IMPROVED GA-CONVEXITY INEQUALITIES

Transcription

1 IMPROVED GA-CONVEXITY INEQUALITIES RAZVAN A. SATNOIANU Corresodece address: Deartmet of Mathematcs, Cty Uversty, LONDON ECV HB, UK; e-mal: web: Abstract We cosder a class of algebrac equaltes for fuctos of varables deedg o arameters that geeralse the case of GA-cove fuctos. The fuctos ths class are GA-cove oly a subdoma of defto yet the equalty for GAcovety stll holds o the whole doma f sutable codtos are satsfed by the arameters. The method s elemetary ad allows us to gve further etesos to a large class of fuctos. As a alcato we show the valdty of a -dmesoal geeralzato of a cojectured equalty related to a roblem gve at the 4 d IMO held at Washgto DC (USA. Key words: cove fuctos, GA-covety, symmetrc fuctos, IMO cometto. Mathematcs Subject Classfcato: 6D5, 6D, 39B6, 5A4.. Itroducto The roerty of covety of a gve fucto f : I R J R s oe of the most owerful tools establshg a wde rage of aalytc equaltes. As show [] deedg of whch tye of arthmetc (A or geometrc (G mea we cosder resectvely o the doma ad the codoma of defto for f four classes of cove fuctos are dstgushed. These are the AA-covety (the usual cove fuctos, AG, GA or GG-covety. Although a more geeral settg ca be aled the followg, due to the geometrc mea we shall assume throughout that I, J (,. To be secfc, AG cove fuctos (or log-cove fuctos are those fuctos f : I (, such that α α (( α + αy f ( f ( y, y I, α f (. Ths s equvalet that log f s cove.

2 GG-cove fuctos (or multlcatvely cove fuctos are those fuctos f : I (, such that α α α α ( y f ( f ( y, y I, α f (. Fally, GA-cove fuctos are those fuctos f : I (, such that, y I, α f α α ( y ( α f ( + α f ( y. (.3 As ca be checked radly every secod order dfferetable fucto satsfyg f ''( + f '( o ts doma (.4 s GA-cove. I artcular ths s true f f s a cove ad creasg fucto. I [] C.P. Nculescu dscussed the beautful class of equaltes, whch arse from the oto of GG-covety for fuctos. Clearly, a smlar le of qury ca be followed to aalyse the class of equaltes arsg by cosderg the remag tyes of covety such as GA ad AG-covety. I ths aer we wsh to eted the case of GAcovety for secod order dfferetable fuctos for whch equalty (.4 s ot satsfed ther etre doma of defto. Clearly to do so there must be some etra codtos mosed. Here we establsh such codtos for the case whe the fuctos deed also o etra arameters that obey gve costrats. These cases lead us to a geeralsato of the GA-covety mlctly furshg aalytc equaltes, whch caot be establshed by the use of a drect method such as (.4. Moreover, these results ca rcle be eteded to the other tye of mea-covety dscussed above. I the frst art we reset the geeral result. As a llustrato we establsh a -dmesoal geeralsato of a algebrac roblem, whch for the artcular case of three varables, has aeared as a cojecture relato to a roosed roblem at the 4 d IMO held Washgto DC, USA []. The three varable cojecture has also aeared recetly as roosal 944 the Amerca Mothly [3].. The ma result Suose that f : (, (, s secod order dfferetable fucto wth f ''( o ts doma. Let g : (, (,, g ( = f ' '( + f '(. Suose that there s > r > wth g(r = such that g < o (, r ad g o ( r,. Further cosder h : (, (, by h(,..., = f ( k f (, for all,... > wth k = k k= =. (.

3 Fally assume that the crtcal ots of h subject to (. ca take at most two dfferet values. That s there are a b such that {,..., } = { a, b} at the crtcal ots of h. have Theorem. If the above codtos are satsfed ad, for every k =,,, we lm h(,..., k (. k k= the h (,..., for all,... > wth =. Proof. Frst ote that h s a cotuous fucto defed o a bouded set from below therefore there s a value m k k= = > such that h m for all,... wth >. We shall show that m whch wll rove the theorem. Moreover, from (., ths s certaly true alog the boudary of the doma,.e. the lmt whe for some k =,,. Let k K = (,...,,..., >, k =. To ed the k= roof t remas to establsh the asserto the teror of K. To do so we shall look at the etremum ots of h. These are foud from the crtcal ots, whch by hyothess ca take at most two dfferet values. That s h =, =,..., { },..., = { a, b } at the crtcal ots. Due to symmetry we ca assume wthout restrcto that... ad a b. Therefore there ests some = q such that =... = q = a ad q = q+ =... = = b (whe = q = we use the coveto that. Note that (. mles that b. Also ote that f q = the there s othg to rove as ths case the cocluso follows by alyg codto (.4 to the mmum ot (or drectly va (.. Net cosder h ( a, b = ( q f ( a + ( q + f ( b f ( (.3 Note that va (. we have that q q+ a b = (.4 We shall show that h ( a, b for all a, b > satsfyg (.4. Va (.4 ths s equvalet to showg that 3

4 + q q h ( b = ( q f b + ( + q f ( b f ( (.5 + q q q b = ( h' b b A smle calculato gves that h '( b = ff h '. Because b ad f ' s creasg the last equalty s ossble oly whe b = whch case (.5 h whch follows from q becomes a equalty. Moreover '' ( = ( f '' ( + f '( codto (.4 aled to g at = ad the fact that r <. Ths shows that b = s a mmum ot for h ad that (.5 s true at ths ot. Therefore t s true for all other ots b. Fally, ths establshes that the asserto s true at the mmum ots of f ad cosequetly ths roves that the cocluso s true for all the teror ots of the doma K. We have already verfed t o the boudary of K so the roof s fshed. 3. A alcato I a recet ote [4] we gave a soluto to a cojectured equalty three ostve varables whch tur s a geeralsato of the d roblem gve at the 4 d IMO held at Washgto DC (USA []. The statemet of the IMO roblem was: Problem. Prove that a a b c bc b + 8ca c + 8ab (3. for all ostve real umbers a, b ad c. At the ed of the offcal IMO soluto the author of the above roosed roblem cojectured the followg more geeral equalty: Cojecture. For ay a, b, c > ad λ 8, the followg equalty holds a a b c λ bc b + λca c + λab + λ (3. Usg a drect calculatory method [4] we establshed the valdty of (3.. The same equalty has also bee recetly ublshed as a roosal AMM [3]. Recetly we leared about a algebrac soluto to (3. that was obtaed by Sava Grozdev (the team 4

5 leader of the Bulgara IMO team [5]. However, hs soluto s very artcular to the case of three ostve umbers ad so caot be eteded to the geeral case of varables. I ths drecto we have roosed [4] the followg eteso of (3. to the -dmesoal case. Cojecture. = ( + λ (3.3 + λ k k for all, >, =,, ad ay λ. Iequalty (3.3 has attracted terest [6]. I [6] Lagrage's method s used to show the valdty of (3.3 but aga the method s ot ameable to further geeralsato. Here we shall show that Cojecture follows aturally from our ma result above. However, before we do ths t s useful to arecate the stregth of (3.3. Frst oe ca roceed as [] ad elot the roerty that the left had sde (3.3 s homogeeous the - varables. Therefore wth the atural trasformato reduce the roblem to showg that Theorem. y k =, =,,, oe ca λ y λ = + + (3.4 for all ad y >, =,, wth the roerty y = ad ay λ. = There are some obvous suggestos to tackle (3.4. A ave aroach would be to aly the AM-GM equalty whch would gve that the AM of the left had sde of (3.3 s larger tha ( + λ /( ( + λ = /( (. However, the last eresso s less tha rather tha bgger to t (whch s what we would have eeded order to obta (3.4 as ca be easly checked by alyg oce more the AM-GM equalty. Drect use of covety roertes does ot aear too sred ether. For eamle the fucto geeratg the geeral term of the left had sde (3.4 s cove. So Jese's equalty 5

6 yelds that the left had sde (3.4 s larger tha = /( + ( λ /. However, the AM-GM equalty wth y = yelds that ( /( + + ( λ/ = /( λ so (3.4 caot be establshed ths smle way ether. Note that whe =, (4 s trval ad for = 3 the valdty of (3.4 was establshed [4,5] as dscussed above. I ths ote we shall establsh the valdty of equalty (3.4 geeral. Proof of Theorem. The cases =, are mmedate ad we leave them as a eercse for the reader to attemt. I the followg we shall dscuss the case whe >. For ay > ad, λ > /( let ( = ( + = f λ. It s easy to see that f s a decreasg ad cove fucto of > for ay λ >. Ideed we have that: f ' /( ( + λ f ( = λ ( < (3.5 '' ( /( ( + λ > ( = λ ( (3.6 for all > ad for ay λ >, >. Furthermore t s easy to see that f '' ( + f ' ( /( ( + λ ( = λ ( + λ (3.7 λ Therefore o the terval J =, f s GA-cove. From the hyothess we also λ have that y =. (3.8 = Therefore (3.4 becomes hh( y, y,..., y = f ( y f ( (3.9 k = for all < y, =,..., satsfyg (3.9 ad all λ. Now t s easy to see that the crtcal ots of hh subject to codto (3.8 must satsfy the equaltes d y = d ( y =... = d ( y, where ( y d( y = ( + λy /(. Now d s 6

7 strctly mootoous o each of J ad R J so we deduce that the crtcal ots of hh (3.9 ca atta at most two dfferet values, let say a ad b. Moreover, (3.8 gves b. At ths stage we see that, wth the ossble eceto of codto (., all the hyothess of Theorem are satsfed our case ad so the cocluso follows for all the teror ots of the doma. We stll eed to check the behavour o the froter of the doma, that s the behavour of (3.9 whe a or (equvaletly b. Because lm f ( a = lm f ( b = b /( we have to check that f ( = ( + a q λ whch s obvously true owg to the codto that λ. Equalty takes lace whe q = ad = λ. Ths verfes also that hyothess (. holds our case. These facts the establsh equalty (3.9 for all crtcal ots ad λ. Therefore the roof fshes by alyg theorem. y >, =,...,, Theorem 3. For ay α, β > wth β ( α we have the equalty = α + β k k ( + α β (3. The roof follows easly from Theorem a smlar maer as the roof of Theorem. A sgfcatly eteded verso of Theorem ca fact be establshed. / / Theorem 4. ( + y ( + λ = λ (3. for all, >, y >, =,, such that y = ad ay λ. The roof of ths geeral equalty s absolutely smlar to that Theorem. I fact t ca be doe almost ad ltteram by relacg the eoet (- by the argumets used the roof of Theorem. Theorems ad 4 also mly the valdty of the followg dual form of (3.3 = 7

8 Theorem 5. = / k k + ( α λβ (3. k + λ k for all, >, >, =,,, α, β > ad ay λ. Proof. (3. follows from (3.- va the trasformato /, =,...,. Corollary. If α, β > wth β ( α the = /( /( ( α + β ( α + β (3.3 for all, >, =,, such that =. = Proof. I (3.3 multly both the deomator ad the umerator of each term from the left had sde by, =,...,, resectvely. Refereces [] C.P. NICULESCU, Covety accordg to the geometrc mea, Math. Ieq. ad Alc., 3 (, [] Problem, 4 d IMO Washgto DC (, see htt://mo.wolfram.com/roblemset/imo_soluto.html [3] MARCIN MAZUR, Proosal 944, Amerca Mathematcal Mothly, 9 (, 475 [4] RAZVAN A. SATNOIANU: The roof of the cojectured equalty from the 4 d Iteratoal Mathematcal Olymad, Washgto DC, Gazeta Matematca 6 (, [5] SAVA GROZDEV, rvate commucato,. [6] WALTHER JANOUS, O a cojecture of Razva Satoau, rert,. 8