9 (994), [9] S. S. Drgoir, Two ppings in connection to Hdrd's inequlities, J. Mth. Anl. Appl., 67(99), MR:934:638, ZBL No. 758:64. [] S. S. Drgoir nd R. P. Agrwl, Two inequlities for dierentile ppings nd pplictions to specil ens ofrel nuers nd to trpezoidl forul, Appl. Mt. Lett. (998), [] S. S. Drgoir nd R. P. Agrwl, Two new ppings ssocited with Hdrd's inequlities for convex functions, Appl. Mth. Lett., (998), [] S. S. Drgoir nd C. Buse, Reneents of Hdrd's inequlity for ultiple integrls, Utilits Mth (Cnd), 47(995), [3] S. S. Drgoir, Y. J. Cho nd S. S. Ki, Inequlities of Hdrd's type for Lipschitzin ppings nd their pplictions, J. of Mth. Anl. Appl., 45 (), [4] S. S. Drgoir nd S. Fitzptrick, The Hdrd's inequlity for s;convex functions in the rst sense, Deonstrtio Mth., 3 (998), [5] S. S. Drgoir nd S. Fitzptrick, The Hdrd's inequlity for s;convex functions in the second sense, Deonstrtio Mth., 3 (999), [6] S. S. Drgoir nd N. M. Ionescu, On soe inequlities for convex-dointed functions, Anl. Nu. Theor. Approx., 9 (99), -8. MR 936: 64 ZBL No. 733 : 6. [7] S. S. Drgoir nd N. M. Ionescu, Soe integrl inequlities for dierentile convex functions, Coll. Pp. of the Fc. of Sci. Krgujevc (Yugoslvi), 3(99), -6, ZBL No. 77. [8] S. S. Drgoir, D. S. Milosevic nd J. Sndor, On soe reneents of Hdrd's inequlities nd pplictions, Univ. Belgrd, Pul. Elek. Fk. Sci. Mth., 4(993), -4. [9] S. S. Drgoir nd B. Mond, On Hdrd's inequlity for clss of functions of Godunov nd Levin, Indin J. Mth., 39 (997), -9. [3] S. S. Drgoir nd C. E. M. Perce, Qusi-convex functions nd Hdrd's inequlity, Bull. Austrl. Mth. Soc., 57 (998), [3] S. S. Drgoir, C. E. M. Perce nd J. E. Pecric, On Jessen's nd relted inequlities for isotonic suliner functionls, Act Mth. Sci. (Szeged), 6(995), [3] S. S. Drgoir, J. E. Pecric nd L. E. Persson, Soe inequlities of Hdrd type, Soochow J. of Mth. (Tiwn), (995), [33] S. S. Drgoir, J. E. Pecric nd J. Sndor, A note on the Jensen-Hdrd inequlity, Anl. Nu. Theor. Approx., 9(99), -8. MR 93 : 6 4.ZBL No. 733 : 6. [34] S. S. Drgoir nd G. H. Toder, Soe inequlities for ;convex functions, Studi Univ. Bes-Bolyi, Mth., 38(993), -8. [35] A. M. Fink, Aest possile Hdrd inequlity, Mth. Ineq. & Appl., (998), 3-3. [36] A. M. Fink, Hdrd inequlities for logrithic concve functions, Mth. Coput. Modeling, to pper. [37] A. M. Fink, Towrd theory of est possile inequlities, Nieuw Archief von Wiskunde, (994), 9-9. [38] A. M. Fink, Two inequlities, Univ. Beogrd Pul. Elek. Fk. Ser. Mt., 6 (995), [39] B. Gvre, On Hdrd's inequlity for the convex ppings dened on convex doin in the spce, Journl of Ineq. in Pure & Appl. Mth., (), No., Article 9, [4] P. M. Gill, C. E. M. Perce nd J. Pecric, Hdrd's inequlity for r;convex functions, J. of Mth. Anl. nd Appl., 5(997),

10 54 S. S. DRAGOMIR [4] G. H. Hrdy, J. E. Littlewood nd G. Poly, Inequlities, nd Ed., Cridge University Press, 95. [4] K.-C. Lee nd K.-L. Tseng, On weighted generlistion of Hdrd's inequlity for G-convex functions, Tsui Oxford Journl of Mth. Sci., 6(), 9-4. [43] A. Lups, The Jensen-Hdrd inequlity for convex functions of higher order, Octogon Mth. Mg., 5 (997), no., 8-9. [44] A. Lups, A generlistion of Hdrd inequlities for convex functions, ONLINE: ( [45] A. Lups, The Jensen-Hdrd inequlity for convex functions of higher order, ONLINE: ( [46] A. Lups, A generlistion of Hdrd's inequlity for convex functions, Univ. Beogrd. Pul. Elektrotehn. Fk. Ser. Mt. Fiz., No ,(976), 5-. [47] D. M. Mkisiovic, A short proof of generlized Hdrd's inequlities, Univ. Beogrd. Pul. Elektrotehn. Fk. Ser. Mt. Fiz., (979), No {8. [48] V. G Mihesn, A generlistion of the convexity, Seinr on Functionl Equtions, Approx. nd Convex., Cluj-Npoc, Roni, 993. [49] D. S. Mitrinovic nd I. Lckovic, Herite nd convexity, Aequt. Mth., 8 (985), 9{ 3. [5] D. S. Mitrinovic, J. E. Pecric nd A.M. Fink,Clssicl nd New Inequlities in Anlysis, Kluwer Acdeic Pulishers, Dordrecht/Boston/London. [5] B. Mond nd J. E. Pecric, A copnion to Fink's inequlity, Octgon Mth. Mg., to pper. [5] E. Neun, Inequlities involving generlised syetric ens, J. Mth. Anl. Appl., (986), [53] E. Neun nd J. E. Pecric, Inequlities involving ultivrite convex functions, J. Mth. Anl. Appl., 37 (989), [54] E. Neun, Inequlities involving ultivrite convex functions II, Proc. Aer. Mth. Soc., 9(99), [55] C. P. Niculescu, A note on the dul Herite-Hdrd inequlity, The Mth. Gzette, July. [56] C. P. Niculescu, Convexity ccording to the geoetric en, Mth. Ineq. & Appl., 3 (), [57] C. E. M. Perce, J. Pecric nd V. siic, Stolrsky ens nd Hdrd's inequlity, J. Mth. Anl. Appl., (998), [58] C. E. M. Perce nd A. M. Ruinov, P;functions, qusi-convex functions nd Hdrdtype inequlities, J. Mth. Anl. Appl., 4(999), 9-4. [59] J. E. Pecric, Rerks on two interpoltions of Hdrd's inequlities, Contriutions, Mcedonin Acd. of Sci. nd Arts, Sect. of Mth. nd Technicl Sciences, (Scopje), 3, (99), 9-. [6] J. Pecric ndv. Culjk, On Hdrd inequlities for logrithic convex functions, suitted. [6] J. Pecric, V. Culjk nd A. M. Fink, On soe inequlities for convex functions of higher order, suitted. [6] J. Pecric nd S. S. Drgoir, A generliztion of Hdrd's integrl inequlity for isotonic liner functionls, Rudovi Mt. (Srjevo), 7(99), 3-7. MR 94: 66. BL [63] J. Pecric, F. Proschn nd Y. L. Tong, Convex Functions, Prtil Orderings nd Sttisticl Applictions, Acdeic Press, Inc., 99.

11 NEW INEQUALITIES OF HERMITE-HADAMARD TYPE 55 [64] F. Qi nd Q.-M. Luo, Reneents nd extensions of n inequlity, Mthetics nd Infortics Qurterly, 9() (999), 3-5. [65] F. Qi, S.-L. Xu, nd L. Denth, A new proof of onotonicity for extended en vlues, Intern. J. Mth. Mth. Sci., (999), 45{4. [66] A. W. Roerts nd P. E. Vrerg, Convex Functions, Acdeic Press, 973. [67] F. Sidi nd R. Younis, Hdrd nd Fejer-type Inequlities, Archiv der Mthetik., to pper. [68] J. Sndor, A note on the Jensen-Hdrd inequlity, Anl. Nuer. Theor. Approx., 9 (99), [69] J. Sndor, An ppliction of the Jensen-Hdrd inequlity, Nieuw-Arch.-Wisk., 8 (99), [7] J. Sndor, On the Jensen-Hdrd inequlity, Studi Univ. Bes-Bolyi, Mth., 36 (99), 9-5. [7] G. H. Toder, Soe generlistions of the convexity, Proc. Colloq. Approx. Opti, Cluj- Npoc (Roni), 984, [7] P. M. Vsic, I. B. Lckovic nd D. M. Mksiovic, Note on convex functions IV: On- Hdrd's inequlity for weighted rithetic ens, Univ. Beogrd Pul. Elek. Fk., Ser. Mt. Fiz., No (98), [73] G. S Yng nd M. C. Hong, A note on Hdrd's inequlity, Tkng J. Mth., 8 (997), [74] G. S Yng nd K. L. Tseng, On certin integrl inequlities relted to Herite-Hdrd inequlities, J. Mth. Anl. Appl., 39(999), School of Counictions nd Infortics, Victori University of Technology, PO Box 448, Melourne City MC, 8, Victori, Austrli. E-il ddress: sever@tild.vu.edu.u URL: