Iteatioal Mathematical Foum, Vol. 8, 23, o., 5-5 HIKARI Ltd, www.m-hikai.com Some Itegal Mea Estimates fo Polyomials Abdullah Mi, Bilal Ahmad Da ad Q. M. Dawood Depatmet of Mathematics, Uivesity of Kashmi Siaga-96, Idia mabdullah mi@yahoo.co.i dabilal67@gmail.com qdawood@gmail.com Copyight c 23 Abdullah Mi et al. This is a ope access aticle distibuted ude the Ceative Commos Attibutio Licese, which pemits uesticted use, distibutio, ad epoductio i ay medium, povided the oigial wok is popely cited. Abstact. I this pape, we establish some itegal mea estimates fo polyomials P (z) = a z + a υ z υ, μ, havig all its zeos i z k, k. Ou esults ot oly geealize ad efie some kow polyomial iequalities, but also a vaiety of iteestig esults ca be deduced fom these by a faily uifom pocedue. Mathematics Subject Classificatio: 3A, 3C, 3C5 Keywods: Polyomial, Maximum modulus, Itegal mea estimates. INTRODUCTION AND STATEMENT OF RESULTS Let P (z) be a polyomial of degee ad P (z) be its deivative. If P (z) has all its zeos i z, the it was show by Tua [9] that () Max z = P (z) 2 Max z = P (z). Iequality () is best possible with equality fo P (z) =αz +β, whee α = β. As a extesio of () Malik [8] poved that if P (z) has all its zeos i z k whee k, the (2) Max z = P (z) +k Max z = P (z). Aziz [] obtaied a geealizatio of (2) i the sese that the ight had side of (2) is eplaced by a facto ivolvig the itegal mea of P (z) o z =.
52 A. Mi, B. A. Da ad Q. M. Dawood I fact he poved that if P (z) has all its zeos i z k, k, the fo each >, P (e iθ ) (3) P (e iθ ) +ke iθ. Aziz ad Shah [3] geealized (3) i a diffeet diectio ad poved that, if P (z) =a z + a υ z υ, μ, is a polyomial of degee havig all its zeos i z k, k, the fo each >, P (e iθ ) (4) P (e iθ ) +k μ e iθ I this pape, we shall also coside the class of polyomials P (z) =a z + a υ z υ, μ, havig all zeos i z k, k, ad theeby obtai cetai itegal iequalities fo these polyomials. As we will see ou esults ot oly geealize ad efie the iequalities (), (2), (3) ad (4), but also a vaiety of iteestig esults ca be deduced fom these by a faily uifom pocedue. Moe pecisely, we pove the followig esults: Theoem. If P (z) =a z + a υ z υ, μ, is a polyomial of degee havig all its zeos i z k, k, the fo evey eal o complex umbe β with β ad fo each >, P (e iθ e )+βma i( )θ μ k (5) +A P (e iθ ) μ e iθ, whee (6) ad A μ = ( a m )k 2μ + μ a μ k μ ( a m )k μ + μ a μ m = Mi z =k P (z). The esult is best possible ad equality i (5) holds fo P (z) =(z μ + k μ ) μ, whee is a multiple of μ. If we do ot have the kowledge of Mi z =k P (z), we obtai the followig esult which is a special case of Theoem. Coollay. If P (z) =a z + a υ z υ, μ, is a polyomial of.
Some itegal mea estimates fo polyomials 53 degee havig all its zeos i z k, k, the fo each >, P (e iθ ) (7) P (e iθ ) +s μ e iθ, whee (8) s μ = a k 2μ + μ a μ k μ a k μ + μ a μ The esult is best possible ad equality i (7) holds fo P (z) =(z μ + k μ ) μ, whee is a multiple of μ. Remak. If P (z) =a z + a υ z υ, μ, has all its zeos i z k, k, the (fo example see Lemma stated i sectio 2) μ a μ a k μ, which ca also be take as equivalet to s μ k μ. Hece iequality (7) is a impovemet of iequality (4). Sice P (e iθ ) Max z = P (z) fo θ<2π, the followig esult easily follows fom Theoem. Coollay 2. If P (z) =a z + a υ z υ, μ, is a polyomial of degee havig all its zeos i z k, k, the fo evey eal o complex umbe β with β ad fo each >, (9) P (eiθ )+ βma μe i( )θ +A μ e iθ Max z = P (z), whee m ad A μ is as defied i Theoem. The esult is best possible ad equality i (9) holds fo P (z) =(z μ + k μ ) μ, whee is a multiple of μ. If we let i (9) ad choose agumet of β with β = suitably, we get () ( Max z = P (z) +A μ ) Max z = P (z) + ma μ ( + A μ ). Iequality () is best possible ad equality holds fo P (z) =(z μ +k μ ) μ, whee is a multiple of μ. It ca be easily veified fo example by the fist deivative test that the fuctio +x Max z = P (z) + mx ( + x)
54 A. Mi, B. A. Da ad Q. M. Dawood is a o-iceasig fuctio of x whee k. If we combie this fact with Lemma 5 (stated i sectio 2), accodig to which A μ k μ, we get Max z = P (z) { Max +k μ z = P (z) + } () k Mi z =k P (z), μ which is a geealizatio of a esult due to Govil [6]. Remak 2. Iequality () was also poved by Aziz ad Shah [3]. Fially, we use Holde s iequality to establish the followig esult which is a geealizatio of Coollay 2 ad also povides a efiemet of a esult [4, T heoem.3] due to Dewa, Mi ad Yadav. Theoem 2. If P (z) =a z + a υ z υ, μ, is a polyomial of degee havig all its zeos i z k, k ad m = Mi z =k P (z), the fo evey eal o complex umbe β with β, >,p >,q > with p + q =, (2) P (eiθ )+ βma μe i( )θ p +A μ e iθ p P (e iθ ) q whee A μ is defied by (6). If we take β =, we get the followig esult. Coollay 3. If P (z) =a z + a υ z υ, μ, is a polyomial of degee havig all its zeos i z k, k ad m = Mi z =k P (z), the fo >,p>,q > with p + q =, p q (3) P (e iθ ) +A μ e iθ p P (e iθ ) q, whee A μ is defied by (6). Remak 3. Sice we have fom Lemma 5 (stated i sectio 2), A μ k μ fo μ, it follows fom Coollay 3 that if P (z) =a z + a υ z υ, μ, is a polyomial of degee havig all its zeos i z k, k, the fo >,p>,q > with p + q =, p q (4) P (e iθ ) +k μ e iθ p P (e iθ ) q. This esult was poved by Dewa, Mi ad Yadav [4]. 2. LEMMAS. Fo the poof of these theoems we eed the followig lemmas. q,
Lemma. If P (z) =a z + Some itegal mea estimates fo polyomials 55 a υ z υ, μ, is a polyomial of degee havig all its zeos i z k, k, ad q(z) =z P ( ) the z { } a k 2μ + μ a μ k μ (5) P (z) q (z), fo z = a k μ + μ a μ ad μ a μ (6) a kμ. The above lemma is due to Aziz ad Rathe [2]. Lemma 2. If P (z) = a υ z υ is a polyomial of degee havig all its zeos i z k, k >, the υ= ad i paticula q(z) m fo z k (7) a > m, whee m = Mi z =k P (z) ad q(z) =z P ( ). z The above lemma is due to Dewa, Sigh ad Mi [5]. Lemma 3. The fuctio s μ (x) = xk2μ + μ a μ k μ (8), xk μ + μ a μ whee k ad μ, is a o-iceasig fucyio of x. Poof. The poof follows by cosideig the fist deivative test fo s μ (x). Lemma 4. If P (z) =a z + a υ z υ, μ, is a polyomial of degee havig all its zeos i z k, k, ad q(z) =z P ( ), the fo z z =, (9) whee (2) ad (2) q (z) A μ P (z) ma μ, A μ = ( a m ) k 2μ + μ a μ k μ ( a m ) k μ + μ a μ μ a μ ( a m ) kμ
56 A. Mi, B. A. Da ad Q. M. Dawood with m = Mi z =k P (z). Poof. By hypothesis, the polyomial P (z) =a z + a υ z υ, μ, has all its zeos i z k, k. If P (z) has a zeo o z = k, the m = ad the esult follows fom Lemma. Hecefoth, we assume that all the zeos of P (z) lie i z <k,k, so that m>. Sice m P (z) fo z = k, theefoe, if λ is ay eal o complex umbe with λ <, the mλz < P (z) fo z = k. Sice all the zeos of P (z) lie i z <k, it follows by Rouche s theoem that all the zeos of P (z) mλz also lie i z <k,k. Hece by Guass-Lucas theoem, the polyomial (22) P (z) mλz also has all its zeos i z <k,k, fo evey λ with λ <. This implies (23) P (z) m z fo z k, k. Because if (23) is ot tue, the thee is a poit z = z with z k such that P (z ) < m z. We choose λ = k P (z ), so that λ < ad with this choice of λ, fom (22), mz we have P (z ) mλz =, whee z k, which cotadicts the fact that all the zeos of P (z) mλz lie i z <k,k. Now, we ca apply iequality (5) of Lemma to the polyomial ad get, (24) whee (25) s μ P (z) mλz P (z) mλz q (z), fo z = s μ = a mλ k 2μ + μ a μ k μ. a mλ k k μ + μ a μ
Some itegal mea estimates fo polyomials 57 Sice fo evey λ with λ <, we have (26) a mλ k a m λ a m ad a > m by Lemma 2. Now combiig (25), (26) ad Lemma 3, we get fo evey λ with λ <, s μ = a mλ k 2μ + μ a μ k μ a mλ k k μ + μ a μ ( a m )k 2μ + μ a μ k μ (27) ( a m )k k μ + μ a μ Theefoe usig (27) ad (24), we get = A μ. (28) A μ P (z) mλz q (z),fo z =. If i (28), we choose the agumet of λ such that P (z) mλz = P (z) m λ which easily follows fom (23), we get A μ P (z) m λ A μ (29) q (z),fo z =. Fially lettig λ i (29), we obtai A μ P (z) q (z) + ma μ fo z =, which poves (9). To pove (2), we apply iequality (6) of Lemma to the polyomial P (z) mλz, ad get μ a μ (3) a mλ kμ fo evey eal o complex umbe λ with λ <. Sice by Lemma 2, we have a > m, we ca choose agumet of λ such that a mλ k = a m λ ad with this choice of the agumet of λ, we get fom (3) that (3) μ a μ ( a m λ ) kμ.
58 A. Mi, B. A. Da ad Q. M. Dawood Iequality (2) ow follows by makig λ i (3). Lemma 5. If P (z) =a z + a υ z υ, μ, is a polyomial of degee havig all its zeos i z k, k, the (32) A μ k μ, whee A μ is defied as i Theoem. Poof. We have fom iequality (2) of Lemma 4, μ a μ ( a m )kμ, which implies, {μ a μ ( a m } )kμ, which is equivalet to (k μ k μ ) {μ a μ ( a m } )kμ, that is, ( a m )k2μ + μ a μ k μ (μ a μ + ( a m )kμ ) k μ, fom which iequality (32) follows. 3. Poof of the Theoem Poof of Theoem. Sice P (z) has all its zeos i z k, it follows by Lemma 4 that q (z) + ma μ (33) A μ P (z) fo z =, μ. Also q(z) =z P ( ) so that P (z) z =z q( ), we have z Equivaletly, P (z) =z q( z ) z 2 q ( z ). zp (z) =z q( z ) z q ( z ), which implies (34) P (z) = q(z) zq (z) fo z =. Now usig (33) ad (34), we get fo evey eal o complex umbe β with β, (35) q (z)+β ma μ q (z) + ma μ A μ P (z) = A μ q(z) zq (z) fo z =.
Some itegal mea estimates fo polyomials 59 Sice P (z) has all its zeos i z k, by the Guass-Lucas theoem all the zeos of P (z) also lie i z. This implies that the polyomial z P ( z )=q(z) zq (z) has all its zeos i z. Theefoe, it follows fom (35) that the fuctio k ( ) z q (z)+ mβaμ k (36) W (z) = A μ (q(z) zq (z)) is aalytic i z ad W (z) fo z. Futhemoe, W () =. Thus the fuctio +A μ W (z) is subodiate to the fuctio +A μ z i z. Hece by a well-kow popety of subodiatio [7], we have fo >ad fo θ<2π, (37) Also fom (36), we have o (38) +A μ W (e iθ ) +A μ W (z) = +A μ e iθ. q(z)+ mβaμz q(z) zq (z) q(z)+mβa μz = +A μw (z) q(z) zq (z). Usig (34) i (38), we get fo z =, z P ( z )+mβa μz = +A μw (z) P (z), o P (z)+mβa μz (39) = +A μw (z) P (z) fo z =. Fom (37) ad (39), we deduce fo each >, P (e iθ )+ mβaμei( )θ k P (e iθ ) = +A μ W (e iθ ) +A μ e iθ
5 A. Mi, B. A. Da ad Q. M. Dawood which completes the poof of Theoem. Poof of Theoem 2. Poceedig similaly as i the poof of Theoem, we get fom iequality (39) that fo z =, P (z)+mβa μz = +A μw (z) P (z). Cosequetly, this implies fo >, P (eiθ )+ mβa μe i( )θ (4) = +A μ W (e iθ ) P (e iθ ) Now applyig Holde s iequality fo p>,q > with p + q = to (4), we get P (eiθ )+ mβa μe i( )θ p +A μ W (e iθ ) p P (e iθ ) q q o (4) P (eiθ )+ mβa μe i( )θ p +A μ W (e iθ ) p P (e iθ ) q q Fially, usig iequality (37) i (4), we get P (eiθ )+ mβa μe i( )θ p +A μ e iθ p P (e iθ ) q q which completes the poof of Theoem 2.
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