INEQUALITIES USING CONVEX COMBINATION CENTERS AND SET BARYCENTERS

Joural of Mathematcal Scece: Advace ad Alcato Volume 24, 23, Page 29-46 INEQUALITIES USING CONVEX COMBINATION CENTERS AND SET BARYCENTERS ZLATKO PAVIĆ Mechacal Egeerg Faculty Slavok Brod Uverty of Ojek Trg Ivae Brlć Mañurać 2 35 Slavok Brod Croata e-mal: Zlatko.Pavc@fb.hr Abtract The artcle demotrate the mortace of cove combato ceter ad et baryceter creatg Jee equalte. Ceter ad baryceter roerte are ued to the formulato of Jee equalty for fucto that are cove at oe de of t doma.. Itroducto Throughout th aer, I R wll be a terval wth the o- emty teror I. Ma uort for the work wll be the famou Jee equalte. The dcrete or bac form (ee [3]) ay that every cove fucto f : I R atfe the equalty 2 Mathematc Subject Clafcato: 26A5, 26D5, 28A, 28A25. Keyword ad hrae: cove combato, baryceter, Jee equalty, quaarthmetc mea. Receved November 3, 23 23 Scetfc Advace Publher

3 ZLATKO PAVIĆ f f ( ), (.) for all cove combato from I. The cotuou or tegral form (ee [4]) ay that every cove µ -tegrable fucto f : I R atfe the equalty for all f tdµ () t f () t dµ (), t (.2) µ ( A) A µ ( A) A µ -meaurable ubet A I wth µ ( A ) >. 2. Cove Combato Ceter The ma reult th ecto Theorem 2.2 for rght cove ad rght cocave fucto the dcrete cae wthout retrcto o coeffcet. If I are ot, ad [, ] are coeffcet uch that, the the um c belog to I, ad t called the cove combato from I. The umber c telf called the ceter of the cove combato. For a cotuou fucto f : I R, the cove combato f ( ) belog to f ( I ). A cove hull of a et X wll be deoted by cox. The followg theorem related to cove combato wth commo ceter: Theorem A. Let,, I be ot uch that co{,, } for k,,, where k. Let α,, α R be o- egatve umber uch that < k α α < β α. k

INEQUALITIES USING CONVEX COMBINATION 3 If oe of the equalte α k α β α β α k α, (2.) vald, the the double equalty α k α f ( ) α f ( ) α f ( ), β β α k (2.2) k hold for every cove fucto f : I R. Theorem A wa realzed [7, Prooto 2]. Corollary 2.. Let,, I be ot uch that co{,, k } for k,,, where k. Let α,, α R be o- k egatve umber uch that < α α. If the equalty k α k α, (2.3) vald, the the equalty k α f ( ) α f ( ), k (2.4) hold for every cove fucto f : I R. The et theorem (weghted rght cove fucto theorem) wa reeted ad roved [2] a the ma reult. Theorem B (WRCF-theorem). Let f ( u) be a fucto defed o a real terval I ad cove for u I, ad let, 2,, be otve real umber uch that m{, 2,, }, 2.

32 ZLATKO PAVIĆ The equalty f ( ) f ( ) f ( ) f ( ), 2 2 2 2 hold for all ad oly f, 2,, I atfyg 22 f f ( ) ( ) f ( y) f ( ), for all, y I uch that y ad ( ) y. I a mlar way, WLCF-theorem (weghted left cove fucto theorem) ad WHCF-theorem (weghted half cove fucto theorem) were lted [2]. The ma defcecy of thee three theorem are cove combato cotag re-determed coeffcet. Eecally, the bomal cove combato clude the fed coeffcet m{, 2,, }. Two ubterval of I ecfed by the umber wth I wll be labelled I rg lef { t I t } ad I { t I t }. We formulate the Jee equalty for rght cove ad cocave fucto. Theorem 2.2 (Dcrete cae of rght covety ad cocavty). Let f : I R be a fucto. If f cove o rg I for ome I, the the equalty f f ( ), (2.5) hold for all cove combato from I atfyg f ad oly f the equalty

INEQUALITIES USING CONVEX COMBINATION 33 f ( qy) f ( ) qf ( y), (2.6) hold for all bomal cove combato from I atfyg qy. If f cocave o rg are vald (2.5) ad (2.6). I for ome I, the the revere equalte Proof. Let u rove the uffcecy for the cae of rght covety. The roof wll be doe by ducto o the teger. The bae of ducto. Take the bomal cove combato from I uch that qy. If qy, the the equalty (2.5) for rg 2 hold by the aumto (2.6). If qy > ad, y I, the we aly the dcrete form of Jee equalty to get the equalty (2.5) for 2. Therefore, uoe qy > wth < ad >. Let the ot be defed by the equato q The we have < qy ad < y, ad o, y co{ qy, }. We wat to aly Corollary 2.. Frt, we olve the equato. α ( qy) α α α, (2.7) 2 3 4 y wth the ukow α α,, ad α, uder the codto α, 2 α3 4 α 2 α 3 α 4. Takg α, we get α2 q, α3, ad α 4 q. Sce the codto (2.3) atfed, t ale f ( qy) qf ( ) f ( ) qf ( y), (2.8) by the equalty (2.4). Sce q, t follow f ( ) f ( ) qf ( ) by aumto. After orderg the equalty (2.8), we get f ( qy) f ( ) qf ( y).

ZLATKO PAVIĆ 34 The te of ducto. Let 2 be a fed teger. Suoe that the equalty (2.5) true for all correodg -membered cove combato. Let be a cove combato from I, ad wthout lo of geeralty, uoe all. > If all, the the equalty (2.5) follow from Jee equalty for cove fucto. Otherwe, f, the. Ug the ducto bae ad reme, t follow: ( ) f f ( ) ( ) f f ( ) ( ) ( ) f f ( ), f whch ed the roof of the theorem. Theorem for left cove ad left cocave fucto ca be formulated o the model of Theorem 2.2. We ed th ecto wth the formulato of the theorem for half cove ad half cocave fucto. Theorem 2.3 (Dcrete cae of half covety ad cocavty). Let R I f : be a fucto. If f cove o rg I or lef I for ome, I the the equalty

INEQUALITIES USING CONVEX COMBINATION 35 f f ( ), (2.9) hold for all cove combato from I atfyg f ad oly f the equalty f ( qy) f ( ) qf ( y), (2.) hold for all bomal cove combato from I atfyg qy. If f cocave o rg I or lef equalte are vald (2.9) ad (2.). I for ome I, the the revere 3. Set Baryceter The ma reult th ecto Theorem 3.3 for rght cove ad rght cocave fucto the tegral cae. Itegral geeralzato of the cocet of arthmetc mea the fte meaure ace are the baryceter of meaurable et, ad the tegral arthmetc mea of tegrable fucto. The bac reult o the tegral arthmetc mea jut the tegral form of Jee equalty. See [6, age 44-45]. For a gve fte meaure µ o I wll be aumed that all ubterval of I, ad therefore the ot, are µ -meaurable. Let A I be a µ -meaurable et wth µ ( A ) >, ad A be the detty fucto o A. If the fucto A µ -tegrable, the the µ -baryceter of A defed by If a fucto mea of f o A defed by B ( A, µ ) tdµ (). t µ ( A) (3.) A f : I R µ -tegrable o A, the the µ -arthmetc

36 ZLATKO PAVIĆ M ( f, A, µ ) f () t dµ (). t µ ( A) (3.2) Note that M (, A, µ ) B( A, µ ). If A the terval, the t A µ -baryceter B ( A, µ ) belog to A. If A the terval ad f cotuou o A, the t µ -arthmetc mea o A belog to f ( A). Theorem C. Let µ be a fte meaure o I. Let A B I be a µ -meaurable et ad A B be a bouded terval uch that < µ ( A ) <. If oe of the equalte vald, the the double equalty B ( A, µ ) B( B, µ ) B( B\ A, µ ), (3.3) M ( f, A, µ ) M( f, B, µ ) M( f, B\ A, µ ), (3.4) hold for every µ -tegrable cove fucto f : I R. The vero of Theorem C for bouded cloed terval A [ a, b] ad B wa roved [7, Prooto ]. A meaure µ o I ad to be cotuou, f µ ({ t }) for every ot t I (accordg to the defto the book [9, age 49]). I the ret of the aer, we wll ue the cotuou fte meaure µ o I, whch otve o the terval, that, whch atfe µ ( A ) > for every odegeerate terval A I. If µ a cotuou fte meaure o I that otve o the terval, the the fucto lef tdµ () t B( I, µ ), lef lef µ ( ) (3.5) I I

INEQUALITIES USING CONVEX COMBINATION 37 trctly creag cotuou o I. The related fucto wth I rg tead of I lef ha the ame roerte. We ca alo ue ay terval A I for the fucto defto (3.5) whch cae the reultg fucto oberved o A. Lemma 3.. Let µ be a cotuou fte meaure o I that otve o the terval. I If a a ot, the the decreag ere ( A ) of bouded terval A I et o that B ( A, µ ) a ad A { a}. Proof. Take a ot a I. Let u how the bac ad the teratve te. I the frt te, we chooe the ot, y I uch that < a < y, ad determe the µ -baryceter of the terval [, y ]: a (). ([, ]) tdµ µ y [ y ] t, If a a, the we take A [, y ]. If a > a, the we oberve the g : a, y defed by fucto [ ] R g( ) td () t a. ([, ] ) µ µ [ ], Sce g cotuou, g ( a) < ad g ( y ) >, there mut be a ot y < a, y > uch that g ( y ). I th cae, we ca take A [, y ]. If a < a, the we creae utl we obta oe of the revou two cae.

38 ZLATKO PAVIĆ I the et te, f A [, y ], we take the ot 2 a 2 ad y2 a y 2, ad reeat the revou rocedure to determe A 2. Lemma 3.2. Let µ be a cotuou fte meaure o I that otve o the terval. Let b B ( B, µ ). B I be a terval wth the µ -baryceter If a B a ot dfferet from b, the the terval A B et o that the bomal cove combato b µ ( A) µ ( B \ A) a a, (3.6) hold wth a B ( A, µ ) ad a B ( B \ A, µ ). Proof. I the cae a < b, we ue the fucto g( ) tdµ () t a, (3.7) lef lef µ ( B ) B o the doma B. The fucto g trctly creag cotuou o, B ad ha the uque zero-ot. Takg A B, the equalty rg (3.6) evdet. Alo, B \ A B \ { } B. rg a lef I the cae a > b, we ue the fucto g wth the et lef rg rg B tead of the et B, ad take A B. I th cae, B \ A B \ { } B. Theorem 3.3 (Itegral cae of rght covety ad cocavty). Let µ be a cotuou fte meaure o I that otve o the terval, ad f : I R be a µ -tegrable fucto. lef lef a

INEQUALITIES USING CONVEX COMBINATION 39 If f cove o rg I for ome I, the the equalty hold for all terval f ad oly f the equalty hold for all bouded terval If f cocave o f tdµ () t f () t dµ (), t (3.8) B B B I atfyg tdµ () t, (3.9) B f tdµ () t f () t dµ (), t (3.) µ ( A) A µ ( A) A A I atfyg tdµ () t. (3.) µ ( A) A rg are vald (3.8) ad (3.). I for ome I, the the revere equalte Proof. Let u rove the uffcecy for the cae of rght covety. Suoe B a terval from I uch that t µ -baryceter b B tdµ () t. If B rg I whch cae B, the equalty (3.8) follow from the tegral form of Jee equalty for cove fucto. Suoe B. If b, the ug Lemma 3., we ca determe the bouded terval A B wth the µ -baryceter. Ug the aumto (3.), ad the equalty (3.4) uder the codto B ( A, µ ) B( B, µ ), we get

4 ZLATKO PAVIĆ f tdµ () t f tdµ () t B µ ( A) A µ ( A) f () t dµ (). t B A f () t dµ () t If < b, the ug Lemma 3.2, we ca determe the terval S B o that the bomal cove combato b µ ( S) µ ( B \ S), (3.2) rg hold wth B ( S, µ ) ad B ( B \ S, µ ). Sce,, we ca aly the dcrete form of Jee equalty to the equalty (3.2), ad get µ ( S) µ ( B \ S) f ( b) f () f ( ). (3.3) I Sce B rg rg \ S B I alo vald, we may aly the tegral form of Jee equalty to the baryceter, ad have the etmate for f ( ): f ( ) f tdµ () t µ ( B \ S) B\ S µ ( B \ S) B\ S f () t dµ (). t Now, we clude Lemma 3. to determe the bouded terval A S wth the µ -baryceter. Ug the aumto (3.), ad the equalty (3.4) uder the codto B ( A, µ ) B( S, µ ), we obta the etmate for f (): f () f tdµ () t µ ( A) A µ ( A) f () t dµ (). t µ ( S) S A f () t dµ () t

INEQUALITIES USING CONVEX COMBINATION 4 After orderg the equalty (3.3), t follow: f tdµ () t f () t dµ () t B S B\ S f () t dµ (), t S f () t dµ () t ad the roof doe. Ug the revou theorem, we ca ere the theorem for left cove ad left cocave fucto the tegral cae. Combg thee cae, we get the theorem for half cove ad half cocave fucto the tegral cae. Theorem 3.4 (Itegral cae of half covety ad cocavty). Let µ be a cotuou fte meaure o I that otve o the terval, ad f : I R be a µ -tegrable fucto. If f cove o hold for all terval rg I or f ad oly f the equalty lef hold for all bouded terval I for ome I, the the equalty f tdµ () t f () t dµ (), t (3.4) B B B I atfyg tdµ () t, (3.5) B f tdµ () t f () t dµ (), t (3.6) µ ( A) A µ ( A) A A I atfyg tdµ () t. (3.7) µ ( A) A

42 ZLATKO PAVIĆ If f cocave o rg I or lef equalte are vald (3.4) ad (3.6). I for ome I, the the revere 4. Alcato to Qua-Arthmetc Mea I the alcato of covety to qua-arthmetc mea, we ue trctly mootoe cotuou fucto ϕ, v / : I R uch that v/ cove wth reect to ϕ( v / ϕ-cove ), that, f v/ ϕ cove (accordg to the termology the book [8, Defto.9]). A mlar otato ued for cocavty. Very geeral form of dcrete ad tegral qua-arthmetc mea, ad t refemet, were tuded [5]. A good aroach to mea ca be ee []. 4.. Dcrete cae Let be a cove combato from I. The dcrete ϕ -qua- arthmetc mea of the ot wth the coeffcet the umber M ( ; ) ( ) ϕ ϕ ϕ, (4.) whch belog to I, becaue the cove combato ϕ( ) belog to ϕ ( I ). If ϕ the detty fucto o I, that, ϕ, the the I dcrete I -qua-arthmetc mea jut the cove combato. The followg the theorem for rght covety ad rght cocavty for qua-arthmetc mea the dcrete cae: Theorem 4. (Dcrete cae of rght covety ad cocavty for quaarthmetc mea). Let ϕ, v / : I R be trctly mootoe cotuou fucto, ad J ϕ( I ).

INEQUALITIES USING CONVEX COMBINATION 43 If v/ ether ϕ -cove o rg J ϕ ( ) for ome ϕ () J ad creag o I or ϕ -cocave o rg J ϕ ( ) ad decreag o I, the the equalty M ( ) M ( ; ), (4.2) ϕ ; v/ hold for all cove combato from J atfyg ϕ( ) ϕ( ) ad oly f the equalty f M (, y;, q) M (, y;, q), (4.3) ϕ v/ hold for all bomal cove combato from J atfyg ϕ ( ) qϕ( y) ϕ(). If v/ ether ϕ -cove o rg J ϕ ( ) for ome ϕ () J ad decreag o I rg or ϕ -cocave o J ϕ ( ) ad creag o I, the the revere equalte are vald (4.2) ad (4.3). Proof. We rove the cae whch the fucto v/ ϕ -cove o the terval rg J ϕ ( ) ad creag o the terval I. Put f v/ ϕ. Frt, f we aly DRCF-theorem to the fucto rg o the terval J ( ), ϕ the we have f : J R cove f ϕ( ) f ( ϕ( )), hold for all cove combato from J atfyg ϕ( ) ϕ( ) ad oly f f f ( ϕ ( ) qϕ( y) ) f ( ϕ( ) ) qf ( ϕ( y) ), hold for all bomal cove combato from J atfyg ϕ ( ) qϕ( y) ϕ().

44 ZLATKO PAVIĆ Secod, after alyg the creag fucto /v o the above equalte, t follow: M ϕ ( ; ) ϕ ( ) v v( ) v/ ( ; ), ϕ / / M hold for all bomal cove combato from J atfyg ϕ( ) ϕ( ) f ad oly f (, y;, q) ϕ ( ϕ( ) qϕ( y) ) v/ ( ϕ( ) qϕ( y) ) M (, y;, q), M ϕ v/ hold for all, y I atfyg ϕ ( ) qϕ( y) ϕ( ). 4.2. Itegral cae Let A I be a µ -meaurable et wth µ ( A ) >, ad ϕ : I R be a trctly mootoe cotuou fucto that µ -tegrable o A. The tegral ϕ -qua-arthmetc mea of the et A wth reect to the meaure µ the ot (, µ M ) ϕ ϕ() µ (). µ ( ) ϕ A t d t (4.4) A A If A the terval, the t ϕ -qua-arthmetc mea M ϕ ( A, µ ) belog to A becaue the ot ϕ() t dµ () t µ ( A ) belog to ϕ ( A). If A ot A coected, the M ( A µ ) may be outde of A. If ϕ, the the ϕ, tegral I -qua-arthmetc mea of A jut the baryceter B ( A, µ ). Theorem for rght covety ad rght cocavty for qua-arthmetc mea the tegral cae a follow: Theorem 4.2 (Itegral cae of rght covety ad cocavty for qua-arthmetc mea). Let µ be a cotuou fte meaure o I that I

INEQUALITIES USING CONVEX COMBINATION 45 otve o the terval. Let mootoe cotuou fucto, ad J ϕ( I ). If v/ ether ϕ -cove o ϕ, v / : I R be µ -tegrable trctly rg J ϕ ( ) for ome ϕ () J ad creag o I or ϕ -cocave o rg J ϕ ( ) ad decreag o I, the the equalty hold for all terval f ad oly f the equalty hold for all bouded terval If v/ ether M ( B, ) M ( B, µ ), (4.5) ϕ µ v/ B I atfyg ϕ() t dµ () t ϕ(), (4.6) B M ( A, ) M ( A, µ ), (4.7) ϕ µ v/ A I atfyg ϕ() t dµ () t ϕ(). (4.8) µ ( A) A ϕ -cove o rg J ϕ ( ) for ome ϕ () J ad decreag o I rg or ϕ -cocave o J ϕ ( ) ad creag o I, the the revere equalte are vald (4.5) ad (4.7). Referece [] P. S. Bulle, D. S. Mtrovć ad P. M. Vać, Mea ad ther Iequalte, Redel, Dordrecht, NL, 988. [2] V. Crtoaje ad A. Baeu, A eteo of Jee dcrete equalty to half cove fucto, Joural of Iequalte ad Alcato 2 (2), 3; Artcle. [3] J. L. W. V. Jee, Om koveke Fuktoer og Ulgheder mellem Mddelværder, Nyt Tdkrft for Matematk. B. 6 (95), 49-68. [4] J. L. W. V. Jee, Sur le focto covee et le égalté etre le valeur moyee, Acta Mathematca 3 (96), 75-93.

46 ZLATKO PAVIĆ [5] J. Mćć, Z. Pavć ad J. E. Pečarć, The equalte for quaarthmetc mea, Abtract ad Aled Aaly 22 (22), 25; Artcle ID 2345. [6] C. P. Nculecu ad L. E. Pero, Cove Fucto ad ther Alcato, Caada Mathematcal Socety, Srger, New York, USA, 26. [7] Z. Pavć, J. E. Pečarć ad I. Perć, Itegral, dcrete ad fuctoal varat of Jee equalty, Joural of Mathematcal Iequalte 5(2) (2), 253-264. [8] J. E. Pečarć, F. Procha ad Y. L. Tog, Cove Fucto, Partal Orderg, ad Stattcal Alcato, Academc Pre, New York, USA, 992. [9] W. Rud, Real ad Comle Aaly, McGraw-Hll, New York, USA, 987. g