For a polynomial p(z) of degree n, it holds the Zygmund\u27s inequality. We have obtained inequalities in the reverse direction for the polynomials having a zero of order m at the origin

V. K. Jain

English

Hrčak - Portal of scientific journals of Croatia

GLASNIK MATEMATIČKIVol. 37(57)(2002), 83 – 88INTEGRAL INEQUALITIES FOR POLYNOMIALS HAVINGA ZERO OF ORDER m AT THE ORIGINV. K. JainKharagpur, IndiaAbstract. For a polynomial p(z) of degree n, it is known that2π∫0∣∣∣p′(eiθ)∣∣∣sdθ1/s≤ n2π∫0∣∣∣p(eiθ)∣∣∣sdθ1/s, s ≥ 1.We have obtained inequalities in the reverse direction for the polynomialshaving a zero of order m at the origin.1. Introduction and statement of resultsLet p(z) be a polynomial of degree n. Zygmund [3] has shown that fors ≥ 1(1.1)2π∫0∣∣p′(eiθ)∣∣s dθ1/s≤ n2π∫0∣∣p(eiθ)∣∣s dθ1/s.In this paper, we have obtained similar type of integral inequalities, but inthe reverse direction, for polynomials having a zero of order m at the origin.More precisely, we proveTheorem 1.1. Let p(z) be a polynomial of degree n, having a zero oforder m at z = 0. Then for s ≥ 1(1.2)2π∫0∣∣p′(eiθ)∣∣s dθ1/s≥ m2π∫0∣∣p(eiθ)∣∣s dθ1/s.2000 Mathematics Subject Classification. 30C10, 30A10.Key words and phrases. Inequalities, zero of order m, polynomials.8384 V. K. JAINBy letting s→ ∞ in (1.2), we obtainCorollary 1.2. Let p(z) be a polynomial of degree n, having a zero oforder m at z = 0. Thenmax|z|=1|p′(z)| ≥ mmax|z|=1|p(z)|.Theorem 1.3. Let p(z) be a polynomial of degree n, having all its zeros in|z| ≤ k, k ≤ 1, with a zero of order m at z = 0. Then for β with |β| < kn−mand s ≥ 1( 2π∫0∣∣∣∣p′(eiθ)+mm′knβei(m−1)θ∣∣∣∣sdθ)1/s≥(1.3)≥{n− (n−m)C(k)s}2π∫0∣∣∣∣p(eiθ)+m′knβeimθ∣∣∣∣sdθ1/s,where(1.4) m′ = min|z|=k|p(z)|,(1.5) C(k)s = k/ 12π2π∫0∣∣1 + keiα∣∣s dα1/s.By taking k = 1 and β = 0 in Theorem 1.3, we obtainCorollary 1.4. If p(z) is a polynomial of degree n, having all its zerosin |z| ≤ 1, with a zero of order m at z = 0, then for s ≥ 1(1.6)2π∫0∣∣p′(eiθ)∣∣s dθ1/s≥ {n− (n−m)Ds}2π∫0∣∣p(eiθ)∣∣s dθ1/s,where(1.7) Ds = 1/ 12π2π∫0∣∣1 + eiα∣∣s dα1/s.Inequality (1.6), with m = 0, is also true for self–inversive polynomials.In other words we haveTheorem 1.5. If p(z) is a polynomial of degree n such that(1.8) p(z) = znp(1/z),INTEGRAL INEQUALITIES FOR POLYNOMIALS 85then for s ≥ 1,(1.9)2π∫0∣∣p′(eiθ)∣∣s dθ1/s≥ n(1 −Ds)2π∫0∣∣p(eiθ)∣∣s dθ1/s,where Ds is as in Corollary 1.4.By letting s→ ∞ in Theorem 1.3, we obtainCorollary 1.6. Let p(z) be a polynomial of degree n, having all its zerosin |z| ≤ k, k ≤ 1, with a zero of order m at z = 0. Then for β with |β| < kn−m(1.10) max|z|=1∣∣∣∣p′(z) +mm′knβzm−1∣∣∣∣ ≥(n+mk1 + k)max|z|=1∣∣∣∣p(z) +m′knβzm∣∣∣∣ ,where m′ is, as in Theorem 1.3.By choosing argument of β suitably and letting |β| → kn−m in Corollary1.6, we obtainCorollary 1.7. If p(z) is a polynomial of degree n, having all its zerosin |z| ≤ k, k ≤ 1, with a zero of order m at z = 0, then(1.11) max|z|=1|p′(z)| ≥(n+mk1 + k)max|z|=1|p(z)| +(n−m1 + k)m′km,where m′ is as in Theorem 1.3.2. LemmasFor the proofs of the theorems, we require the following lemmas.Lemma 2.1. If p(z) is a polynomial of degree n, having no zeros in |z| < k,k ≥ 1, then for s ≥ 12π∫0∣∣p′(eiθ)∣∣s dθ1/s≤ nE(k)s2π∫0∣∣p(eiθ)∣∣s dθ1/s,whereE(k)s = 1/ 12π2π∫0∣∣k + eiα∣∣s dα1/s.This lemma is due to Govil and Rahman [2].Lemma 2.2. If p(z) is a polynomial of degree n such thatp(z) = znp(1/z),86 V. K. JAINthen for s ≥ 12π∫0∣∣p′(eiθ)∣∣s dθ1/s≤ nDs2π∫0∣∣p(eiθ)∣∣s dθ1/s,where Ds is as in Corollary 1.4.This lemma is due to Dewan and Govil [1].3. Proofs of the theoremsProof of Theorem 1.1. We obviously have(3.1) p(z) = zmφ(z),where φ(z) is a polynomial of degree n−m, with the property thatφ(0) 6= 0.Thenq(z) = znp(1/z)(3.2)= zn−mφ(1/z),is also a polynomial of degree n−m. Hence we have for s ≥ 1,(3.3)2π∫0∣∣q′(eiθ)∣∣s dθ1/s≤ (n−m)2π∫0∣∣q(eiθ)∣∣s dθ1/s.But by (3.2), we have for 0 ≤ θ ≤ 2π∣∣q′(eiθ)∣∣ =∣∣np(eiθ) − eiθp′(eiθ)∣∣∣∣q(eiθ)∣∣ =∣∣p(eiθ)∣∣ ,which, by (3.3), imply that for s ≥ 1,(3.4)2π∫0∣∣np(eiθ)− eiθp′(eiθ)∣∣s dθ1/s≤ (n−m)2π∫0∣∣p(eiθ)∣∣s dθ1/s.INTEGRAL INEQUALITIES FOR POLYNOMIALS 87Now, by Minkowski inequality, we have for s ≥ 1n( 2π∫0∣∣p(eiθ)∣∣s dθ)1/s≤≤2π∫0∣∣np(eiθ)− eiθp′(eiθ)∣∣s dθ1/s+2π∫0∣∣eiθp′(eiθ)∣∣s dθ1/s≤ (n−m)2π∫0∣∣p(eiθ)∣∣s dθ1/s+2π∫0∣∣p′(eiθ)∣∣s dθ1/s, (by (3.4))and Theorem 1.1 follows.Proof of Theorem 1.3. The polynomial q(z), given by (3.2) will haveno zeros in |z| < 1k . Now if(3.5) m0 = min|z|= 1k|q(z)| = min|z|= 1k∣∣∣znp(1/z)∣∣∣ = m′kn, by (1.4)).then, by Rouché’s theorem, the polynomialq(z) +m0βzn−m, |β| < kn−m,of degree n−m, will also have no zeros in |z| < 1k , 1k ≥ 1. Hence, by Lemma2.1, we have for s ≥ 1 and |β| < kn−m2π∫0∣∣∣∣q′(eiθ)+m′knβei(n−m−1)θ(n−m)∣∣∣∣sdθ1/s≤(n−m)C(k)s2π∫0∣∣∣∣q(eiθ)+m′knβei(n−m)θ∣∣∣∣sdθ1/s,i.e.,2π∫0∣∣∣∣np(eiθ)− eiθp′(eiθ)+ βm′kn(n−m)eimθ∣∣∣∣sdθ1/s≤(n−m)C(k)s2π∫0∣∣∣∣p(eiθ)+ βm′kneimθ∣∣∣∣sdθ1/s, (by (3.2)).(3.6)88 V. K. JAINNow by Minkowski inequality, we have for s ≥ 1 and |β| < kn−mn2π∫0∣∣∣∣p(eiθ)+m′knβeimθ∣∣∣∣sdθ1/s≤2π∫0∣∣∣∣np(eiθ)+m′knβ(n−m)eimθ − eiθp′(eiθ)∣∣∣∣sdθ1/s+2π∫0∣∣∣∣eiθp′(eiθ)+mm′knβeimθ∣∣∣∣sdθ1/s,and Theorem 1.3 follows, by (3.6).Proof of Theorem 1.5. The polynomialq(z) = znp(1/z)is a polynomial of degree n, with the propertyq(z) = znq(1/z), (by (1.8)).Hence, by Lemma 2.2, we have for s ≥ 12π∫0∣∣q′(eiθ)∣∣s dθ1/s< nDs2π∫0∣∣q(eiθ)∣∣s dθ1/s.Now Theorem 1.5 follows on lines, similar to those of Theorem 1.1.References[1] K. K. Dewan and N. K. Govil, An inequality for self inversive polynomials, J. Math.Anal. Appl. 95 (1983), 490.[2] N. K. Govil and Q. L. Rahman, Functions of exponential type not vanishing in a halfplane and related polynomials, Trans. Amer. Math. Soc. 137 (1969), 501-517.[3] A. Zygmund, A remark on conjugate series, Proc. Lond. Math. Soc. (2) 34 (1932),392-400.Mathematics Department,I.I.T., Kharagpur-721302,IndiaReceived : 09.02.2000

Integral inequalities for polynomials having a zero of order m at the origin

Abstract

Similar works

Full text

Available Versions

Hrčak - Portal of scientific journals of Croatia