A Note on a Theorem of T.J.Rivlin

Abdullah Mir,Ajaz Wani and Imtiaz Hussain

Department of Mathematics,University of Kashmir,Srinagar,190006,India

Abstract.In this paper,we obtain a result that improves the results of Govil and Nwaeze,Qazi and the classical result of Rivlin.

Key Words:Polynomial,maximum modulus,zeros.

1 Introduction and statement of results

For an arbitrary entire function f(z),letFor a polynomial P(z)=of degree n,it is known that

Inequality(1.1)is due to Varga[7]who attributed it to Zarantonello.

It is noted that equality holds in(1.1)if and only if P(z)has all its zeros at the origin,so it is natural to seek improvement under appropriate assumption on the zeros of P(z).It was shown by Rivlin[6]that ifin|z|<1,then(1.1)can be replaced by

As a generalization of(1.2),Govil[2]proved that if P(z)0 in|z|<1,then for 0

In 1992,Qazi[4]generalized(1.3)in a different direction and proved that if P(z)=a0+is a polynomial of degree n not vanishing in|z|<1 then for 0

More recently,Govil and Nwaeze[3]besides proving some other results,also proved the following generalization and refinement of(1.3).

Theorem 1.1.Letin,then for 0

Some more results related to inequalities that compares the growth of a polynomial on|z|=r and|z|=R,where r

In this note,we present the following extension of Theorem 1.1.As we shall see our result provides refinements of(1.2),(1.3)and(1.4)as well.

Theorem 1.2.Let1 ≤μ

Remark 1.1.Forμ=1,Theorem 1.2 reduces to Theorem 1.1.Taking k=1 in Theorem 1.2 we get the following refinement of(1.4).

Corollary 1.1.Let1≤μ

If we take R=1 in Theorem 1.2,we get

Corollary 1.2.Let1≤μ

The following extension and refinement of inequality(1.2)due to Rivlin[6]immediately follows from Corollary 1.2 by taking k=1 in it.

Corollary 1.3.Let1≤μ

2 Lemmas

For the proof of Theorem 1.2,we need the following lemmas.

Lemma 2.1.Let1≤μ

The above lemma is due to Pukhta[5].

Lemma 2.2.Let1≤μ

The above lemma is due to Bidkham and Dewan[1].

3 Proof of the theorem

Proof of Theorem 1.2.Let 0

which implies

Now for 0

which implies on using Lemma 2.2,

which gives for 0

Hence from(3.2),we get

which is equivalent to

which is(1.6)and this completes the proof of Theorem 1.2.