1 Introduction and statement of results

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

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

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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 if in|z|<1,then(1.1)can be replaced by

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

where

Proof of Theorem 1.2.Let 0Also k≥1,therefore,Since in|z|where Using Lemma 2.1 to the polynomial T(z),we get

Theorem 1.1.Let in,then for 0

Theorem 1.2.Let 1 ≤µin|z|

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

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.

where

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

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.Let 1≤µ0 in|z|<1,then for 0

where

Corollary 1.2.Let 1≤µ0 in|z|

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

式中：表示径流量观测值；表示BP神经网络对径流量的模拟值；表示该场洪水径流量的平均值；N表示观测值的个数。NSE的取值为-,1，NSE的值越接近1则表示模型的模拟度越高，模型的可信度越高。

where

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.Let 1≤µ0 in|z|<1,then for 0

where

2 Lemmas

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

Lemma 2.1.Let 1≤µ in|z|

The above lemma is due to Pukhta[5].

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Lemma 2.2.Let 1≤µ in|z|

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

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3 Proof of the theorem

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

which implies

Now for 0

农业用水主要是灌溉，从2005～2017年间，绥中县灌溉面积年平均增长率为2.5%，目前水田灌溉面积1万hm2，旱田3万hm2，蔬菜0.57万hm2。预测在2020～2030年间，绥中县灌溉面积年平均增长率为1.5%，到2030年，水田灌溉面积达1.16万hm2，旱田3.48万hm2，蔬菜0.66万hm2。

which implies on using Lemma 2.2,

which gives for 0

Hence from(3.2),we get

which is equivalent to

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

References

[1]M.Bidkham and K.K.Dewan,Inequalities for a polynomial and its derivative,J.Math.Anal.Appl.,166(1992),319–324.

[2]N.K.Govil,On the maximum modulus of polynomials,J.Math.Anal.Appl.,112(1985),253–258.

[3]N.K.Govil and Eze R.Nwaeze,Some sharpening and generalizations of a result of T.J.Rivlin,Anal.Theory Appl.,33(2017),219–228.

[4]M.A.Qazi,On the maximum modulus of polynomials,Proc.Amer.Math.Soc.,115(1992),337–343.

[5]M.S.Pukhta,Extremal Problems for Polynomials and on Location of Zeros of Polynomials,Ph.D Thesis submitted to the Jamia Millia Islamia,New Delhi,1995.

[6]T.J.Rivlin,On the maximum modulus of polynomials,Amer.Math.Monthly,67(1960),251–253.

[7]R.S.Varga,A comparision of the successive overrelaxation method and semi-iterative methods using Chebyshev polynomials,J.Soc.Indust.Appl.Math.,5(1957),39–46.

[8]A.Zireh,E.Khojastehnejhad and S.R.Musawi,Some results concerning growth of polynomials,Anal.Theory Appl.,29(2013),37–46.