A Note On A Result Due To Ankeny And Rivlin

A Note On A Result Due To Ankeny And Rivlin

Eze Raymond Nwaeze

Received 27 November 2015

Abstract

Letp(z) =a0+a1z+a2z²+a3z³+ +anzⁿbe a polynomial of degreenhaving no zeros in the unit disk. Then it is well known that forR 1;max_jzj=Rjp(z)j

Rⁿ+ 1

2 max_jzj=1jp(z)j:In this paper, we consider polynomials with gaps, hav- ing all its zeros on the circleS(0; K) :=fz:jzj=Kg; 0< K 1; and estimate the value of max_jzj=Rjp(z)j

max_jzj=1jp(z)j

s

for any positive integers:

1 Introduction

Letp(z) =Pn

j=0a_jz^j be a polynomial of degreen:We will denote M(p; r) := max

jzj=rjp(z)j; r >0;

jjpjj:= max

jzj=1jp(z)j; and

D(0; K) :=fz:jzj< Kg; K >0:

Bernstein observed the following result, which in fact is a simple consequence of the maximum modulus principle (see [8, p. 137]). This inequality is also known as the Bernstein’s inequality.

THEOREM 1. Letp(z) =Pn

j=0a_jz^jbe a polynomial of degreen:Then forR 1;

M(p; R) Rⁿjjpjj: Equality holds for p(z) = zⁿ; being a complex number.

For polynomial of degreennot vanishing in the interior of the unit circle, Ankeny and Rivlin [1] proved the following result.

Mathematics Sub ject Classi…cations: 15A18; 30C10; 30C15; 30A10.

yDepartment of Mathematics, Tuskegee University, Tuskegee, AL 36088, USA.

170

THEOREM 2 (Ankeny and Rivlin [1]). Letp(z) =Pn

j=0a_jz^j6= 0inD(0;1):Then forR 1;

M(p; R) Rⁿ+ 1

2 jjpjj: (1)

Here equality holds forp(z) = + zⁿ

2 ;wherej j=j j= 1:

In 2005, Gardner, Govil and Musukula [3] proved the following generalization and sharpening of Theorem 2.

THEOREM 3. Letp(z) =a0+Pn

j=tajz^j; 1 t n;be a polynomial of degreen andp(z)6= 0 inD(0; K); K 1:Then forR 1;

M(p; R) Rⁿ+s₀

1 +s₀ jjpjj Rⁿ 1

1 +s₀ m n 1 +s₀

"

(kpk m)² (1 +s₀)²ja_nj² (jjpjj m)

#

( (R 1)(kpk m)

(kpk m) + (1 +s₀)ja_nj ln

"

1 + (R 1)(kpk m) (kpk m) + (1 +s₀)ja_nj

#)

;(2) where m= min_jzj=Kjp(z)j;and

s0=K^t+1

t n jatj

ja0j mK^{t 1}+ 1

t n

jatj

ja0j mK^t+1+ 1:

Several research monographs have been written on this subject of inequalities (see for example Govil and Mohapatra [4], Milovanovi´c, Mitrinovi´c and Rassias [7], Rahman and Schmeisser [9], and recent article of Govil and Nwaeze [5]).

While trying to obtain an inequality analogous to (1) for polynomials not vanishing in D(0; K); K 1;Dewan and Ahuja [2] were able to prove this only for polynomials having all the zeros on the circleS(0; K) :=fz:jzj=Kg; 0< K 1:

THEOREM 4. Let p(z) = Pn

j=0ajz^j be a polynomial of degree n having all its zeros on S(0; K); K 1:Then forR 1 and for every positive integers;

fM(p; R)g^s K^{n 1}(1 +K) + (R^ns 1)

K^{n 1}+Kⁿ fM(p;1)g^s:

Fors= 1, Theorem 4 reduces to the following Corollary 5.

COROLLARY 5. Letp(z) =Pn

j=0ajz^j be a polynomial of degreenhaving all its zeros on S(0; K); K 1:Then forR 1;

M(p; R)

"

K^{n 1}(1 +K) + (Rⁿ 1) K^{n 1}+Kⁿ

#

M(p;1): (3)

2 Main Results

THEOREM 6. Let

p(z) =z^m 2

4a_{n m}z^{n m}+

n mX

j=

an m jz^{n m j} 3 5;

where1 n mand0 m n 1;be a polynomial of degreen;havingm f old zeros at origin and remaining n m zeros on S(0; K); K 1: Then for R 1 and every positive integer s;

[M(p; R)]^s L( ;K; m; n; s)[M(p;1)]^s; where

L( ;K; m; n; s) = 1

n(K^{n m 2 +1}+K^{n m +1})

"

n(K^{n m 2 +1}+K^{n m +1})

+(R^ns 1)[n+mK^{n m 2 +1}+mK^{n m +1} m]

# :

Form= 0;by Theorem 6, we have COROLLARY 7. Letp(z) =a_nzⁿ+Pn

j= a_{n j}z^{n j}; 1 n;be a polynomial of degree n; having all zeros onjzj= K; K 1:Then for R 1 and every positive integer s;

[M(p; R)]^s L( ;K; n; s)[M(p;1)]^s; where

L( ;K; n; s) = Kⁿ (K¹ +K) + (R^ns 1) K^{n 2 +1}+K^{n +1} :

If we set = 1 into Corollary 7, we get the following result of Dewan and Ahuja [2].

COROLLARY 8. Let p(z) =Pn

j=0ajz^j; be a polynomial of degree n; having all zeros on jzj=K; K 1:Then forR 1 and every positive integers;

[M(p; R)]^s L(1;K; n; s)[M(p;1)]^s; where

L(1;K; n; s) = K^{n 1}(1 +K) + (R^ns 1) K^{n 1}+Kⁿ :

3 Lemmas

For the proof of Theorem 6, we will need the following lemmas. The …rst lemma is due to Kumar and Lal [6].

LEMMA 9. Let

p(z) =z^m 2

4a_{n m}z^{n m}+

n mX

j=

a_{n m j}z^{n m j} 3 5;

where1 n mand0 m n 1;be a polynomial of degreen;havingm f old zeros at origin and remainingn mzeros onjzj=K; K 1:

maxjzj=1jp⁰(z)j n+m(K^{n m 2 +1}+K^{n m +1} 1) K^{n m 2 +1}+K^{n m +1} max

jzj=1jp(z)j:

The next lemma is the Bernstein inequality given in Theorem 1.

LEMMA 10. Letp(z)be a polynomial of degreen:Then forR 1;

M(p; R) RⁿM(p;1):

We now turn our attention to proof of the main result.

PROOF OF THEOREM 6. By Lemma 9, we have

maxjzj=1jp⁰(z)j n+m(K^{n m 2 +1}+K^{n m +1} 1) K^{n m 2 +1}+K^{n m +1} max

jzj=1jp(z)j:

Applying Lemma 10 to the polynomial p⁰(z)which is of degree n 1; it follows that for allR 1 and 2[0;2 );

p⁰(Reⁱ ) max

jzj=Rjp⁰(z)j R^{n 1}max

jzj=1jp⁰(z)j

R^{n 1} n+m(K^{n m 2 +1}+K^{n m +1} 1) K^{n m 2 +1}+K^{n m +1} max

jzj=1jp(z)j: So for each 2[0;2 )andR 1;we obtain

p(Reⁱ ) ^s p(eⁱ ) ^s= Z R

1

d p(teⁱ ) ^s dt dt

= Z R

1

s p(teⁱ ) ^{s 1}p⁰(eⁱ )eⁱ dt:

This implies that

p(Reⁱ )^s p(eⁱ )^s+s Z R

1

p(teⁱ )^{s 1} p⁰(eⁱ ) dt:

Therefore, [M(p; R)]^s

[M(p;1)]^s+s Z R

1

[tⁿM(p;1)]^{s 1} p⁰(eⁱ ) dt

[M(p;1)]^s+s Z R

1

t^{ns n}[M(p;1)]^{s 1}t^{n 1}n+m(K^{n m 2 +1}+K^{n m +1} 1) K^{n m 2 +1}+K^{n m +1} M(p;1)dt

= [M(p;1)]^s+s n+m(K^{n m 2 +1}+K^{n m +1} 1)

K^{n m 2 +1}+K^{n m +1} [M(p;1)]^s Z R

1

t^{ns 1}dt

= [M(p;1)]^s+ [M(p;1)]^s n+m(K^{n m 2 +1}+K^{n m +1} 1)

K^{n m 2 +1}+K^{n m +1} sR^ns 1 ns

= [M(p;1)]^s 8<

:1 + h

n+m(K^{n m 2 +1}+K^{n m +1} 1) i

(R^ns 1) n(K^{n m 2 +1}+K^{n m +1})

9=

;:

This yields

[M(p; R)]^s [M(p;1)]^s

n(K^{n m 2 +1}+K^{n m +1}) n K^{n m 2 +1}+K^{n m +1} +h

n+m(K^{n m 2 +1}+K^{n m +1} 1)i

(R^ns 1) : This completes the proof.

Acknowledgment. Many thanks to the anonymous referee for his/her valuable comments.

References

[1] N. C. Ankeny and T. J. Rivlin, On a theorem of S. Bernstein, Paci…c J. Math., 5(1955), 849–852 .

[2] K. K. Dewan and A. Ahuja, Growth of polynomials with prescribed zeros, J. Math.

Inequal., 5(2011), 355–361.

[3] R. B. Gardner, N. K. Govil and S. R. Musukula, Rate of growth of polynomials not vanishing inside a circle, JIPAM. J. Inequal. Pure Appl. Math., 6(2005), article 53.

[4] N. K. Govil and R. N. Mohapatra, Markov and Bernstein Type inequalities for Polynomials, J. of Inequal. and Appl., 3(1999), 349–387.

[5] N. K. Govil and E. R. Nwaeze, Bernstein type inequalities concerning growth of polynomials, Mathematical Analysis, Approximation Theory and their Applica- tions, Springer International Publishing, Switzerland, (2016), 293–316.

[6] S. Kumar and R. Lal, Generalizations of some polynomial inequalities, Int. Electron.

J. Pure Appl. Math., 3(2011), 111–117.

[7] G. V. Milovanovic, D. S. Mitrinovic and Th. M. Rassias, Topics in Polynomials:

Extremal Problems, Inequalities, Zeros, World Scienti…c Publishing Co. Pte. Ltd., (1994).

[8] G. Pólya and G. Szeg½o, Aufgaben und Lehrsätze aus der Analysis,4^thed., Springer Verlag, Berlin, 1(1970).

[9] Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, Oxford Univer- sity Press, New York, (2002).