Letp(z)be a polynomial of degreen

INTEGRAL MEAN ESTIMATES FOR POLYNOMIALS WHOSE ZEROS ARE WITHIN A CIRCLE

K.K. DEWAN¹, NARESH SINGH¹, BARCHAND CHANAM¹AND ABDULLAH MIR²

1DEPARTMENT OFMATHEMATICS

FACULTY OFNATURALSCIENCES, JAMIAMILLIAISLAMIA(CENTRALUNIVERSITY)

NEWDELHI-110025, INDIA

nareshkuntal@yahoo.co.in

2POSTGRADUATEDEPARTMENT OFMATHEMATICS, UNIVERSITY OFKASHMIR

HAZRATBAL, SRINAGAR-190006, INDIA

Received 17 October, 2007; accepted 19 December, 2008 Communicated by G.V. Milovanovi´c

ABSTRACT. Letp(z)be a polynomial of degreen. Zygmund [11] has shown that fors≥1 Z 2π

0

|p⁰(e^iθ)|^sdθ ^1/s

≤n Z 2π

0

|p(e^iθ)|^sdθ ^1/s

.

In this paper, we have obtained inequalities in the reverse direction for the polynomials having a zero of ordermat the origin. We also consider a problem for the class of polynomialsp(z) = a_nzⁿ+

n

P

ν=µ

a_n−νz^n−νnot vanishing outside the disk|z|< k,k≤1and obtain a result which, besides yielding some interesting results as corollaries, includes a result due to Aziz and Shah [Indian J. Pure Appl. Math., 28 (1997), 1413–1419] as a special case.

Key words and phrases: Polynomials, Zeros of orderm, Inequalities, Polar derivatives.

2000 Mathematics Subject Classification. 30A10, 30C10, 30D15.

1. INTRODUCTION ANDSTATEMENT OFRESULTS

Letp(z) be a polynomial of degreen and p⁰(z) its derivative. It was shown by Turán [10]

that ifp(z)has all its zeros in|z| ≤1, then

(1.1) max

|z|=1|p⁰(z)| ≥ n 2 max

|z|=1|p(z)|.

The work of second author is supported by Council of Scientific and Industrial Research, New Delhi, under grant F.No.9/466(78)/2004- EMR-I..

315-07

More generally, if the polynomial p(z) has all its zeros in |z| ≤ k, k ≤ 1, it was proved by Malik [5] that the inequality (1.1) can be replaced by

(1.2) max

|z|=1|p⁰(z)| ≥ n

1 +k max

|z|=1|p(z)|.

Malik [6] obtained a L^p analogue of (1.1) by proving that if p(z) has all its zeros in|z| ≤ 1, then for eachr >0

(1.3) n

Z 2π

0

|p(e^iθ)|^rdθ ¹_r

≤ Z 2π

0

|1 +e^iθ|^rdθ ¹_r

max

|z|=1|p⁰(z)|.

As an extension of (1.3) and a generalization of (1.2), Aziz [1] proved that if p(z) has all its zeros in|z| ≤k,k ≤1, then for eachr >0

(1.4) n

Z 2π

0

|p(e^iθ)|^rdθ ¹_r

≤ Z 2π

0

|1 +ke^iθ|^rdθ ¹_r

max

|z|=1|p⁰(z)|.

If we letr → ∞in (1.3) and (1.4) and make use of the well known fact from analysis (see for example [8, p. 73] or [9, p. 91]) that

(1.5)

Z 2π

0

|p(e^iθ)|^rdθ ¹_r

→ max

0≤θ<2π

p(e^iθ)

as r→ ∞, we get inequalities (1.1) and (1.2) respectively.

In this paper, we will first obtain a Zygmund [11] type integral inequality, but in the reverse direction, for polynomials having a zero of ordermat the origin. More precisely, we prove Theorem 1.1. Let p(z) = z^mPn−m

j=0 ajz^j be a polynomial of degree n, having all its zeros in

|z| ≤k,k≤1, with a zero of ordermatz = 0. Then forβwith|β|< k^n−m ands ≥1

(1.6)

Z 2π

0

p⁰(e^iθ) + mm⁰ kⁿ

βe¯ ^i(m−1)θ

s

dθ ¹s

≥

n−(n−m)C_s^(k)

Z 2π

0

p(e^iθ) + m⁰ kⁿ

βe¯ ^imθ

s

dθ ¹_s

,

wherem⁰ = min

|z|=k|p(z)|,

C_s^(k) = 1

2π Z 2π

0

|S_c+e^iθ|^sdθ ⁻¹_s

and Sc =

1 n−m

an−m−1

an−m

+ 1 k² + _n−m¹

a_n−m−1 an−m

.

By takingk= 1andβ = 0in Theorem 1.1, we obtain:

Corollary 1.2. Ifp(z)is a polynomial of degreen, having all its zeros in|z| ≤ 1, with a zero of ordermatz = 0, then fors ≥1

(1.7)

Z 2π

0

|p(e^iθ)|^sdθ ¹_s

≥

n−(n−m)C_s⁽¹⁾

Z 2π

0

|p(e^iθ)|^sdθ ¹_s

,

where

C_s⁽¹⁾ = 1 1

2π R2π

0 |1 +e^iθ|^sdθ ¹_s .

By lettings→ ∞in Theorem 1.1, we obtain Corollary 1.3. Letp(z) =z^mPn−m

j=0 a_jz^j be a polynomial of degreen, having all its zeros in

|z| ≤k,k≤1, with a zero of ordermatz = 0. Then forβwith|β|< k^n−m

(1.8) max

|z|=1

p⁰(z) + mm⁰ kⁿ

βz¯ ^m−1

≥

m+nS_c 1 +S_c

max

|z|=1

p(z) + m⁰ kⁿ

βz¯ ^m ,

wherem⁰ andS_c are as defined in Theorem 1.1.

By choosing the argument ofβ suitably and letting|β| →k^n−min Corollary 1.3, we obtain the following result.

Corollary 1.4. Letp(z) =z^mPn−m

j=0 a_jz^j be a polynomial of degreen, having all its zeros in

|z| ≤k,k≤1, with a zero of ordermatz = 0. Then

(1.9) max

|z|=1|p⁰(z)| ≥

m+nS_c 1 +Sc

max

|z|=1|p(z)|+ (n−m)S_c 1 +Sc

m⁰ k^m, wherem⁰ andS_c are as defined in Theorem 1.1.

LetD_αp(z)denote the polar derivative of the polynomialp(z)of degreenwith respect to the pointα. Then

Dαp(z) = np(z) + (α−z)p⁰(z).

The polynomialD_αp(z)is of degree at most(n−1)and it generalizes the ordinary derivative in the sense that

(1.10) lim

α→∞

Dαp(z)

α =p⁰(z).

Our next result generalizes as well as improving upon the inequality (1.4), which in turns, gives a generalization as well as improvements of inequalities (1.3), (1.2) and (1.1) in terms of the polar derivatives ofL^pinequalities.

Theorem 1.5. If p(z) = anzⁿ +Pn

j=µan−jz^n−j, 1 ≤ µ ≤ n, is a polynomial of degree n, having all its zeros in |z| ≤ k, k ≤ 1, then for every real or complex numbersα and β with

|α| ≥k^µand|β| ≤1and for eachr >0 (1.11) max

|z|=1|Dαp(z)|

≥ n(|α| −k^µ) R2π

0 |1 +k^µe^iθ|^rdθ¹_r

Z 2π

0

p(e^iθ) + βm⁰

k^n−µe^i(n−1)θ

r

dθ ¹_r

+ n k^n−µm⁰,

wherem⁰ = min

|z|=k|p(z)|.

Dividing both sides of (1.11) by|α|, letting|α| → ∞and noting that (1.10), we obtain Corollary 1.6. If p(z) = a_nzⁿ +Pn

j=µa_n−jz^n−j, 1 ≤ µ ≤ n, is a polynomial of degree n, having all its zeros in|z| ≤k,k ≤1, then for every real or complex numberβwith|β| ≤1, for eachr >0

(1.12) max

|z|=1|p⁰(z)| ≥ n

R2π

0 |1 +k^µe^iθ|^rdθ¹_r

Z 2π

0

p(e^iθ) + βm⁰

k^n−µe^i(n−1)θ

r

dθ ¹_r

,

wherem⁰ = min

|z|=k|p(z)|.

Remark 1. Lettingr → ∞in (1.12) and choosing the argument ofβ suitably with|β| = 1, it follows that, ifp(z) = a_nzⁿ+Pn

j=µan−jz^n−j,1≤µ≤n, is a polynomial of degreen, having all its zeros in|z| ≤k,k ≤1, then

(1.13) max

|z|=1|p⁰(z)| ≥ n (1 +k^µ)

max|z|=1|p(z)|+ 1 k^n−µ min

|z|=k|p(z)|

.

Inequality (1.13) was already proved by Aziz and Shah [2].

2. LEMMAS

For the proofs of these theorems we need the following lemmas.

Lemma 2.1. Let p(z) = Pn

j=0a_jz^j be a polynomial of degree n having no zeros in |z| < k, k ≥1. Then fors≥1

(2.1)

Z 2π

0

|p⁰(e^iθ)|^sdθ ¹_s

≤nSs

Z 2π

0

|p(e^iθ)|^sdθ ¹_s

,

where

S_s = 1

2π Z 2π

0

|S_c⁰ +e^iθ|^sdθ ¹_s

and S_c⁰ = k²h

1 n

a1

a0

+ 1i 1 + _n¹

a1

a0

k²

.

The above lemma is due to Dewan, Bhat and Pukhta [3].

The following lemma is due to Rather [7].

Lemma 2.2. Let p(z) = a_nzⁿ +Pn

j=µan−jz^n−j, 1 ≤ µ ≤ n, be a polynomial of degree n having all its zero in|z| ≤k,k≤1. Then

(2.2) k^µ|p⁰(z)| ≥ |q⁰(z)|+ n k^n−µ min

|z|=k|p(z)| for |z|= 1, whereq(z) =zⁿp ¹_¯z

.

3. PROOFS OFTHETHEOREMS

Proof of Theorem 1.1. Let

p(z) = z^m

n−m

X

j=0

ajz^j =z^mφ(z), (say) whereφ(z)is a polynomial of degreen−m, with the property that

φ(0)6= 0. Then

q(z) =zⁿp 1

¯ z

=z^n−mφ 1

¯ z

is also a polynomial of degreen−mand has no zeros in|z|< _k¹, ¹_k ≥1. Now if m₀ = min

|z|=¹

k

|q(z)|= min

|z|=¹

k

zⁿp 1

¯ z

= 1 kⁿ min

|z|=k|p(z)|= m⁰ kⁿ , then, by Rouche’s theorem, the polynomial

q(z) +m₀βz^n−m, |β|< k^n−m,

of degreen−m, will also have no zeros in|z|< ¹_k, _k¹ ≥1. Hence, by Lemma 2.1, we have for s≥1and|β|< k^n−m

Z 2π

0

q⁰(e^iθ) + m⁰

kⁿβe^{i(n−m−1)θ}(n−m)

s

dθ ¹_s

≤(n−m)C_s^(k) Z 2π

0

q(e^iθ) + m⁰

kⁿβe^i(n−m)θ

s

dθ ¹_s

,

which implies (3.1)

Z 2π

0

np(e^iθ)−e^iθp⁰(e^iθ) + ¯βm⁰

kⁿ(n−m)e^imθ

s

dθ ¹_s

≤(n−m)C_s^(k) Z 2π

0

p(e^iθ) + m⁰ kⁿ

βe¯ ^imθ

s

dθ ¹_s

.

Now by Minkowski’s inequality, we have fors≥1and|β|< k^n−m n

Z 2π

0

p(e^iθ) + m⁰ kⁿ

βe¯ ^imθ

s

dθ ¹_s

≤ Z 2π

0

np(e^iθ) + m⁰ kⁿ

β(n¯ −m)e^imθ −e^iθp⁰(e^iθ)

s

dθ ¹_s

+ Z 2π

0

e^iθp⁰(e^iθ) + mm⁰ kⁿ

βe¯ ^imθ

s

dθ ¹_s

,

which implies, by using inequality (3.1) n

Z 2π

0

p(e^iθ) + m⁰ kⁿ

βe¯ ^imθ

s

dθ ¹_s

≤(n−m)C_s^(k) Z 2π

0

p(e^iθ) + m⁰ kⁿ

βe¯ ^imθ

s

dθ ¹_s

+ Z 2π

0

p⁰(e^iθ) +mm⁰ kⁿ

βe¯ ^i(m−1)θ

s

dθ ¹_s

,

and the Theorem 1.1 follows.

Proof of Theorem 1.5. Sinceq(z) = zⁿp ¹_z¯

so thatp(z) = zⁿq ¹_¯z

, therefore, we have

(3.2) p⁰(z) =nzⁿ⁻¹q

1

¯ z

−zⁿ⁻²q⁰ 1

¯ z

,

which implies

(3.3) |p⁰(z)|=|nq(z)−zq⁰(z)| for |z|= 1. Using (3.2) in (2.2), we get for1≤µ≤n

|q⁰(z)|+ m⁰n

k^n−µ ≤k^µ|nq(z)−zq⁰(z)| for |z|= 1.

Now, from the above inequality, for every complexβ with|β| ≤1, we get, for|z|= 1

q⁰(z) + ¯βm⁰n k^n−µ

≤ |q⁰(z)|+ m⁰n k^n−µ

≤k^µ|nq(z)−zq⁰(z)|. (3.4)

For every real or complex numberαwith|α| ≥k^µ, we have

|D_αp(z)|=|np(z) + (α−z)p⁰(z)|

≥ |α| |p⁰(z)| − |np(z)−zp⁰(z)|, which gives by interchanging the roles ofp(z)andq(z)in (3.3) for|z|= 1

|D_αp(z)| ≥ |α||p⁰(z)| − |q⁰(z)|

≥ |α||p⁰(z)| −k^µ|p⁰(z)|+ m⁰n

k^n−µ (using (2.2)).

(3.5)

Sincep(z)has all its zeros in|z| ≤ k ≤ 1, by the Gauss-Lucas theorem, all the zeros of p⁰(z) also lie in|z| ≤1. This implies that the polynomial

zⁿ⁻¹p⁰ 1

¯ z

=nq(z)−zq⁰(z)

has all its zeros in|z| ≥ ¹_k ≥1. Therefore, it follows from (3.4) that the function

(3.6) w(z) =

zq⁰(z) + ¯β m⁰n k^n−µz k^µ(nq(z)−zq⁰(z))

is analytic for|z| ≤ 1and|w(z)| ≤ 1for|z| ≤ 1. Furthermore w(0) = 0. Thus the function 1 +k^µw(z)is a subordinate to the function1 +k^µzin|z| ≤1. Hence by a well-known property of subordination [4], we have forr >0and for0≤θ <2π,

(3.7)

Z 2π

0

|1 +k^µw(e^iθ)|^rdθ ≤ Z 2π

0

|1 +k^µe^iθ|^rdθ . Also from (3.6), we have

1 +k^µw(z) =

nq(z) + ¯βm⁰n k^n−µz nq(z)−zq⁰(z) , or

nq(z) + ¯β m⁰n k^n−µz

=|1 +k^µw(z)||p⁰(z)| for |z|= 1, which implies

(3.8) n

p(z) +β m⁰ k^n−µzⁿ⁻¹

=|1 +k^µw(z)||p⁰(z)| for |z|= 1.

Now combining (3.7) and (3.8), we get n^r

Z 2π

0

p(e^iθ) +β m⁰

k^n−µe^i(n−1)θ

r

dθ ≤ Z 2π

0

|1 +k^µe^iθ|^r|p⁰(e^iθ)|^rdθ .

Using (3.5) in the above inequality, we obtain n^r(|α| −k^µ)^r

Z 2π

0

p(e^iθ) +β m⁰

k^n−µe^i(n−1)θ

r

dθ

≤ Z 2π

0

|1 +k^µe^iθ|^rdθ

max|z|=1|D_αp(z)| − nm⁰ k^n−µ

r

,

from which we obtain the required result.

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