GLASNIK MATEMATIcKI Vol._18 (38) (1983), 77 8S. ON THE PETROVIC INEQUALITY FOR CONVEX FUNCTIONS J. E. Pecaric, Beograd Abstr~t. I this paper we give some geeralizatios of wel1-kow Petrovic's iequality for covex fuetios. M. Petrovic [14] (see also [10, p. 22]) has proved the followig theorem: THEOREM A. If fis a covex fuctio o the segmet I ~ [O, al, tl XI E I U = 1,,.,' ) ad XI +... + X El, the f(xi) +...+f(x) ~ f(xl +...+ X) + ( - l)f(o). (1) T. Popoviciu [15] has cosidered the more geeral iequality ad he ohtaied that (2) holds for Pi > O ad XI ~ O(i = 1,..., ). P. M. Vasic [17] (see also [18]) showed that the result of T. Popoviciu is ot valid, ad proved that (2) h01ds if pl ~ 1 ad XI ~ O (1 ~ i ~ ). F. Giaccardi [7] has geeralized the iequalities (1) ad (2). The coditios for validity of such a iequality were weakeed by P. M. Vasic ad Lj. Stakovic [19] (see also [9]). Their result is the followig theorem: THEOREM B. Let Pl ~ O ad XI (i = 1, ""' ) be real umbers which satisfy (XI - xo) ( r Pk Xk - Xi) ~ O (i = 1,..., ), ~ Pk Xk :fi Xo. (3) k=l k=1 The, for every covex fuctio f, Mathematic$ subject classificcititms (1980): Primary 26 D 99. Ke, fljordsad phrases: Covex fuetio, Petrovic's iequality.
78 J. E. Pecaric where " "" II A = ( ~ P, X, - Xo ~ P,)!( ~ P, X, - Xo), B = ~ Pi Xi!( ~ P, Xi - 1=1 i=l i=l Xo). If Xo = O, the (4) 'reduces to (2). (5) I [19], the followig resu1t was proved: THEOREM C. Let Xi' P, (i = 1,..., ) be real umbers such that II -I X ( ~ P, XI _ X) ~ O, P ~ O, X ~ P, X, ~ O, ~ Pi Xi :f:. O. (6) The, lor every covex luctio I, where P (p, x;f) ~ P-1 (p, x;f) (7) II II II P (p, x;f) =/( ~ P, Xi) - '2..Pt.!(Xi) + ( ~ P! - 1)/(0).. For some further geeralizatios of the iequalities (2) ad (4) see [8], [11], [17] ad [19]. I. 01ki [12] (see also [4], [5]) has proved the followig result: THEOREM D. Let 1 ~ hi ~... ~ h ~ O ad al ~...~a ~ ~ O. Let I be a covex luctio o [O, ad. The Resu1ts, which prevised this iequality ([2], [3], [16], [20], [21]), are give i [10, pp. 112 ad 113]. R. E. Barlow, A. W. Marshall ad F. Proscha [1] have proved a geeralizatio of Theorem D (see Remark' 2). I [13] the followig ge.eralizatio of Theorem D was give: 'theorem E. Let the real umbers Ph ''''P ad Xl'~. ~ ~ X ~ Xo ~ O satisly the coditios (8) O ~ P" ~ 1 (k = 1,..., ), ~ PI Xi > Xo (if Xo > O), k where P" =.~P" 11I is a covex luctio, the the reverse iequahty. i (4) ho/ds.. 11
O the Petr()vic iequa1ity for covex fuctios 79 I this.paper we shall prove, startig with the Fuchs geeraw lizatio of the Majorizatio theorem (see [6]) some geeralizatios of the previous results. Fuchs' result ca be writte i the followig form: THEOREM F..-Let al ~... ~ a$' h~...~b$ ad qu..,qs be real umbers such that. hold. The, for every co.vexfuctio f, the iequality is valid. Now, we shali prove the followig theorem: (9) THEOREM 1. Let Xl.~. ~ Xm ~ xo ~ Xm+l ~ ~ XII (m E E (O, 1,..., )) ad PU..., Pil be real umbers such that X, ~ I (i = = O, 1,... ; ; I is a iterval i R), ~ Pi X, E I ad k ~ Pi ( ~ Pr Xr -,=1 Xi) ~ O cl ~ k~ m), ~ P, ( ~ Pr ~r - i=k.,=1 Xi) ~ O (m + 1~ k ~ ). (10) The, for every covex fuctio f o I, the iequality (4) hblds~11 the reverse iequalitt'eshold i (10), the the reverse ~'equality~'(4} holds. Proof. By substitutios s=+i; aj =X,, qj=pj(i= 1,.,:m); am+l=xo, qm+l=b(l-p,.); al = X'-l' ql = P'-l (i = m + 2,..., + 1); b, = ~ p,x, (i =.1,...,. + 1); ad " s=+i; a,= ~P,x, (i=i,...,+l);'b,=x" q, ' p,(i~ = 1,., m); bm+1= xo, qm+l = J3 (1 - P,.); b, = x,-u ql = P'-l (i = m +2,..., + i); from Theorem F, we g~t Theorem 1.
80 J. E. Pecaric Remaik 1. For Xl ~ ~ X ~ Xo the coditios (10) 'be~ome ad for Xo li?; XI li?; li?; X k ~ p, ( ~ P,X, - XI) ~ O (k = 1,.., ), (11) i l,,,,,1 ~ P, ( ~ P, x, - X,) ~ O (k -:. 1,..., ). (12) i=k T""1 If P,li?; O(i = 1,..., ), the coditios (11) ad (12) ca be replaced o by the coditios ~ P, X, li?; XI ad ~ P, X, ~ X, which are com- 1=1 plete1y equivalet to the coditio (3). From Theorem 1, for Xo = O we get: THEOREM 2. Let the real umbers X 1 li?; li?; X m li?; O ~ li?; xm+1 li?;.. li?; X (m E (0,1,.,» ad Pu...,P satisfy the co. ditiem x, El (i = 1,..., j O E 1), ~ P, X, El ad (10). The, for i=i every covex luct~'o I o 1, the iequality (2) holds. II the reverse iequalities i (10) hold, the the reverse iequality i (2) is 'Valid. Now, we shall give' some speciai cases of Theorems 1 ad 2. THEOREM 3. Let Xl li?;. li?; Xm li?; Xo li?; Xm+1 li?;.. li?; X (m E E (O, 1,.,,'» be real umbers such that XI E I (i = O, 1,., ), Xm li?; O ad Xm+1 ~ O. (a) 1/ P is real -tple such that O ~ P" ~ 1 (k = 1,..,' m), O ~ ~ ~ 1 (k = m = 1,..., ) (p" = P - P"-l)' (13) the for every covex fuctt'o f :1 ~ R, the reverse iegua1ity i (4) helds. or (b) 1/ ~ P, X, E 1ad il either 1=1 the (4) helds. P" li?; 1 (k = 1,, m), P" ~ O (k = m + 1,., )j (14) P" ~ O (k = 1,., m), P" li?; 1 (k = m + 1,..., ); (15)
O the Petrovic iequality for covex fuctios 81 Proo!. (a) For k = 1,..., m we have k k-l ~ Pi ( ~ Pr Xr - Xi) = ( ~ Pr Xr - Xk) Pk + ~ Pi (XI+l - Xi) = r=1 r=1 m-l = Pk (Xm P - Xk + ~ Pi (Xi - Xi+l) + ~ ii; (XI - Xi-I)) + ;=m+l k-l + ~ Pi (Xi+! -Xi) =Pk(XmP -Xk) + (1-Pk) ~ PI (Xi+l -Xi) + m-l + Pk ~ Pi (Xi - Xi+l) + Pk ~ ii; (Xi - Xi-I) = ;=k ;=m+l k-l = Pk (P - 1) Xm + (1 - Pk) ~ Pi (XI+l - Xi) + m-l + Pk ~ (Pi-l)(Xj-Xi+l)+Pk ;=k ~ ii;(xi-xi-l)=pk(pm+lxm+l ;=m+l + k-l + (Pm - 1) Xm) + (1 - Pk) ~ Pi (Xi+l - Xi) + m-l + Pk ~ (Pi - 1) (Xi - XI+l) + Pk ~ ii; (XI - XI-I) ~ O. j=k ;=m+2 Aalogously, for k = m + 1,..., we have ~ Pi ( ~ PrXr - XI) = Pk(Xm+l (Pm+1-1) + xmpm) + ;=k r=1 k-l m Usig (13) we also have XI;?; ~ Pi Xi ;?;O ad O;?; ~ PI Xi ;?; ;=m+l ;?;X wherefrom we have XI ;?; ~ Pi Xi ;?;X Usig the above idetities we ca prove (b). From Theorem 3, for Xo = O we get: THEOREM 4. Let XI ;?; o ;?;Xm ;?;O;?; Xm+l ;?;... ;?;X (m E E (O, 1, o, " )), Xi E I (i = 1, ",' ) ad Pl'. o o, P be real umbers.
82 J. E. Pecaric (i) If (13) holds, the for every covex fuctio f : I -7- R the re- ~'erse iequality i (2) is valid. (ii) If the coditios of Theorem 3 (b) are fulfilled, the (2) holds. Remark 2. I [1], the fol1owig result was give: Let XI ~ ~ Xm ~ O ~ Xm+l ~ 00. ~ X". (a) Reverse iequality holds i (2), if ad oly if (13) holds. (b) (2) holds if ad oly if there exists j ~z such that Pi ~ O, i <j; Pi ~ 1, j ~z ~ m, ~ ~ O, i ~m + 1, or there exists j ~ m such that Pi ~ O, i ~ m; Pi ~ 1, 111 + 1 ~ i ~j; Pl ~ O, i >j. For j = 1 ad j =, from (b), we get (ii) from Theorem 4. Usig Theorem 2, we ca exted the coditios uder which the iequality (7) holdso Namely, from it we obtai that iequality f (ql al + q2 a2) - qi!(al) - qd(a2) + (ql + q2 - l)f(o) ~ O (16) holds if: 01' O ~ al ~ a2 ad q2 (ql al + q2 a2 - a2) ~ O, ql (ql al + q2 a2 - al) + q2 (ql al + (17) (18) ql (ql al + q2 a2 - al) ~ O, q2 (ql al + q2 a2 - a2) ~ O. (19) The reverse iequality i (16) holds if i (17), (18) ad (19) the reverse iequalities holds. -I Usig the substitutios: al = XII' a2 = ~ Pi Xi' ql = Pil' q2 = 1; -I ad al = ~ Pl Xi' a2 = XII' ql = 1, q2 = Pil' we get that (7) holds if 01' -I ~ Pi Xi ~ XII ~ O, ~ Pi Xi ~ ~ Pi Xi ~ O; (20) -I XII ~ O ~ ~ Pi Xi' XII ~ ~ Pi Xi ~ ~ Pi Xi; (21) 01' if either of the coditios (20) ad (21) hold, with the reverse iequalities.
O the Petrovic iequality for eovex fuetios 83 or The reverse iequality i (7) holds if -I -I ~ Pi Xi ~ X ~ O, ~ Pi Xi = ~ Pi Xi ~ O; -I JZ Iz-l X ~ O ~ ~ Pi Xi' X ~ ~ Pi Xi ~ ~ Pi Xi; ;~1 (22) (23) or if either of the coditios (22) ad (23) hold, with the reverse iequalities. By combiig the coditio (21) ad the correspodig coditio with the reverse iequalities, we obtai (6). The author is grateful to Professors 1. 01ki ad P..M. Vasic for the useful suggestios. REFERENCES: [1] R. E. Bal'lo'ilJ, A. W. iviars!zall ad F. Prosc!zall, Some iequalities for starshaped ad eovex fuetios, Pacif. ]. Math. 29 (1969), 19-42. [2] R. BelimalJ, O a iequality of Weiberger, Amer. Math. (1953), 402. Mothly 60 [3] M. Boeracki, Sur les iegalities remplies par des expressios dot les termes ot des siges alteres, A. Uiv. Mariae Curie-Sklodowska 7 A (1953), 89-102. [4] H. D. Bruk, O a iequality for eovex fuetio, Proe. Amer. Math. Soe. 7 (1956), 817-824. [5],Itegral iequalities for fuetio with odeereasig ieremets, Pacif. J. Math. 14 (1964), 783-793. [6] L. Fuchs, A ew proof of a iequality of Hardy-Littlewood-P6Iya, Mat. Tidsskr. B. 1947, 53-54. [7] F. Giaccardi, Su alcue disuguagliaze, Gior. Mat. Fiaz. 1 (4) (1955), 139-153. [8] J. D. Keckic ad l. B. Lackovic, Geeralizatio of some iequa1itiesof P. M. Vasic, Mat. Vesik 7 (22) (1970), 69-72. [9] P. Lah ad M. Ribaric, Coverse of Jese's iequality for eovex fuetios, Uiv. Beograd. Pub1. Elektroteh. Fak. Ser. Mat. Fiz. No 412-No 460 (1973), 201-205. [10] D. S. Mitriovic (I eooperatio with P. M. Berli-Heidelberg-New York, 1970. Vasic), Aalytie Iequalities, [11] A. W. Marshall, 1. Olki ad F. Proseha, Mootoicity of ratios of meas ad other applicatios of majorizatio, I: Iequalities, edited by O. Shisha. New York-Lodo 1967, 177-190. [12] l. Olki, O iequalities of Szego ad Bellma, Proe. Nat. Acad. Sci. USA 45 (1959), 230-231. [13] J. E. Pecaric, O the Jese-Steffese iequality, Ui\". Beograd. Publ. Elektroteh. Fak. Ser. Mat. Fiz. No 634-No 676 (1979).
84 J. E. Pecaric [14] M. Petrovic, Sur ue foetioelle, Publ. Math. Uiv. Belgrade 1 (1932), 149-156. [15] T. PopoviciII, Les foetios eovexes, Aetualites Sci. Id. No 922, Paris, 1945. [16] G. Szeg6, Uber eie Verallgemeierug des Dirieh1etsehe Itegrals, Math. Z. 52 (1950), 676-685. [17] P. M. Vasic, O jedoj ejedakosti M. Petrovica za kovekse fukcije, Matematicka biblioteka, No 38 (1968), Beograd, 101-104. [18], Sur ue iegalite de M. Petrovic, Mat. Vesik 5 (20) (1968),473-478. [19] ad Lj. Stakovic, O some iequalities for eovex ad g-eovex fuetios, Uiv. Beograd. Publ. Elektroteh. Fak. Ser. Mat. Fiz. No 412-No 460 (1973), 11-16. [20] H. F. W'eiberger, A iequa1ity with alteratig sigs, Proe. Nat. Acad. Sci. USA 38 (1952), 611-613. [21] E. M. l'('rigllt A iequality for eovex fuetios, Amer. Math. Mothly 61 (1954), 620-622. (Reeeived Mareh 7, 1979) (Revised August 5, 1979 ad Jue 13, 1980) Gradeviski fakultet pp 895 11000 Beograd, Yugoslavia o PETROVICEVOJ NEJEDNAKOSTI ZA KONVEKSNE FUNKCIJE J. E. Pecaric, Beograd Sadržaj U claku je dokazaa slijedeca geeralizacija pozate Petroviceve ejedakosti za kovekse fukcije TEOREMA 1. Neka su X1 ~ ~ Xm ~ Xo ~ Xm+1 ~ ~ X (m E (0,1,..., )) i Pl"'" P takvi reali brojevi da XI El (i = O, 1,..., ), LPIXI El i da važi (10). Tada za svaku fukciju f koveksu a l važi 1=1 (4), gdje su A i B defiisai sa (5). Ako u (10) važe suprote ejedakosti, tada suprota ejedakost važi i u (4). U specijalom slucaju dobija se: TEOREMA 3. Neka su X1 ~ ~ Xm ~ Xo ~ Xm+l ~. ~ X (m E (O, 1,..., )) takvi reali brojevi da XI El (i= O, 1,..., ), Xm ~ O, Xm+ 1~ O. (a) Ako je P takva reala -torka da važi (13), tada za svaku koveksu fukciju f : l ~R, važi suprota ejedakost u (4)
O the Petrovic iequality for covex fuctios 85 (h) Ako ~ P, X, E I i ako važi ili (14) ili (15) tada važi (4). Za Xo = O, iz Teoremali 3 dobijaju se respektivo Teoreme 2 i 4, a takode su dobijei eki rezultati koji se odose a rafiiraje Petroviceve ejedakosti.