On an Operator Preserving Inequalities between Polynomials

Transcription

1 Applied Mathematics ublished Olie Jue ( O a Operator reservig Iequalities betwee olyomials Nisar Ahmad Rather Mushtaq Ahmad Shah Mohd Ibrahim Mir ost-graduate Departmet of Mathematics Kashmir Uiversity Sriagar Idia {drarather mushtaqa}@gmailcom Received April ; revised May 8 ; accepted May 6 ABSTRACT Let be the class of polyomials of degree ad B a family of operators that map ito itself For B B we ivestigate the depedece of o the maximum modulus of R BRB r Br r o for arbitrary real or complex umbers with ad R > r ad preset certai sharp operator preservig iequalities betwee polyomials Keywords: Compoet olyomials; B-Operator; Complex Domai Itroductio to the Statemet of Results Let deote the space of all complex polyomials j a j of degree If the coj cerig the estimate of the maximum of o the uit circle ad the estimate of the maximum of o a larger circle R we have ad max max () R max R max () Iequality () is a immediate cosequece of S Berstei s theorem (see [-3]) o the derivative of a trigoometric polyomial Iequality () is a simple deductio from the maximum modulus priciple (see [4 p 346] or [5 p 58]) If we restrict ourselves to the class of polyomials havig o ero i the () ad () ca be replaced by max max (3) ad R max max (4) R Iequality (3) was cojectured by Erdös ad later verified by Lax [6] Akey ad Rivli [7] used Iequality (3) to prove Iequality (4) As a compact geeraliatio of Iequalities () ad () Ai ad Rather [8] have show that if the for every real or complex umber with R > ad R R max (5) The result is sharp As a correspodig compact geeraliatio of Iequalities (3) ad (4) they [8] have also show that if ad for the for every real or complex umber with R R R max for The result is sharp ad equality i (6) holds for a b a b Cosider a operator B which carries a polyomial ito B! (7)! where ad are such that all the eros of (6) Copyright SciRes

2 558 N A RATHER ET AL lie i the half plae u C C (8) (9) As a geeraliatio of the Iequalities () ad () QI Rahma [9] proved that if the for ad if B B max for the for B B max () () (see [9] Iequality (5) ad (53)) I this paper we cosider a problem of ivestigatig the depedece of R BRB r Br r o the maximum modulus of o for arbitrary real or complex umbers with ad R > r ad develop a uified method for arrivig at these results I this directio we first preset the followig iterestig result which is compact geeraliatio of the Iequalities () () (5) ad () Theorem If the for arbitrary real or complex umbers ad with R > r ad where max B R R r B r R R r r B R Rr r () The result is best possible ad equality i () holds for ; Remark For from Iequality () we have for ad R r B r B R R r B max (3) Remark For ad Iequality () reduces to max B R R B B R max (4) for ad R > which cotais Iequality () as a special case Remark 3 For Iequality () yields R BR Br r R R r B max r (5) for R r ad If we choose i () ad otig that all the eros of u defied by (8) lie i the half plae (9) we get: Corollary If the for all real or complex umbers a d with R > r ad R r R R r r R Rr r max (6) is defied as i Theorem The For the case B from () we obati for all real or complex umbers ad wit h R > r ad where Rr result is sharp ad equality i (6) holds for R r r R r R R r max (7) Iequality (7) is equivalet to the Iequality (5) for ad For ad Iequality (7) icludes Iequality () as a special case Next we use Theorem to prove the followig result Theorem If the for arbitrary real or complex umbers ad with R > r ad R B R r B r B Q R R r BQ r R R r r B Rr max where Q ad Rr (8) is defied as i Theorem The result is sharp ad equality i (8) holds for Remark 4 Theorem icludes some well kow polyomial iequalities as special cases For example iequality (8) reduces to a result due to Q I Rahma ([8] Iequality (5) with ad ) Also for Copyright SciRes

3 N A RATHER ET AL 559 Iequality (8) gives B R B r BQ R BQ r R r B max (9) for R > r ad If we take i (8) we get: Corollary If the for all real or com plex umbers ad with R > r ad R r r rqr R R r r RQ R R () R Rr r max where Rr is defied as i Theorem The result is sharp ad equality i () holds for For ad Theorem icludes a result due to A Ai ad Rather [] as a special case Iequality () ca be sharpeed if we restrict ourselves to the class of polyomials havig o eros i I this directio we ext prove the followig result which is a compact geeraliatio of the Iequalities (3) (4) ad (6) Theorem 3 If ad for the for arbitrary real or complex umbers ad with R > r ad B B R R r r R R r r B where Rr Rr max () is defied as i Theorem The result is sharp ad equality i () holds for Remark 5 Iequality () is a special case of the Iequality () for ad If we choose i () ad ote that all the eros of u defied by (8) lie i the half plae defied by (9) it follows that if ad for the for R r ad R R R r r r R R r r max Settig i () we obtai for () R R R r r r max R R r r (3) for R > r ad Takig i () we obtai for ad R > R R max (4) which i particular gives Iequality (3) Next choosig i () we immediately get for R > r ad R R r r R R r r Rr max (5) which is a compact geeraliatio of the Iequalities (3) (4) ad (6) The result is sharp ad equality i (5) holds for a b a b If we put i (5) we get the followig result Corollary 3 If ad for the for every real or complex umber with R > r ad r R A polyomial R r max (6) is said to be self-iversive if where Q Q It is kow [6 ] that if is a self-iversive polyomial the max max (7) Here fially we establish the followig result for self-iversive polyomials Theorem 4 If is a self-iversive polyomial the for arbitrary real or complex umbers ad with R > r ad B R R r B r where Rr R R r r B Rr ma x (8) is defied as i Theorem The result is sharp ad equality i () holds for The followig result is a immediate cosequece of Copyright SciRes

4 56 N A RATHER ET AL Theorem 4 Corollary 4 If is a self-iversive polyomial the for arbitrary real or complex umbers ad with R > r ad R R r r R R r r (9) Rr max w Rr is defied as i Theorem The result is best possible For th e Iequality (9) reduces to here r R R r max (3) Remark 6 Iequality (6) is a special case of the Iequality (3) May other iterestig results ca be deduced f rom Theorem 4 i the same way as we have deduced from Theorem ad Theorem Lemmas For the proofs of these theorems we eed the followig lemmas The first lemma ca be easily proved Lemma If ad has all its eros i the for every R > r ad R r r R (3) The ext Lemma follows from corollary 83 of [ p 65] Lemma If ad has all its eros i the all the eros of B also lie i Lem ma 3 If ad does ot vaish i the for arbit rary real or co mplex umbers ad with R > r ad B R R r B r BQ R R r BQ r (3) w Q R r is defied as i Theorem The result is sharp ad equality i (3) holds for roof of Lemma 3 Sice the polyomial has all its eros i for every real or complex umber with g Q here ad the polyomial where Q atleast oe ero i has all its eros i with so that we ca write i te g h where < h is a polyomial of degree havig all its eros i Applyig lemma to the polyomial h we obtai for R > r a d < π t ad This implies for i Re i Re i e i Re g t h Re i te i R i r R > r ad < π h re rt i R gre g i re (33) Rt r i Sice R r t so that g Re for < π ad r r t from Iequality (33) we R R t obtai for R > r ad < π g Equivaletly R r i i Re gre g R r R g r (34) for ad R> r Hece for every real or complex umber with ad R > r we have for g R g r g R g r R r g r Also Iequality (34) ca be writte as i i gre g re for every R > r ad < π (35) r (36) R Sice i r g Re ad from iequality (36) we R obtai for < π ad R > r Equivaletly i i g g re Re gr gr for Sice all the eros of g R lie i R a direct applicatio of Rouche s theorem shows that the polyomial g R g r has all its eros i for every real or complex umber with Applyig Rouche s theorem agai it follows from (35) that for arbitrary real or complex umbers with Copyright SciRes

5 N A RATHER ET AL 56 ad F R > r all the eros of the polyomial grr r gr R QR Rr R ( ) QR RR r r QRR r Qr lie i with Applyig Lemma to the polyomial F ad otig that B is a liear operato r it follows that all the eros of the polyomial T BF B R R r B r BQRR r BQr lie i for all real or complex umbers with ad R > r This implies R r B r B R BQ R R r BQ r for ad R > r If Iequality (38) is ot true the there is a poit w with w such that B R R r B r R r B Qr w BQ R w (37) But all the eros of Q lie i therefore it follows (as i case of g ) that all the eros of Q R R r Q r lie i Hece by Lemma all the eros of BQ R R r B Q r lie i so that BQRR r BQr = We take BR R r Br BQRR r BQr w w w the is a well defied real or complex umber with ad with this choice of from (37) we obtai Tw where w This cotradicts the fact that all the eros of T lie i Thus B R R r B r B Q R R r BQ r for ad R > r This proves (38) ad hece Lemma 3 3 roofs of the Theorems roo f of Theorem Let M max the M for By Rouche s Theorem it follows that all the eros of the polyomial H M lie i for every real or complex umber with therefore as before (as i Lemma 3) we coclude that all the eros of the polyomial R H G H R r r lie i for all real or complex umbers ad with ad Hece by Lemm a the polyomial T BG B H R R r BH r B R R r B r R R r r B M has all its eros i for every real or complex umber with This implies for every real or complex umbers ad with ad R > r B R R r B r R R r r B M for (38) If Iequality (4) is ot true the there is a poit w with w such that B R R r B r R R r r B M Sice = B we take w w B R R r B r w R R r r B M w w so that is a w ell defied real or complex umber with ad with this choice of from (39) we get Tw where w This cotradicts the fact that all the eros of T lie i Thus for every real or complex umbers ad with ad R > r Copyright SciRes

6 56 N A RATHER ET AL B R R r B r R R r r B M for This completes the proof of Theorem roof of Theorem Let M max the M for If is ay real or complex umber with the by Rouche s Theorem the polyomial f M does ot vaish i Applyig Lemma 3 to the polyomial f ad usig the fact that B is a liear operator it follows that f or all real or complex umbers ad with R > r ad for where Bf R R r Bf r r B f R R r B f QM f f M Usig the fact that B Q obtai B R R r B r Rr BQ R R r BQ r R R r r B M we for all real or complex umbers ad with R > r ad Now choosig the argumet of such that BQR R r BQr R R r r B M = R R r r B M BQ R R r BQ r which is possible by Theorem we get for B B R R r r Rr M R R r r B M BQ R R r BQ r R > r ad This implies B R R r B r B Q R R r BQ r R R r r B Rr M for R > r ad Lettig we obtai B R R r B r B Q R R r BQ r R R r r B Rr M which is iequality ( 8) ad this proves Theorem roof of Theorem 3 Lemma 3 ad Theorem together yields for all real or complex umbers ad with R > r ad B R R r B r B R R r B r B B R R r r R r B r B R which gives B Q R R r BQ r R R r r B Rr M B R R r B r R R r r B Rr M which is the Iequality () ad this completes the proof of Theorem 3 roof of Theorem 4 Sice is a self-iversive polyomial of degree therefore Q for all C This implies i particular that for all real or complex umbers ad with R > r ad B R R r B r B Q R R r BQ r Combiig this with Theorem the desired result fol- Copyright SciRes

7 N A RATHER ET AL 563 lows immediately This completes the proof of Theorem 4 4 Ackowledgemets Authors are thakful to the referee for his suggestios REFERENCES [] G V Milovaovic D S Mitri ovic ad Th M Rassias Topics i olyomials: Extremal roperties Iequalities Zeros World scietific ublishig Co Sigapore 994 [] Q I Rahma ad G Schmessier Aalytic Theory of olyomials Claredo ress Oxford [3] A C Schaffer Iequalities of A Markoff a d S Ber- stei for olyomials ad Related Fuctios Bulleti of the America Mathematical Society Vol 47 No 94 pp doi:9/s [4] M Ries Uber Eie Sat des Herr Serge Berstei Acta Mathematica Vol 4 96 pp doi:7/bf4855 [5] G ólya ad G Segö Aufgabe ud Lehrsäte aus der Aalysis Spriger-Verlag Berli 95 [6] D Lax roof of a Cojecture of Erdös o the Derivative of a olyomial Bulleti of the America Mathematical Society Vol 5 No pp doi:9/s [7] N C Akey ad T J Rivli O a Theorm of S Berstei acific Joural of Mathematics Vol pp [8] A Ai ad N A Rather O a Iequality of S Berstei ad Gauss-Lucas Theorem Aalytic ad Geometriv Iequalities Kluwer Academic ub Dordrecht doi:7/ _3 [9] Q I Rahma Fuctios of Expoetial Type Trasactios of the America Mathematical Society Vol pp doi:9/s x [] E B Saff ad T Sheil-Small Coefficiet ad Itegral Mea Estimates for Algebraic ad Trigoometric olyomials with Restricted Zeros Joural of the Lodo Mathematical Society Vol 9 No 975 pp 6- [] M Marde Geometry of olyomials Surveys i Mathematics No Copyright SciRes