Cmjv01i02p0121 by Journal of Computer and Mathematical Sciences

J. Comp. & Math. Sci. Vol. 1(2), 121-126 (2010).

On Maximum Modulus of Polynomials M.S. PUKHTA Div. of Agri. Engineering S. K. University of Agricultural Sciences & Technology of Kashmir, Srinagar-111921 (India) E-mail: mspukhta_67@yahoo.co.in

ABSTRACT In this paper we obtain certain inequalities for polynomials having zeros in the closed exterior or closed interior of a circle. Our result generalize as well as improve upon some well known results. Key words : Polynomials, Derivative, Zeros, Inequalities, Extremal problems

1. INTRODUCTION AND STATEMENT OF RESULTS

1  r  M p ,r    M  p , 1 , r  1 (1.1)  2 

Let p(z) be a polynomial of degree n and let M  p , R   max p  z  , z R

then we have, as a simple deduction from maximum modulus principle [7, p.158 problem III 267 and 269]

M  p , R   r n M  p ,1 , r  1 with equality only for

p  z    z n ,   1. For polynomials not vanishing in z  1, Rivlin8 has obtained a stronger inequality

Here the equality holds for n p  z      z  ,

  .

Aziz 1 obtained the following generalization of inequality (1.1) for polynomials not vanishing in z  K ,

K  0. Theorem 1.1. If p  z  is a polynomial of degree n, having no zeros in

z  K , K  0 then

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M.S. Pukhta, J.Comp.&Math.Sci. Vol.1(2), 121-126 (2010). n

 r  K M p , r    M  p ,1 , for K  1 1 K 

& r  1 or K  1 & r  K 2 . Here equality holds for the polynomial

p z   z  K  . n

On applying Theorem 1.1 to the

1 polynomial z p   Aziz 1 obtained z n

Theorem 1.2. If p  z  is a polynomial of degree n, having all its zeros in z  K,

K  0 then n

 R K M  p , R    M  p ,1, for K  1  1 K 

& R  1 or K  1 & R  K 2 .

 R K M  p , R  R s   M  p ,1,for s  n .  1 K  Where s is the order of a possible zero

of p  z  at z  0 . Recently Dewan,Harish & Yadav5 proved the following improvement of the above theorem. Theorem 1.4. Let p  z  be a polynomial of degree n, having all its zeros in z  K, K  1 then for K  R  K 2 , then n

 R K M  p , R  R   M  p ,1+  1 K  s

 R  K  1  n   m R  R s  n  K   1  K  Where m  min p  z  and s is the order z K

of possible zero of p  z  at Here equality holds for the polynomial p  z    z  K  . n

Jain4 proved the following improvement of the above theorem. Theorem 1.3. If p  z  is a polynomial of degree n, having all its zeros

In this paper we have obtained the refinement of theorem 1.3 which is turn also improves theorem 1.4. Theorem 1.5. Let p  z  be a polynomial of degree n, having all its zeros in z  K, K  1 then for K  R  K 2 , then n

in z  K , K  1 then for K  R  K 2 , then

z  0.

 R  K   M  p , R   R   1  M  p ,1  K  R s

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M.S. Pukhta, J.Comp.& Math.Sci. Vol.1(2), 121-126 (2010).

   1  n s 1  R  K  mp , K  + n  R  R     K   R  R K  Where s is the order of a possible zero

its zeros in z  K , K  0 then for r  K and R  K ,

M p ,r 

of p  z  at z  0 . 2. Lemmas : For the proof of theorem, we require the following lemmas. n

j Lemma 2.1. If p z   a0   a j z

123

r  1 r n  K  r n 1 M  p , R, r  1 R n  K  R n 1 1   n.

Proof of Lemma 2.3. Let r  k ,

R  k then the polynomial F  z   prz  has no zeros in z 

j

K K ,  1. r r

is a polynomial of degree n, having no Since

zeros in z  K , K  1 then





M p / ,1 

n M  p ,1 1 K 

K  1, we have by lemma 1, r

M F , 1  /

nr M F , 1 r  K 

The above lemma is due to Chan and Malik 3. Lemma 2.2. If p z   a0 



 a

n r  1 Or M p , r   M  p , r  (2.1) r K

z ,



j

1    n , is a polynomial of degree n,

As p /  z  is a polynomial of degree

having no zeros in z  K , K  1 then

n 1, we have by Maximum Modulus Principle 6.





M p / ,1 

n  p z    M  p ,1  min z k  1 K  

The above lemma is due to Govil6 (see also2). Lemma 2.3. If p z   a0 





M p/ , r r n 1





, tr



 1

n 1

  n r t r K r 



n 1

M  p , r , t  r (2.3)

j

is a polynomial of degree n, having all

(2.2)

Combining (2.1) & (2.2), we get

M p/ ,t 

 a



M p/ ,t t n 1

Now for 0    2 & r  K , R  K , we e

Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

124

M.S. Pukhta, J.Comp.&Math.Sci. Vol.1(2), 121-126 (2010).

have

1    n is a polynomial of degree n ,





 

having all its zeros in z  K , K  0

 

p R e i  p re i   p / te i e i dt

then for r  K and R  K ,

r R



or p R e

i

  p re    p te e i

i

M p,r M  p , R  n  1   n 1 r r K r R r  K  R n 1 n

 1

which on using (2.3) gives





 1 n 1

    nr t r  K r

p Re i  p re i



M  p , r  dt

n 1

  1 1  m  n  1  n  1  n 1  n 1  R r K R  r r  K r

R  r r  r  K r n



Where m  min p  z  .

 1



n 1

Proof of Lemma 2.4. The proof of lemma follows on the same lines as the proof of lemma 2.3, by using lemma 2.2 instead of lemma 2.1. We omit the details.

Thus

R  r r r  K r n



p Re

i



 1



n 1

z K

M p , r

 

 p re i

Proof of Theorem1.5 : The poly-

Which implies





 R n r n r  1  M  p , R    n 1   1  M p , r  r r  K 





r  1 R n  K  R n 1 Or M  p , R   n   1 M p , r, r  K  r n 1 rR

M p,r M  p , R   1 n Or  1 n  n 1 r r K r r R  K  R n 1 Which completes the proof of lemma 2.3. Lemma 2.4. If p z   a0 

 a j

z j,

1 z

n nomial q z   z p  has all its zeros

in z 

1 1 ,  1 and is of degree n-s.. K K

On applying lemma 2.4 to the polynomial q z  with R  1 , we have ve

 ns  1  1  ns 1   r r    r  K     max qz max qz    z 1 z r 1   r 1       K   

Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

M.S. Pukhta, J.Comp.&Math.Sci. Vol.1(2), 121-126 (2010).

   1   r ns r 1     r ns1  K    min q z + 1  ,  z 1 1   r  1       K K   

1 1 R 2 K K  ns1   1  r r     K   r n max p z  p  z  max 1 z 1 1 1 z r  r    K      1   r ns r 1     r ns1  1 K   min p z + n 1   z K 1 K r 1     K   Or

   max pz  z 1  

   1   r ns r 1     r ns1  1  K   min p z + n n 1   z K 1 r K  1 r     K  

1 1 r 2 K K 1 Replacing r by , in the above inequaR for

lity we get

   max pz  z 1  

 R  1 1       n s R R R K   + n 1  min pz 1   z K K   R        K   R  n

Which is equivalent to

1   r   1 K max pz  s1  1 r  r 1  1 z r   K

1  1    max pz  Rs1  R K z R  1 1  1   R K

125

 R  K    max pz + max pz  Rs   1   z R  R  K  z 1



Rs 1 R  K  1  n  R  K n  R K   R

  min pz  

z K

for

K  R  K2 This completes the proof of the Theorem 1.5. REFERENCES 1. Aziz, A., Growth of polynomial whose zeros are within or outside a circle, Bull. Aust. Math. Soc. 35, 247-256 (1987). 2. A. Aziz and N. A. Rather, New Lq inequalities for polynomials, Mathematical Inequalities & Applications 1, 177-191 (1998). 3. T.C. Chan and M.A. Malik, On ErdosLax theorem, Proc. Indian Acad. Sci. (Math Science) 92 (3), 191 - 193 (1983).

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M.S. Pukhta, J.Comp.&Math.Sci. Vol.1(2), 121-126 (2010).

4. Jain. V. K., On polynomials having zeros in closed exterior or closed interior of a circle, Indian J. Pure & Appl. Math. 30, 153-159 (1999). 5. K.K. Dewan, Harish Singh and R.S. Yadav, Inequalities concerning polynomials having zeros in closed exterior or closed interior of a circle, Southeast asian Bulletin of Mathematics 27,

591-597 (2003). 6. N.K. Govil, On growth of polynomials, J. of Inequal. & Appl. 7 (5), 623-631 (2002). 7. Polya, G. and Szego, G., Problems & Theorems in Analysis, 1, Berlin (1972). 8. Rivlin T.J., On the Maximum Modulus of polynomials, Amer. Math. Monthly 67, 251-253 (1960).

Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)