International Journal of Mathematical Archive-3(4), 2012, Page: Available online through ISSN

Transcription

1 Iteratioal Joural of Mathematical Archive-3(4,, Page: Available olie through ISSN INEQUALITIES CONCERNING THE B-OPERATORS N. A. Rather, S. H. Ahager ad M. A. Shah* P. G. Departmet of Mathematics, Uiversity of Kashmir, Haratbal, Sriagar- 96, Idia dr.arather@gmail.com, ahagarsaad@gmail.com, mushtaqa@gmail.com (Received o: -4-; Accepted o: 3-4- ABSTRACT I this paper we cosider a operator B which carries a polyomial of degree ito ]= λ + λ (/P (/! + λ (/ P (/! Where λ, λ ad λ are such that all the eros of U(= λ + C(, λ + C(, λ lie i the half plae -/ ad ivestigate the depedece of R] α r] o the miimum ad the maximum modulus of o for every real or complex umber α with α, R > r with restrictio o the eros of the polyomial ad establish some ew operator preservig iequalities betwee polyomials. Mathematics subect classificatio (: 3A6, 3C. Keywords ad phrases: polyomials, B-operator, iequalities i the complex domai.. INTRODUCTION TO THE STATEMENT OF RESULTS. Let P ( deote the space of all complex polyomials famous result kow as Berstei s iequality (for referece see[4, 7,], / ( Max P ( Max = = = a of degree. If P P, the accordig to a as cocerig the maximum modulus of o a larger circle = R >, we have ( Max R Max = R> = (for referece see [8, p. 58 problem 69] or [, p. 346] Equality i ( ad ( holds for P ( = λ, λ. = For the class of polyomials P P havig all their ero i, we have / (3 Mi P ( Mi = = ad (4 Mi R Mi. = R> = Iequalities (3 ad (4 are due to A. Ai ad Q. M. Dawood [ ]. Both the results are sharp ad equality i (3 ad (4 holds for P ( = λ, λ. For the class of polyomials P P havig o ero i <, we have / (5 Max P ( Max = = ad R + (6 Max P (. = *Correspodig author: M. A. Shah*, * mushtaqa@gmail.com Iteratioal Joural of Mathematical Archive- 3 (4, April 544

2 N. A. Rather, S. H. Ahager ad M. A. Shah*/ Iequalities Cocerig The B-operators/ IJMA- 3(4, April-, Page: Equality i (5 ad (6 holds for P ( = λ + µ, λ = µ =. Iequality (5 was coectured by P. Erdӧs ad later verified by P. D. Lax [5]. Akey ad Rivili [] used (5 to prove (6. A. Ai ad Q.M. Dawood [] improved iequalities (5 ad (6 by showig that if P ( i <, the / (7 Max P ( Max Mi = = = ad R + R (8 Max Max Mi. = R> = = As a compact geeraliatio of iequalities (5 ad (6, Ai ad Rather [3] have show that if P ( for <, the for every real or complex umber α with α ad R, P R α P R α + α Max P for. (9 ( ( { } ( The result is sharp ad equality i (7 holds for = a + b, a = b =. Rahma [9] (see also Rahma ad Schmeisser[, p.538] itroduced a class polyomial P P ito P ( P ( ( ] : = λ + λ + λ!! λ, λ ad λ are such that all the eros of P P ad B of operators B that carries a ( = λ + λc(, + λc(,, C(, r =!/ r!( r!, r, u( lie i the half plae ( /. As a geeraliatio of the iequalities ( ad (, Q.I. Rahma [9] proved that if (3 ] ] Max for P P, the (see [9],iequality ( 5. ad if P, P ( for < P, the (4 ] { ] + λ } Max for, B (see [8], iequality (5. ad (5.3. B I this paper we ivestigate the depedece of R] r] o the miimum ad the maximum of, IJMA. All Rights Reserved 545 α modulus of o for every real or complex umber α with α, > r R ad obtai certai compact geeraliatios of some well-kow polyomial iequalities. I this directio we first preset the followig iterestig result which is a compact geeraliatio of iequalities (, ( ad (3. Theorem : If F P has all its eros i ad is a polyomial of degree at most such that for =, the for every real or complex umber αα with α ad R > r

3 N. A. Rather, S. H. Ahager ad M. A. Shah*/ Iequalities Cocerig The B-operators/ IJMA- 3(4, April-, Page: for (5 R] α r] R] r], B B. The followig result immediately follows from Theorem by takig ( M M Max = F = =. Corollary : If P, the for every real or complex umber α with, R > r P α, (6 [ ( ] [ ( ] [ ] ( B P R α B P r R αr B Max P for B.The result is best possible ad equality i (6 holds for = a, a B. Remark : For α =, Corollary reduces to the iequality (3. Next if we choose λ = λ = i (6 ad ote that i this case all the eros of u( defied by ( lie i regio defied by (, we obtai for every real or complex umber α with α, R > r, (7 R r R αr Max α for. For α =, iequality (7 icludes iequality ( as a special case. Further, if we divide both sides of the iequality (7 by R - r with α = ad make R r, we get P ( r r Max for, which, i particular, yields iequality ( as a special case. Next we preset the followig result, which is a compact geeraliatio of the iequalities (3 ad (4. Theorem :. If P P ad has all its eros i, the for every real or complex umber αα with α ad R > r B P R α B P r R αr B Mi P for (8 [ ( ] [ ( ] [ ] (, B. The result is best possible ad equality i (8 holds for = a, a. B Remark : For αα =, from iequality (8, we have for ad > (9 R] R ] Mi = R ] Mi, R, B B.The result is sharp. Next, takig λ = λ = i (8 ad otig that all the eros of uu( defied by ( lie i the half plae (, we get Corollary : If P has all its eros i, the for every real or complex umber α with, R > r P ( RP ( R rp ( r R αr Mi α for The result is sharp ad the extremal polyomial is = λ, λ. α,, IJMA. All Rights Reserved 546

4 N. A. Rather, S. H. Ahager ad M. A. Shah*/ Iequalities Cocerig The B-operators/ IJMA- 3(4, April-, Page: If we divide the two sides of ( by R - r with α = ad let R r, we get for, P ( r + rp ( r Mi. The result is sharp. For the choice λ = λ = i (8, we obtai for every real or complex umber α with α, R > r, ( R r R αr Mi α for. For α =, iequality ( icludes iequality (4 as a special case. If we divide both sides of the iequality ( by R - r with α = ad make R r, we get for ( P ( r r Mi =, which, i particular, yields iequality (3 as a special case. P P Corollary ca be sharpeed if we restrict ourselves to the class of polyomials, havig o ero i <. I this directio, we ext preset the followig compact geeraliatio of the iequalities (7, (8 ad (9, which also iclude refiemets of the iequalities (3 ad (4 as special cases. Theorem 3: If P P ad P ( for <, the for every real or complex umber α with α, R > r ad, (3 [ ] R αr B R] r] { R αr ] + α λ } Max Mi = = α λ B.The result is sharp ad equality i (3 holds for = a + b, a = b =. B Remark 3: For α =, iequality (3 yields refiemet of Iequality (4. If we choose λ = λ = i (3 ad ote that all the eros of u ( defied by ( lie i the half plae defied by (, we get for, R > r ad α, (4 RP ( R α rp ( r R αr Max Mi. Settig α = i (4, we obtai for ad R>, P ( R R which,i particular, gives iequality (7. Max Mi Next choosig λ =λ = i (3, we immediately get the followig result, which is a refiemet of iequality (9. Corollary 3: If P P ad P ( for < ad,, the for every real or complex umber α with α, R > r { } { } ( 5 R α r R αr + α Max R αr α Mi = =. The result is sharp ad equality i (5 holds for = a + b, a = b =. Iequality (5 is a compact geeraliatio of the iequalities (7 ad (8., IJMA. All Rights Reserved 547

5 N. A. Rather, S. H. Ahager ad M. A. Shah*/ Iequalities Cocerig The B-operators/ IJMA- 3(4, April-, Page: LEMMAS For the proofs of these theorems, we eed the followig lemmas. Lemma : If P P ad P ( has all its eros i R+ k (6 R r. r+ k k k, the for every R r ad =, Proof of Lemma : Sice all the eros of lie i k k, we write = i ( = Cos r e θ, r k, =,,,. Now for θ <π, R r, we have Hece / Re re R + r RrCos( θ θ R + r R+ k = i θ re r r r rrco ( s e θ θ r r + + r+ k. Re Re r = e R + k i re θ = re r e r + k for θ <π, which implies for = ad R r, R+ k R r. r+ k This completes the proof of Lemma. The ext lemma follows from Corollary 8.3 of [6, p. 65]. Lemma : If P P ad P ( has all its eros i, the all the eros of ] also lie i. Lemma 3: If P P ad P ( does ot vaish i <, the for every real or complex umber α with R> r, ad =, (7 R] r] R] α r] Q ( = /.The result is sharp ad equality i (7 holds for = a + b, a = b =. α, Proof of Lemma 3: Let Q ( = /. Sice all the eros of th degree polyomial P ( lie i, therefore, is a polyomial of degree havig all its eros i. Applyig Theorem with replaced by Q (, we obtai for every R> r ad, (8 R] r] R] α r]. This proves Lemma 3. Lemma 4: If P, the for every real or complex umber α with α, R > r ad P,, IJMA. All Rights Reserved 548

6 N. A. Rather, S. H. Ahager ad M. A. Shah*/ Iequalities Cocerig The B-operators/ IJMA- 3(4, April-, Page: (9 R] α r] + R] r] { R αr ] + α λ } Max = /. The result is sharp ad equality i (9 holds for Proof of Lemma 4: Let M Max, = P ( = λ, α. = the M for =. If µ is ay real or complex umber with µ >, the by Rouche s theorem, the polyomial = -µ M does ot vaish i <. Applyig Lemma 3 to the polyomial ad usig the fact that B is a liear operator, it follows that for every real or complex umber α with α, R >r, R] α r] H ( R] H ( r] for, H µ ( = / = / µ M = M. Agai usig the liearity of B ad the fact B [ ] = λ, we obtai (3 ( B P R B P r ( M ( B Q R B Q r ( R r B M [ ( ] α [ ( ] µ α λ [ ( ] α [ ( ] µ α [ ] for every real or complex umber α with α, R > r ad. Now choosig the argumet of µ o the right had side of (3 such that ( ( BQR [ ( ] α BQr [ ( ] µ R α r B [ ] M = µ R αr B [ ] M BQR [ ( ] αbqr [ ( ], which is possible by Corollary, we get, from (3, (3 ( ] [ ( ] [ ] [ ( ] [ ( ] P R αb P r µ α λ M µ R αr B M B Q R αb Q r for α, R> r ad. Lettig µ i (3, we obtai R] α r] + R] r] { R αr ] + λ α λ }M. This proves Lemma 4.. PROOFS OF THE THEOREM Proof of Theorem : By hypothesis F ( is a polyomial of degree havig all its eros i ad P ( is a polyomial of degree at most such that (3 ( P for =, Therefore, if F ( has a ero of multiplicity m at = e, the P ( must have a ero of multiplicity at least m at = e. If / is a costat, the the iequality (5 is obvious. We assume that / is ot a costat, so that by maximum modulus priciple, it follows that P ( < for >. Suppose F ( has m eros o = m so that we write F ( = F ( F ( F ( is a polyomial of degree m whose all eros lie o = ad F ( is a polyomial of degree exactly m havig all its eros i <. This gives with the help of iequality (3 that P ( = P ( F ( P ( is a polyomial of degree at most m. Now, from iequality (3, we get, IJMA. All Rights Reserved 549

7 N. A. Rather, S. H. Ahager ad M. A. Shah*/ Iequalities Cocerig The B-operators/ IJMA- 3(4, April-, Page: P F ( for = ( F ( for =. Therefore, for every real or complex umber λ with λ >, a direct applicatio of Rouche s theorem shows that all the eros of the polyomial P ( λf ( of degree m lie i <. Hece the polyomial = F ( ( P ( λf ( = λ has all its eros i with at least oe ero i <, so that we ca write = ( te H ( t < ad H( is a polyomial of degree - havig all its eros i. Hece with the help of Lemma with k =, we obtai for every R > r ad θ < π, R e = R e te H ( R e = R e R r R r t e This implies for R > r ad θ <π, R + r + R e te r e te R + t r e r + t r + t R + (33 R e r e. R + t r + Sice R > r > t so that ( for R > r ad θ < π, which leads to for H ( r e ( r e. te H ( r e + r r + t G R e for θ <π ad > >, from iequality (33, we obtai + R R + t R + Re > re r + r + Gre ( < GR ( e < GR ( e R + θ < π ad R > r. Equalivaletly, we have (34 G ( r < R for = ad R> r. Sice all the eros of G ( R lie i (/R <, a direct applicatio of Rouche s theorem shows that the polyomial R α r has all its eros i < for every real or complex umber α with α. Applyig Lemma ad usig the liearity of B, it follows that all the eros of the polyomial T ( = R α r] = R r], IJMA. All Rights Reserved 55

8 N. A. Rather, S. H. Ahager ad M. A. Shah*/ Iequalities Cocerig The B-operators/ IJMA- 3(4, April-, Page: lie i < for every real or complex umber α with α ad R> r. Replacig by λ, we coclude that all the eros of the polyomial (35 T ( = ( R] r] λ( R r] lie i < for all real or complex umbers α, λ with α, λ > ad R> r. This implies (36 R] α r] R r] for ad R > r. If iequality (36 is ot true, the there a poit = w with w such that ( B [ R] r] = w > ( R r] = w α, R > r. Sice all the eros of lie i, it follows (as i the case of that all the eros of R α r] lie i. Hece ( [ R r] B α, R > r. =w We choose λ = ( R] r] = w ( R] r] = w so that λ is well defied real or complex umber with λ >, ad with choice of λ, from (35, we get, T(w = with w. This is clearly a cotradictio to the fact that all the eros of T( lie i <. Thus for every real or complex umber α with α ad R > r, This completes the proof of Theorem. R] α r] R r]. Proof of Theorem : The result is clear if has a ero o =, for the (. We ow assume that has all its eros i < so that m > ad m for =. This gives for every λ with λ <, λ m < for = m = Mi By Rouche s theorem, it follows that all the eros of polyomial = λm lie i < for every real or complex umber λ with λ <. Therefore, (as before we coclude that all the eros of polyomial = R α r lie i < for every real or complex α with umber α ad R > r. Hece by Lemma, all the eros of the polyomial = P = (37 S( = ] = R] r] = R] r] λ( R αr lie i < for all real or complex umbers α, λ with α, λ < ad R > r. This implies (38 BPR [ ( ] αbpr [ ( ] R αr B [ ] m for ad R >r. ] m, IJMA. All Rights Reserved 55

9 N. A. Rather, S. H. Ahager ad M. A. Shah*/ Iequalities Cocerig The B-operators/ IJMA- 3(4, April-, Page: If iequality (38 is ot true, the there is a poit = w with w such that Sice { B ]} = w { [ ( ] α [ ( ]} α { [ ]} [, we take BPR BPr < R r B m. = w = w { BPR [ ( ] BPr [ ( ]} / m( R r { B [ ]} λ = α α = w = w so that λ is a well defied real or complex umber with λ < ad with choice of λ, from (37, we get S(w = with w. This cotradicts the fact that all the eros of S( lie i <. Thus for every real or complex umber α with α ad R > r, This completes the proof of Theorem. BPR [ ( ] αbpr [ ( ] R αr B [ ] m for. Proof of Theorem 3: By hypothesis, the polyomial does ot vaish i <, therefore, if m = Mi, the m for. We first show that for every real or complex umber δ with δ, the polyomial = + mδ does ot vaish i <. This is obvious if m = ad for m >, we prove it by a cotradictio. Assume that has a ero i < say at = w with w <, the we have w + m w = w =. This gives δ w = mδ w m w < m, which is clearly a cotradictio( to the miimum modulus priciple. Hece has o ero i < for every δ with δ. Applyig Lemma 3 to the polyomial, we obtai for every real or complex umber α with umber α ad R > r, R] α r] R] r],, = = / = / mδ = mδ. Equivaletly, R] r] mδ R αr ] R] r] mδ α λ (39 ( ( for all real or complex umbers α, δ with umber α, δ ad R > r. Now choosig the argumet of δ such that ( BPR [ ( ] αbpr [ ( ] mδ R αr B [ ] = BPR [ ( ] αbpr [ ( ] + m δ α B [ ], We obtai from (39, for α, δ ad R > r, R] α r] + m δ R αr ] R] α r] + m δ α λ, for, or equivaletly, ( R αr ] α λ m R] ], R] α r] + δ r for α, δ ad R > r. Lettig δ, we get ( R αr ] α λ m R] ], R] r] + r, IJMA. All Rights Reserved 55

10 N. A. Rather, S. H. Ahager ad M. A. Shah*/ Iequalities Cocerig The B-operators/ IJMA- 3(4, April-, Page: for α ad R > r. Combiig this iequality with Lemma 4, we get, for every real or complex umber α with α, R > r ad, R] r] + ( R αr ] α λ m R] r] + R] r] R α r ] + α λ M, which is equivalet to (3 ad this completes the proof of Theorem 3. REFERENCES [] N.C. Akey ad T.J. Rivli, O a Theorem of S. Berstei, Pacific J. Math., 5(955, [] A. Ai ad Q.M. Dawood, Iiequality for polyomials ad its derivative, Approx. Theory, 54(988, [3] A. Ai ad N.A. Rather, O a iequality of S. Berstei ad Gauss-Lucas Theorem, Aalytic ad Geometric iequalities ad their Applicatio,(Th.M.Rassias ad H. M. Sarivastava eds Kluwer Acad. Pub., (999, [4] S. Berstei, Sur ĺ ordre de la meilleure approximatio des foctios par des polyomes de degree doé, Momoires de ĺ Académic Royal de Belgique, 4(9,-3. [5] P.D. Lax, Proof of a coecture of P.Erdӧs o the derivative of a polyomial, Bull. Amer.Math. Soc., 5(944, [6] M. Marde, Geometry of Polyomials, Math. Surveys, No. 3, Amer.Math. Soc. Providece, RI, 949. [7] G.V. Milovaović, D.S. Mitriović ad Th.M. Rassias, Topics i polyomials: Extremal Properties, Iequalities ad Zeros, World Scietific Publishig Co., Sigapore (994. [8] G. Pόlya ad G. Seg, Problems ad Theorems i Aalysis, Vol., Spriger New York, 97. [9] Q. I. Rahma, Fuctios of expoetial type, Tras. Amer. Soc., 35(969, [] Q.I. Rahma ad G. Schmeisser, Aalytic Theory of polyomials, Oxford Uiversity Press, New York,. [] M. Ries, Über eie sat des Herr Serge Berstei, Acta Math., 4(96, [] S. Berstei, Sur ĺ ordre de la meilleure approximatio des foctios par des polyomes de degré doé, Momoires de ĺ Académic Royal de Belgique, 4(9, -3. ****************************, IJMA. All Rights Reserved 553