L Inequalities for Polynomials

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1 Aled Mathematcs 3-38 do:436/am338 Publshed Ole March (htt://wwwscrorg/oural/am) L Iequaltes for Polyomals Abstract Abdul A Nsar A Rather Deartmet of Mathematcs Kashmr Uversty Sragar Ida E-mal: drarather@gmalcom Receved July 9 ; revsed Jauary 4 ; acceted Jauary 7 I ths aer we cosder a roblem of vestgatg the deedece of PR Pr o P for every real or comlex umber wth R > r > ad reset certa comact geeralatos whch besdes yeldg some terestg results as corollares clude some well-kow results artcular those of Zygmud Berste De-Bru Erdös-Lax ad Boas ad Rahma as secal cases Keywords: L -Iequaltes Polyomals Comlex Doma Itroducto Let P = = P defe ad deote the sace of all comlex olyomals a of degree at most For P P P := Pe < P = := max P A famous result kow as Berste s equalty(for referecesee[] or []) states that f P P the P P () whereas cocerg the maxmum modulus of P o the crcle = R > we have P R R P () (for referece see [3]) Iequaltes () ad () ca be obtaed by lettg the equaltes ad P P (3) > > P R R P R (4) resectvely Iequalty (3) was foud by Zygmud [4] whereas equalty (4) s a smle cosequece of a result of Hardy [5] (see also [6]) Sce Iequalty (3) was deduced from MRess terolato formula [7] by meas of Mkowsk s equaltyt was ot clear whether the restrcto o was deed essetal Ths questo was oe for a log tme Fally Arestov [8] roved that (3) remas true for < < as well Both the Iequaltes (3) ad (4) ca be shareed f we restrct ourselves to the class of olyomals havg o ero < I fact f P P ad P < the Iequaltes (3) ad (4) ca be resectvely relaced by P P (5) ad R PR P R > > (6) Iequalty (5) s due to De-Bru [9] for ad Rahma ad Schmesser [] exteded t for < < whereas the Iequalty (6) was roved by Boas ad Rahma [] for ad later t was exteded for < < by Rahma ad Schmesser[] For = the Iequalty (5) was coectured by Erdös ad later verfed by Lax [3] whereas Iequalty (6) was roved by Akey ad Rvl [4] Recetly the Authors [] (see also [5]) vestgated the deedece of o P R P P for R > As a comact geeralato of Iequaltes (3) ad (4) they have show that f P P the for every R > ad Coyrght ScRes

2 3 A AZIZ ET AL P R P R P (7) It s atural to seek the corresodg aalog of (7) for olyomals P P havg o ero < ad whch s a comact geeralato of Iequaltes (5) ad (6) I the reset aer we cosder a more geeral roblem of vestgatg the deedece of o P R P r P for every real or comlex umber wth R > r > ad develo a ufed method for arrvg at these results We frst reset the followg terestg result ad a comact geeralato of Iequaltes (3) ad (4) whch also exteds Iequalty (7) for < < as well Theorem If P P the for every real or comlex umber wth R > r ad > P R P r R r P (8) The result s best ossble ad equalty (8) holds for P= a a Remark For = Theorem reduces to Iequalty (4) ad for = r = t valdates Iequalty (7) for each > If we set = Iequalty (8) we mmedately get the followg geeralato of Iequalty (7) Corollary If P P the for R > r ad > P R P r R r P (9) The result s best ossble ad equalty (9) holds for P= a a If we dvde the two sdes of Iequalty (9) by R r ad let R r we get: Corollary If P P the for r ad > P r r P () Remark For r = Corollary reduces to Zygmud s Iequalty (3) for each > The followg result whch s a comact geeralato of Iequaltes of () ad () follows from Theorem by lettg Iequalty (8) Corollary 3 If P P the for every real or comlex umber wth ad R > r P R P r R r max P for = = () The result s best ossble ad equalty () holds for P= a a Remark 3 For = Corollary 3 reduces to Iequalty () ad for = f we dvde the two sdes of () by R r ad let R r t follows that f P P the for r P r r max P for = () = Iequalty () reduces to Berste s Iequalty () for r = For olyomals P P havg o ero < we ext rove the followg terestg mrovemet of (8) whch amog other thgs clude De-Bru s theorem (Iequalty (5)) ad a result of Boas ad Rahma (Iequalty (6)) as secal cases Theorem If P P P does ot vash ad < the for every real or comlex umber wth R > r ad > R r P R P r P (3) The result s best ossble ad equalty (3) holds for P = a b a = b = For = Theorem reduces to Iequalty (6) A varety of terestg results ca be easly deduced from Theorem Here we meto a few of these The followg corollary mmedately follows from Theorem by takg = Corollary 4 If P P ad P does ot vash < the for R > r ad > R r PRPr P (4) The result s shar ad equalty (4) holds for P ( )= a ba = b= Remark 4 For r = f we dvde the two sdes of (4) by R ad let R we mmedately get De-Bru s theorem (Iequalty (5)) for each > Next we meto the followg comact geeralato of a theorem of Erdös ad Lax (Iequalty (5) for ) ad a result of Akey ad Rvl (Iequalty (5) for ) whch mmedately follows from Theorem by lettg (3) Corollary 5 If P P ad P does ot vash < the for every real or comlex umber wth ad R > r R r PR Pr max P = (5) for = The result s best ossble ad equalty (5) holds P a b a b for = = = Coyrght ScRes

3 A AZIZ ET AL 33 Remark 5 For = f we dvde the two sdes of (5) by R r ad let R r we get P r r max P for = (6) = For r = Iequalty (6) was coectured by Erdös ad later verfed by Lax[] If we take = (5)we mmedately get R PR P R > (7) Iequalty (7) s due to Akey ad Rvl [] A olyomal P P s sad to be self-versve f P = uq for all C where = u ad Q = P( ) It s kow[6 7] that f P P s selfversve olyomal the for every P P (8) Fally we reset the followg result whch clude some well-kow results for self-versve olyomals as secal cases Theorem 3 If P P s self-versve olyomal the for every real or comlex umber wth R > r ad > R r P R P r P (9) The result s best ossble ad equalty (9) holds for P= Remark 6 Takg = Theorem 3 t follows that f P P s self-versve olyomal the for R > ad > P R R P () The result s shar May terestg results ca be deduced from Theorem 3 exactly the same way as we have deduced from Theorem Lemmas For the roofs of these theorems we eed the followg lemmas Lemma If P P ad P has all ts eros k where k the for every R r ad = R k r k Pr P R Proof of Lemma Sce all the eros of k we wrte where have r Hece = P C r e = () P le k Now for < R r we Re re R r RrCos = re r r r rr e Cos R r R k = rr rk = = P Re Re r e P re re r e Rk Rk = = rk rk for < Ths mles for = ad R > r R k PR Pr r k whch comletes the roof of Lemma Lemma If P P ad P does ot vash < the for every real or comlex umber wth R r ad = P R P r Q R P r () where Q = P( ) The result s shar ad equalty () holds for P= Proof of Lemma For the case R = r the result follows by observg that P Q for Heceforth we assume that R > r Sce the olyomal P has all ts eros therefore for every real or comlex umber wth > the olyom al f = P Q where Q= P( ) has all ts eros Alyg Lemma to the olyomal f wth k = we obta for every R > r ad < R f Re f re (3) r Sce f Re for every R > r < ad R> r t follows from (3) that Coyrght ScRes

4 34 A AZIZ ET AL r f Re f Re f re R for every R > r ad < Ths gves > f r < f R for ad R > r Usg Rouche s theorem ad otg that all the eros of f R le < we coclude that the R olyomal T= f R f r (4) = PRP rq RQ r has all ts eros < for every real or comlex umber wth > ad R > r Ths mles P R P r Q R Q r (5) for ad R > r If Iequalty (5) s ot true the exst a ot = w wth w such that P RwP rw QRw Qrw But all the eros of Q le Qr le < Hece Qrw wth w We take follows (as case of Q R Q Rw > therefore t f ) that all the eros of Prw Qrw P Rw = Q Rw the s a well defed real or comlex umber wth > ad wth ths choce of from (4) we obta Tw = where w Ths cotradcts the fact that all the eros of T( ) le < Thus P R P r Q R Q r for ad R > r Ths roves Lemma Next we descrbe a result of Arestov For = ad P = a = P we defe P= a = The oerator s sad to be admssble f t reserves oe of the followg roertes: ) P has all ts eros C: ) P has all ts eros C: The result of Arestov may ow be stated as follows Lemma 3 [8] Let x= logx where s a covex odecreasg fucto o R The for all P P ad each admssble oerator P e d C P e d where C = ax I artcular Lemma 3 ales wth : x x for Therefore we have every P e d C P e d (6) We use (6) to rove the followg terestg result Lemma 4 If P P ad P does ot vash < the for every real or comlex umber wth R > r > ad real PRe Pre e R P e R r P e r d R r e P e d Proof of Lemma 4 Let = (7) Q P Sce P does ot vash < by Lemma for every real or comlex umber wth R > r ad =we have Pr P R = Q R Q r R P R r P r Now(as the roof of Lemma ) the olyomal H = Q R Q r = R P R r P r has all ts eros < for every real or comlex umber wth ad R > r t follows that the olyomal = H R P R r P r has all ts eros > Hece the fucto PR Pr f = RPR rpr s aalytc ad f for = Sce f s ot a costat t follows by the Maxmum Modulus Prcle that or equvaletly f < for < P R P r < R P R r P r for < (8) A drect alcato of Rouche s theorem shows that Coyrght ScRes

5 A AZIZ ET AL 35 = e R P R r P r P P R P r R r e a ( e R r a = does ot vash < for every wth R > r ad real Therefore s admssbe oerator Alyg (6) of Lemma 3 the desred result follows mmedately for each > Ths comletes the roof of Lemma 4 From lemma 4 we deduce the followg more geeral lemma whch s a result of deedet terest wth varety of alcato Lemma 5 If P P the for every real or comlex umber wth R > r > ad real PRe Pre e R P e R r P e r d R r e P e d (9) The result s shar ad equalty (9) holds for P= Proof of Lemma 5 Sce P s a olyomal of degree at most we ca wrte k P = P P = k = = k P le ad all the where all the eros of eros of P le < Frst we suose that P has o ero o = so that all the eros of P le > k Let Q= P ad Q = P the eros of Q le > for = Now cosder the olyomal k g = P Q = the all the eros of = = k the all g le > ad for = g = P Q = P P = P (3) By the Maxmum Modulus Prcle t follows that P g for (3) We clam that the olyomal h= P g does ot vash for every wth > If ths s ot true the h = for some wth Ths gves P g Sce g ad > = t follows that P > g wth whch clearly cotradcts (3) Thus h does ot vash for every wth > so that all the eros of h le for some > ad hece all the eros of h le Alyg h we get (8) to the olyomal hrhr < R h Rr h r for < R > r Takg = e < > ad we get the = < as h Re h re < R h e R r h e r < R > r ad Ths mles h R h r < R h R r h r for = A alcato of Rouche s theorem shows that the olyomal = T h R h r e R h R r h r does ot vash for every real or comlex umber wth R > r ad real Relacg h by P h t follows that the olyomal = g Rg r e R g R r g r T P R P r e R P R r P r (3) does ot vash for every wth ad > Ths mles P R P r e R P R r P r g Rg re R g Rr g r (33) for R > r ad real If Iequalty (33) s ot true the there s a ot = wth such that PRPr e R P Rr P r >( g R g r ) e R g R r g r Sce all the eros of olyomals t follows (as before) that all the eros of olyomal g R g r e R g R r g r also l- g le > Coyrght ScRes

6 36 A AZIZ ET AL e > for every real or comlex umber wth R > r ad real Hece grgr e R g Rr g r wth We take = PRPre R P Rr P r g R g r e R g R r g r so that s a well-defed real or comlex umber wth > ad wth ths choce of from (3) we get T = wth Ths clearly s a cotradcto to the fact that T does ot vash Thus for every wth R > r ad real g R g r e R g R r g r PRPre R P Rr P r for whch artcular gves for each > ad < PRe Pre gre gre e R P e R r P e r d e R g e R r g e r d Usg lemma 4 ad (3) t follows that for every wth R > r > ad real Pre P Re e R P e R r P e r d R r e g e d = R r e P e d Now f P has a ero o = * (34) to the olyomal = (34) the alyg P P t P where t < we get for every wth R > r > ad real * * * * P Re P re e R P e R r P e r d * R r e P e d (35) Lettg t (35) ad usg cotuty the desred result follows mmedately ad ths roves Lemma 5 3 Proofs of the Theorems Proof of Theorem Sce degree at most we ca wrte P s a olyomal of k P = P P = k = = k P le ad all the where all the eros of eros of P le > Frst we suose that all k the eros of P le < Let Q= P the all the eros of Q le < ad Q = P for = Now cosder the olyomal k F = P Q = the all the eros of = = k F le < ad for = F = P Q = P P = P (9) By the Maxmum Modulus Prcle t follows that Sce P F for F for ad > a drect alcato of Rouche s theorem shows that the olyomal H = P F has all ts eros < for every wth > Alyg lemma to the olyomal H we deduce (as before) H r < H R for = ad R > r Sce all the eros of H R le < R we coclude that for every wth ad > all the eros of olyomal G = HR Hr = PRP rf RF r le < Ths mles (as the case of Lemma ) PR Pr FRFr for ad R > r whch artcular gves for R > r ad > P Re P re d F Re F re d (3) Coyrght ScRes

7 A AZIZ ET AL 37 Aga sce all the eros of F le as before F R F r has all ts eros for every real or comlex umber wth Therefore the oerator defed by = F F R F r R r b b = s admssble Hece by (6) of Lemma (3) for each > we have F Re F re d R r F e d (3) Combg Iequaltes (37) ad (38) ad otg that = Pe F e we obta for R > r ad > P Re P re d R r P e d (3) I case P has a ero o = the Iequalty (39) follows by usg smlar argumet as the case of Lemma 5 Ths comletes the roof of Theorem Proof of Theorem By hyothess P P ad P does ot vash therefore by Lemma for every real or comlex umber wth < ad R > r Pre RPe R rpe r P Re Also by Lemma 5 where F e G d (33) (34) R r e P e d = ad F P Re P re G = R P e R r P e r Itegratg both sdes of (4) wth resect to from to we get for each > R > r ad real F e G dd R r e d P e d (35) Now for every real t ad > we have te d e d If F we take t = G F t ad we get F e G d the by (4) G = F e d F G = F e d F G = F e d F F e d For F = ths equalty s trvally true Usg ths (4) we coclude that for every real or comlex umber wth R > r ad real e d P Re P re d R r e d P e d Sce = R r e d = R r e d = R r e d R r e d = R r e d (43) (44) the desred result follows mmedately by combg (43) ad (44) Ths comletes the roof of Theorem Proof of Theorem 3 Sce P s a self-versve olyomal we have = where u = ad = P uq for all C Q P Therefore for every real or comlex umber ad R > r P R P r = Q R Q r for all C so that G F Pre P Re = RPe R rpe r = Coyrght ScRes

8 38 A AZIZ ET AL Usg ths (4) wth ad roceedg smlarly as the roof of Theorem we get the desred result Ths comletes the roof of Theorem 3 4 Refereces [] G V Mlovaovc D S Mtrovc ad T M Rassas Tocs Polyomals: Extremal Proertes Iequaltes Zeros World Scetfc Publshg Comay Sgaore 994 [] A C Schaffer Iequaltes of A Markoff ad S Berste for Polyomals ad Related Fuctos Bullet Amerca Mathematcal Socety Vol 47 No do:9/s [3] G Pólya ad G Segö Aufgabe ud Lehrsäte aus der Aalyss Srger-Verlag Berl 95 [4] A Zygmud A Remark o Cougate Seres Proceedgs of Lodo Mathematcal Socety Vol do:/lms/s-3439 [5] G H Hardy The Mea Value of the Modulus of a Aalytc Fucto Proceedgs of Lodo Mathematcal Socety Vol [6] Q I Rahma ad G Schmesser Les q Ualtués de Markoff et de Berste Presses Uversty Motréal Motréal 983 [7] M Res Formula d terolato Pour la Dérvée d u Polyome Trgoométrque Comtes Redus de l Academe des Sceces Vol [8] V V Arestov O Itegral Iequaltes for Trgoometrc Polymals ad Ther Dervatves Mathematcs of the USSR-Ivestya Vol do:7/im98v8abeh375 [9] N G Bru Iequaltes Cocerg Polyomals the Comlex Doma Nederal Akad Wetesch Proceedg Vol [] Q I Rahma ad G Schmesser L Iequaltes for Polyomals The Joural of Aroxmato Theory Vol do:6/-945(88)973- [] R P Boas Jr ad Q I Rahma L Iequaltes for Polyomals ad Etre Fuctos Archve for Ratoal Mechacs ad Aalyss Vol do:7/bf5397 [] A A ad N A Rather L Iequaltes for Polyomals Glask Matematck Vol 3 No [3] P D Lax Proof of a Coecture of P Erdös o the Dervatve of a Polyomal Bullet of Amerca Mathematcal Socety Vol do:9/s [4] N C Akey ad T J Rvl O a Theorm of S Berste Pacfc Joural of Mathematcs Vol [5] A A ad N A Rather Some Comact Geeralato of Zygmud-Tye Iequaltes for Polyomals Nolear Studes Vol 6 No [6] A A A New Proof ad a Geeralato of a Theorem of De Bru Proceedgs of Amerca Mathematcal Socety Vol 6 No [7] K K Dewa ad N K Govl A Iequalty for Self- Iversve Polyomals Joural of Mathematcal Aalyss ad Alcato Vol 95 No do:6/-47x(83)9- Coyrght ScRes