ONE GENERALIZED INEQUALITY FOR CONVEX FUNCTIONS ON THE TRIANGLE

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1 Joural of Pure ad Appled Mathematcs: Advaces ad Applcatos Volume 4 Number 205 Pages Avalable at DOI: ONE GENERALIZED INEQUALITY FOR CONVEX FUNCTIONS ON THE TRIANGLE Mechacal Egeerg Faculty Slavosk Brod Uversty of Osjek Trg Ivae Brlć Mažurać Slavosk Brod Croata e-mal: zlatko.pavc@sfsb.hr Abstract The am of the paper s to obta a very geeral equalty for covex fuctos o the tragle. To costruct such equalty we clude postve lear fuctoals o the space of real fuctos. That equalty ca be appled to varous forms of mathematcal meas ad ts tegral form cotas the Jese Fejér ad Hermte-Hadamard equalty.. Itroducto.. Covex fuctos o the tragle The tragle wth vertces a b c R that do ot belog to the same le s used permaetly throughout the artcle. It wll be marked by or abc ad ts teror wll be marked by. Each pot x ca be represeted by the uque tromal covex combato 200 Mathematcs Subject Classfcato: 26D5 26B25 46N0. Keywords ad phrases: covex fucto tragle space of real fuctos postve lear fuctoal. Receved August Scetfc Advaces Publshers

2 78 x αa + βb + γc. ( The coeffcets ca be expressed by the rato of areas ar( xbc ar( xac ar( xab α β γ ar( abc ar( abc ar( abc (2 ad thus by the rato of determats x x2 x x2 x x2 b b2 a a2 a a2 α c a c2 a2 β c a c2 a2 γ b a b2 a2. b b2 b b2 b b2 c c2 c c2 c c2 (3 Let f : R be a covex fucto. The at plae of f passes through the respectve graph pots of a b ad c ad ts equato s ar( xbc ar( xac ar( xab f abc ( x f ( a + f ( b + f (. c (4 ar( abc ar( abc ar( abc Let d be a pot of the tragle teror. The port plaes of f at d pass through the respectve graph pot of d ad ther equatos κ d f d + ad deped o the pars of slope coeffcets [ ( ( ] f κ 2 [ f2 ( d f2 ( d+ ]. For the specfed par of coeffcets κ ad κ 2 the correspodg equato s f d ( x x κ ( x d + κ ( x d + f ( d d. ( The port-at plae equalty fd ( x f ( x fabc ( x (6 holds for every x. The at ad port plae of a covex surface are preseted geometrcally Fgure.

3 ONE GENERALIZED INEQUALITY FOR CONVEX 79 Fgure. Secat ad port plae of a covex fucto..2. Postve lear fuctoals o the space of real fuctos Let X be a oempty set ad let F F( X be a subspace of the lear space of all real fuctos o the doma X. We assume that the space F cotas the ut fucto u defed by u ( x for every x X. The the space F cotas every real costat κ wth the meag of κ κu ad F cotas every composte fucto f ( g g 2 of a affe fucto f : R R ad a par of fuctos g g2 F. Specfcally usg the equato f ( x x2 κ x + κ2x2 + κ3 we have the composto whch belogs to the space F. f ( g g κ g + κ g + κ ( u Let L L( F( X be the space of all lear fuctoals o the space F ( X. A fuctoal L L s sad to be postve (oegatve f the equalty L ( g 0 holds for every oegatve fucto g F. If

4 80 g g 2 F are fuctos such that g ( x g2( x for x X the a postve fuctoal L satsfes the equalty L( g L(. (8 g 2 A fuctoal L L s sad to be utal (ormalzed f L ( u. Such fuctoal has the property L ( κ u κ for every real costat κ. 2. Ma Results The frst lemma provdes a basc cluso relatg to the mage of a par of fuctos g g. The proof of lemma cludes a covex 2 F aalytcs through the applcato of covex combatos. Lemma 2.. Let for x X. g g 2 F be fuctos such that ( g ( x g2( x The a postve utal fuctoal L L satsfes the cluso ( L ( g L( g. (9 2 Proof. Takg x X we have the pot ( g ( x g2( x ad ts uque covex combato ( g ( x g ( x α( x( a a + β( x ( b b + γ( x ( c. ( c2 Usg equatos (3 we ca determe fuctos α ( x β( x ad γ ( x showg that they belog to F. For example α( x α g ( x + α g ( x α3 Sce the fuctoal L s postve the umbers L ( α L( β ad L ( γ are oegatve ad sce α ( x + β( x + γ( x u( x t follows that L ( α + L( β + L( γ. Actg wth the fuctoal L to each of the coordates of equato (0 we obta the covex combato ( L( g L( g L( α( a a + L( β ( b b + L( γ ( c ( c2 esurg that the pot ( L ( g L( g 2 belogs to the tragle.

5 ONE GENERALIZED INEQUALITY FOR CONVEX 8 Usg hyperplaes McShae proved that the cluso (9 2 geerally apples to closed covex sets R. That part of hs work (see [7] he called the geometrc formulato of Jese s equalty. The od lemma provdes a basc equalty relatg to a affe fucto. Lemma 2.2. Let fuctoal. g g 2 F be fuctos ad L L be a utal The a affe fucto f : R 2 R satsfes the equalty f ( L( g L( g L( f ( g. (2 2 g2 Proof. Usg the affe equato f ( x x κ x + κ x + ad κ3 applyg the utal property of L we obta f ( L( g L( g2 κ L( g + κ2l( g2 + κ3 ( κ g + κ g + u L 2 2 κ3 L( f ( g g 2 (3 provg the equalty formula (2. Now we ca clude a cotuous covex fucto ad get the ma result. Theorem 2.3. Let g g 2 F be fuctos such that ( g ( x g2( x for x X. Let L L be a postve utal fuctoal. The a cotuous covex fucto equalty f : R satsfes the double f ( L( g L( g L( f ( g g f ( L( g L( (4 2 2 g2 f provded that the composte fucto f ( g g 2 s F. Proof. The pot l ( L( g L( g 2 belogs to the tragle by Lemma 2.. We sketch the proof three steps depedg o the posto of l.

6 82 If l we take ay port plae of f at l. Actg wth the postve fuctoal L to the followg subequalty of the port-at equalty formula (6: we obta f l ( g ( x g ( x f ( g ( x g ( x f ( g ( x g ( x L l ( f ( g g L( f ( g g L( f ( g g. Applyg Lemma 2.2 to affe fuctos f l ad f ad puttg l stead of ( L ( g L( the above equalty takes the form f l g2 where the frst term ( L( g L( g L( f ( g g f ( L( g L( g f l ( L( g L( g f ( L( g L( g. 2 2 If l belogs to the relatve teror of some sde we apply the prevous procedure to the correspodg covex curve by usg ay port le relatg to l ad the at le relatg to edpots of that sde. If l { a b c} we rely o the cotuty of f usg a port plae at a pot of the teror that s close eough to l. Formula (4 ca be expressed the form that cludes the covex combato of the tragle vertces a b ad c. The respectve form of Theorem 2.3 s as follows: Corollary 2.4. Let g g 2 F be fuctos such that ( g ( x g2( x for x X. Let L L be a postve utal fuctoal ad let α a + βb + γc be the covex combato that s equal to ( L ( g L(. g2

7 ONE GENERALIZED INEQUALITY FOR CONVEX 83 The a cotuous covex fucto equalty f : R satsfes the double f ( α a + βb + γc L( f ( g g αf ( a + βf ( b + γf ( (5 2 c f provded that the composte fucto f ( g g 2 s F. Proof. The correspodg terms of formulae (4 ad (5 are the same. As regards the last terms we have f ( L( g L( g2 αf ( a + βf ( b + γf ( c αf ( a + βf ( b + γf ( c because of the affty of f ad ts cocdece wth f at vertces. As for the coeffcets formula (5 let l ( L( g L( ad let g2 lbc 2 lac 3 lab. Puttg x l formula (2 we get ar ( ar( 2 ar( α β γ ar( ar( ar( 3. (6 Fuctoal equaltes for fuctos that are ot ecessarly covex were cosdered [5]. 3. Applcatos to Dscrete ad Itegral Iequaltes We utlze Theorem 2.3 to derve the geeralzatos of mportat equaltes for covex fuctos o the tragle. The followg s the dscrete verso of Theorem 2.3. Corollary 3.. Let g g2 : R be fuctos such that ( g ( x g 2 ( x for x ad let h : R be a postve fucto. Let x x be pots let ( ( ( ( g x h x g2 x h x l (7 ( ( h x h x ad let lbc 2 lac 3 lab.

8 84 The a cotuous covex fucto equalty f g ( x h( x h( x g2 ( x h( x h( x f : R satsfes the double f ( g ( x g ( x h( x 2 h( x ar( ar( ( 2 ar( 3 f a + f ( b + f (. c (8 ar( ar( ar( Proof. Let F be the space of all real fuctos o the doma X. We take the summarzg lear fuctoal defed by q ( x h ( x L( q L( q; h (9 h ( x for every fucto q F. The summarzg fuctoal L s postve ad utal. Applyg L to the par of gve fuctos g ad g 2 we get the pot ( ( ( ( g x h x g2 x h x l ( L( g L( g2 (20 ( ( h x h x ad applyg L to the composto f ( g g we get the sums rato 2 f ( g( x g2( x h( x L( f ( g g2. (2 h( x Usg x ( L( g L( g 2 the equato of the at plae formula (4 we get ar( ( ( ( ar( ( ( 2 ar 3 f L g L g2 f a + f ( b + f (. c (22 ar( ar( ar(

9 ONE GENERALIZED INEQUALITY FOR CONVEX 85 Embeddg the rght-had sdes of Equatos (20 (2 ad (22 to the equalty formula (4 we obta the dscrete equalty formula (8. The ext s the tegral verso of Theorem 2.3. Corollary 3.2. Let g g2 : R be tegrable fuctos such that ( g ( x g2( x for x ad let h : R be a postve tegrable fucto. Let ( ( ( ( g x h x g2 x h x l (23 ( ( h x h x ad let lbc 2 lac 3 lab. The a cotuous covex fucto equalty f f : R satsfes the double g ( x h( x g ( x h( x 2 f ( g( x g2( x h( x h( x h( x h( x ar( ar( ( 2 ar( 3 f a + f ( b + f (. c (24 ar( ar( ar( Proof. Let F be the space of all tegrable fuctos over the doma X. Accordg to the Lebesgue theorem o the Rema tegral the composto f ( g g 2 s tegrable over because s bouded ad cotuous almost everywhere. We take the tegratg lear fuctoal defed by q( x h( x L( q L( q; h h( x (25 for every fucto q F ad further follow the proof of Corollary 3..

10 86 The equalty formula (24 cotas the exteded tegral form of Jese s equalty (see [4] the Fejér equalty (see [] ad cosequetly the Hermte-Hadamard equalty (see [3] ad [2]. The equalty smplfcatos are as follows. Usg the ut fucto h ( x x whch case 2 ( ( g x g2 x l (26 ar( ar( we get the exteded tegral form of Jese s equalty f g ( ( x g2 x f ( g ( x g2( x ar( ar( ar( ar( ar( ( 2 ar( 3 f a + f ( b + f (. c (27 ar( ar( ar( Now we use the projectos g ( x x2 x ad g 2( x x2 x2 ad a postve fucto h ( x x 2 whose baryceter l falls to ( a + b + c 3. Thus l xh( x h( x x2h( x h( x a + b + c 3 (28 dcatg that α β γ 3 ad we have the Fejér equalty a b c f f ( x h( x h( x f ( a + f ( b + f ( c. 3 (29

11 ONE GENERALIZED INEQUALITY FOR CONVEX 87 Puttg the ut fucto h ( x x Fejér s equalty 2 formula (29 we obta the Hermte-Hadamard equalty f ( x a b c f ( a f ( b f ( c f (30 3 ar( 3 Improvemets of the Hermte-Hadamard equalty for covex fuctos o the bouded closed terval of real umbers were cosdered [6]. Ackowledgemets The author would lke to thak Velmr Pavć who has prepared Fgure. Refereces [] L. Fejér Über de Fourerrehe II Math. Naturwss. Az. Ugar. Akad. Wss. 24 ( [2] J. Hadamard Étude sur les proprétés des foctos etères et e partculer d ue focto cosderée par Rema J. Math. Pures Appl. 58 ( [3] Ch. Hermte Sur deux lmtes d ue tégrale défe Mathess 3 ( [4] J. L. W. V. Jese Sur les foctos covexes et les égaltés etre les valeurs moyees Acta Math. 30 ( [5] Z. Pavć Fuctos lke covex fuctos J. Fuct. Spaces 205 (205 Artcle ID [6] Z. Pavć Improvemets of the Hermte-Hadamard equalty J. Iequal. Appl. 205 (205 Artcle ID 222. [7] E. J. McShae Jese's equalty Bull. Amer. Math. Soc. 43 ( g

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