An inequality related to η-convex functions (II)

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1 Int. J. Nonliner Anl. Appl. 6 (15) No., 7-33 ISSN: 8-68 (electronic) An inequlity relted to η-conve functions (II) M. Eshghi Gordji, S. S. Drgomir b, M. Rostmin Delvr, Deprtment of Mthemtics, Semnn University, P.O.Bo , Semnn, Irn b School of Computer Science nd Mthemtics, Victori University, Melbourne, Victori, Austrli (Communicted by M.B. Ghemi) Abstrct Using the notion of η-conve functions s generliztion of conve functions, we estimte the difference between the middle nd right terms in Hermite-Hdmrd-Fejer inequlity for differentible mppings. Also s n ppliction we give n error estimte for midpoint formul. Keywords: η-conve function; Hermite-Hdmrd-Fejer inequlity. 1 MSC: Primry 6A51; Secondry 5D Introduction nd Preliminries The elegnce in shpe nd properties of conve functions mkes it ttrctive to study this kind of functions in mthemticl nlysis. It should be noticed tht in new problems relted to conveity, generlized notions bout conve functions re required to obtin pplicble results. During recently yers mny efforts hve gone on generliztion of notion of conve functions. Most importnt generliztions cn be found in works tht chnge the form of defining of functions to generlized form such s qusi-conve [1], pseudo-conve [7], strongly conve [9], logrithmiclly conve [8], pproimtely conve [5], midconve [6] functions etc. On the other hnd Hermite-Hdmrd-Fejer inequlity, n interesting result relted to conve functions hs been proved in [] s the following: Theorem 1.1. Let f : [, b] R be conve function. Then f( + b ) g()d f()g()d f() + f(b) g()d, (1.1) ( where g : [, b] R + = [, + ) is integrble nd symmetric bout = +b g() = g(+b ), [, b] ). Corresponding uthor Emil ddresses: mdjid.eshghi@gmil.com (M. Eshghi Gordji), sever.drgomir@vu.edu.u (S. S. Drgomir), rostmin333@gmil.com (M. Rostmin Delvr) Received: Jnury 1 Revised: My 15

2 8 Eshghi, Drgomir, Rostmin If in (1.1) we consider g 1 then we obtin Hermite-Hdmrd inequlity: f( + b ) 1 b f()d f() + f(b). (1.) An interesting question in (1.), ws estimting the difference between left nd middle terms nd between right nd middle terms. In [], the difference between middle nd right terms in (1.) hs been estimted s the following: Theorem 1.. Let f : I R R be differentible mpping on I,, b I with < b. If f is conve on [, b], then the following inequlity holds: f() + f(b) 1 b f()d b ( f () + f (b) ). 8 Motivted by these works we introduce the notion of η-conve functions s generliztion of conve functions nd estimte the difference between middle nd left terms in (1.1), when f is n η-conve function. Also s n ppliction we give n error estimte for midpoint formul. Definition 1.3. [3] Let I be n intervl in rel line R. A function f : I R is clled conve with respect to bifunction η : R R R (briefly η-conve), if for ll, y I nd t [, 1]. f(t + (1 t)y) f(y) + tη ( f(), f(y) ) (1.3) In fct bove definition geometriclly sys tht if function is η-conve on I, then it s grph between ny, y I is on or under the pth strting from ( y, f(y) ) nd ending t (, f(y) + η(f(), f(y)) ). If f() should be the end point of the pth for every, y I, then we hve η(, y) = y nd the function reduces to conve one. Note tht by tking = y in (1.3) we get tη(f(), f()) for ny I nd t [, 1] which implies tht η(f(), f()) for ny I. Also if we tke t = 1 in (1.3) we get f() f(y) η ( f(), f(y) ) for ny, y I. If f : I R is conve function nd η : I I R is n rbitrry bifunction tht stisfies η(, y) y for ny, y I, then f(t + (1 t)y) f(y) + t[f() f(y)] f(y) + tη ( f(), f(y) ) showing tht f is η-conve. There re simple emples bout η-conveity of function([3]).

3 An inequlity relted to η-conve functions (II) 6 (15) No., Emple 1.. (1) For conve function f, we my find nother function η other thn the function η(, y) = y such tht f is η-conve. Consider f() = nd η(, y) = + y. Then we hve f ( λ + (1 λ)y ) = ( λ + (1 λ)y ) y + λ + λ(1 λ)y y + λ + λ(1 λ)( + y ) y + λ( + + y ) = y + λ( + y ) = f(y) + λη ( f(), f(y) ) for ll, y R nd λ (, 1). Also the fcts y + ( + y ) nd y y, for ll, y R show the correctness of inequlity for λ = 1 nd λ = respectively which mens tht f is η-conve. Note tht the function f() = is η-conve w.r.t ll η(, y) = + by with 1, b 1 nd, y R. () Consider function f : R R defined s {, ; f() =, <. nd define bifunction η s η(, y) = y, for ll, y R = (, ]. It is not hrd to check tht f is n η-conve function but not conve one. (3) Define the function f : R + R + s f() = { + y, y; R + s η(, y) = ( + y), > y. Then f is η-conve but is not conve. {, 1; 1, > 1. nd bifunction η : R + R + The first result is the fct tht ny η-conve function with bounded bifunction η from bove, stisfies the Lipschitz condition. Two definitions re required. Definition 1.5. [1] A function f : [, b] R is bsolutely continuous on [, b] if corresponding to ny ε > there eists δ > such tht for ny collection { i, b i n 1 of disjoint open intervls of [, b] with n 1 (b i i ) < δ, n 1 f(b i) f( i ) < ε. Definition 1.6. [1] A function f : [, b] R is sid to stisfy Lipschitz condition on [, b] if there is constnt K so tht for ny two points, y [, b], f() f(y) K y. Lemm 1.7. Suppose tht f : I R is n η-conve function nd η is bounded from bove on f(i) f(i). Then f stisfies the Lipschitz condition on ny closed intervl [, b] contined in I, the interior of I. Hence, f is bsolutely continuous on [, b] nd continuous on I. Proof. Let M η be the upper bound of η on f(i) f(i). Consider closed intervl [, b] in I nd choose ε > such tht [ ε, b + ε] belongs to I. Suppose tht, y re distinct points of [, b]. Set z = y + ε (y ) nd t = y. So it is not hrd to see tht z [ ε, b+ε] nd y = tz +(1 t). y ε+ y Then f(y) f() + tη(f(z), f()) f() + tm η. This implies tht where K = Mη ε. f(y) f() tm η = y ε + y M η y M η = K y, ε

4 3 Eshghi, Drgomir, Rostmin Also if we chnge the plce of, y in bove rgument we hve f() f(y) K y. Therefore f(y) f() K y. It follows tht if we choose δ < ε/k, then f is bsolutely continuous. Finlly since [, b] is rbitrry on I, then f is continuous on I. As consequence of Lemm 1.7, n η-conve function f : [, b] R where η is bounded from bove on f([, b]) f([, b]) is integrble.. Min Result The first result of this section is lemm tht is generliztion of Lemm.1 in []. Lemm.1. Suppose tht f : [, b] R is differentible function, g : [, b] R + is continuous function nd f is n integrble function on [, b]. Then f() + f(b) g()d f()g()d = 1 b g(u)f ()dud 1 g(u)f ()dud. Proof. By Leibniz integrl rule nd integrtion by prts we hve ( ) d f()g()d = f() g(u)du = f(b) g(u)du With the sme rgument f()g()d = ( f() Adding reltions (.1) nd (.), gives the result. ) d g(u)du = f() g(u)du + The following lemm is consequence of lemm.1. g(u)f ()dud. (.1) g(u)f ()dud. (.) Lemm.. Suppose tht f : [, b] R is differentible function, g : [, b] R + is continuous function nd symmetric bout +b nd f is n integrble function on [, b]. Then f() + f(b) (b ) 1 1 g()d + b + + b g(u)du + b f()g()d = ) g(u)du b ) f ( 1 + t + 1 t f ( 1 t t b) dt. b) dt+ (.3) Proof. From Lemm.1 we cn see f() + f(b) b I = g()d f()g()d = +b 1 g(u)f ()dud + g(u)f ()dud (.) +b +b g(u)f ()dud +b g(u)f ()dud.

5 An inequlity relted to η-conve functions (II) 6 (15) No., By chnging the vrible = 1+t + 1 t 1 t b nd = + 1+t b in (.) we hve I = b 1 + b g(u)f ( 1 + t + 1 t b)dudt+ (.5) t 1+t + b g(u)f ( 1 t t b)dudt (.6) g(u)f ( 1 + t + b + 1 t b)dudt (.7) g(u)f ( 1 t 1 t 1+t + b t b)dudt. (.8) Consider (.5) with (.7) nd consider (.6) with (.8) together. Then I = b 1 [ + b ] g(u)du g(u)du f ( 1 + t + 1 t b)dt+ (.9) 1 [ 1 t 1+t + b b ] g(u)du g(u)du f ( 1 t t b)dt. Since g is symmetric with respect to +b then nd + b 1 t 1+t + b g(u)du g(u)du g(u)du = g(u)du = Implying (.1) nd (.11) in (.9) we hve (b ) 1 + b ) I = g(u)du + b 1 + b ) g(u)du + b 1 t 1+t + b + b g(u)du, (.1) 1 t 1+t + b + b g(u)du. (.11) f ( 1 + t + 1 t f ( 1 t t b) dt. b) dt+ Remrk.3. Lemm.1 nd. re equivlent to Lemm.1 in [], if we set g 1. Bsed on Lemm., we obtin the min theorem of the pper. Theorem.. Suppose tht f : [, b] R is differentible function, g : [, b] R + is continuous function nd symmetric bout +b nd f is n η-conve function where η is bounded from bove on [, b]. Then f() + f(b) b g()d f()g()d (b ) [ f (b) ( + η f (), f (b) ) ] 1 1 t 1+t + b g(u)dudt. (.1) + b

6 3 Eshghi, Drgomir, Rostmin Proof. From Lemm. nd the fct tht f is η-conve where η is bounded from bove we hve f() + f(b) b g()d f()g()d (b ) 1 1 t 1+t + b [ f g(u) ( 1 + t + b + 1 t b) ( + f 1 t t b) ] dudt (b ) 1 1 t 1+t + b [ g(u) f (b) t + b η( f (), f (b) ) + f (b) + 1 t η( f (), f (b) )] dudt = (b ) [ f (b) + η ( f (), f (b) ) ] 1 1 t 1+t + b g(u)dudt. + b Remrk.5. Theorem. reduces to Theorem 1., if we consider g 1 nd η(, y) = y for ll, y [, b]. Finlly s n ppliction of Theorem., we give n error estimte for midpoint formul tht is generliztion of Proposition.1 in []. Suppose tht d is prtition = < 1 < < n 1 < n = b of intervl [, b]. Consider formul where T (f, g, d) = f()g()d = T (f, g, d) + E(f, g, d), n 1 nd E(f, g, d) is the pproimtion error. i= f( i ) + f( i+1 ) i+1 i g()d Theorem.6. Suppose tht f : [, b] R is differentible function, g : [, b] R + is continuous function nd symmetric with respect to +b nd f is n η-conve function where η is bounded from bove on [, b]. Then n 1 E(f, g, d) i= ( i+1 i ) [ f ( i+1 ) + η ( f ( i ), f ( i+1 ) ) ] 1 1 t i+ 1+t i+1 1+t i+ 1 t i+1 g()ddt. Proof. It is enough to pply Theorem. on the subintervl [ i, i+1 ] (i =, 1,, n 1) of the prtition d for intervl [, b], nd to sum ll chieved inequlities over i nd then using tringle inequlity. References [1] B. Definetti, Sull strtificzioni convesse, Ann. Mth. Pur. Appl., 3 (199) [] S.S. Drgomir nd R.P. Agrwl, Two inequlities for differentible mppings nd pplictions to specil mens of rel numbers nd to trpezoidl formul, Appl. Mth. Lett., 11 (1998) [3] M. Eshghi Gordji, M. Rostmin Delvr nd M. De l Sen, On ϕ-conve Functions, J. Mth. Inequl, In Press.

7 An inequlity relted to η-conve functions (II) 6 (15) No., [] L. Fejer, Uberdie fourierreihen, II, Mth. Nturwise. Anz Ungr. Akd. Wiss., (196) [5] D.H. Hyers nd S.M. Ulm, Approimtely conve functions, Proc. Amer. Mth. Soc., 3 (195) [6] J. Jensen, On konvee funktioner og uligheder mellem middlverdier, Nyt. Tidsskr. Mth. B., 16 (195) [7] O.L. Mngsrin, Pseudo-Conve functions, SIAM Journl on Control, 3 (1965) [8] J.E. Pecric, F. Proschn nd Y.L. Tong, Conve functions, prtil orderings nd sttisticl pplictions, Acdemic Press, Boston, 199. [9] B.T. Polyk, Eistence theorems nd convergence of minimizing sequences in etremum problems with restrictions, Soviet Mth. Dokl., 7 (1966) [1] A.W. Robert nd D.E. Vrbeg, Conve Functions, Acdemic Press, 1973.

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