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International Journal of Engineering Inventions e-ISSN: 2278-7461, p-ISBN: 2319-6491 Volume 2, Issue 3 (February 2013) PP: 49-58

Further Generalizations of Enestrom-Kakeya Theorem M. H. Gulzar Department of Mathematics, University of Kashmir, Srinagar 190006

Abstract:- Many generalizations of the Enestrom –Kakeya Theorem are available in the literature. In this paper we prove some results which further generalize some known results. Mathematics Subject Classification: 30C10,30C15 Keywords and phrases:- Complex number, Polynomial , Zero, Enestrom – Kakeya Theorem.

I.

INTRODUCTION AND STATEMENT OF RESULTS

The Enestrom –Kakeya Theorem (see[6]) is well known in the theory of the distribution of zeros of polynomials and is often stated as follows: Theorem A: Let P ( z ) 

n

a j 0

j

z j be a polynomial of degree n whose coefficients satisfy

a n  a n 1  ......  a1  a0  0 . Then P(z) has all its zeros in the closed unit disk

z  1.

In the literature there exist several generalizations and extensions of this result. Joyal et al [5] extended it to polynomials with general monotonic coefficients and proved the following result: Theorem B: Let P ( z ) 

n

a j 0

j

z j be a polynomial of degree n whose coefficients satisfy

a n  a n 1  ......  a1  a 0 . Then P(z) has all its zeros in

z 

a n  a0  a0 an

.

Aziz and zargar [1] generalized the result of Joyal et al [6] as follows: Theorem C: Let P ( z ) 

n

a j 0

j

z j be a polynomial of degree n such that for some k  1

kan  a n 1  ......  a1  a0 . Then P(z) has all its zeros in

z  k 1 

kan  a0  a0 an

.

For polynomials ,whose coefficients are not necessarily real, Govil and Rahman [2] proved the following generalization of Theorem A: n

Theorem C: If

P ( z )   a j z j is a polynomial of degree n with Re( a j )   j and Im( a j )   j , j 0

j=0,1,2,……,n, such that

 n   n 1  ......   1   0  0 ,

where

 n  0 , then P(z) has all its zeros in 2 n z  1  ( )(   j ) .  n j 0

Govil and Mc-tume [3] proved the following generalisations of Theorems B and C:

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Further Generalizations of Enestrom-Kakeya Theorem Theorem D: Let P ( z ) 

n

a

z j be a polynomial of degree n with Re( a j )   j and Im( a j )   j ,

j

j 0

k  1, k n   n 1  ......   1   0 ,

j=0,1,2,……,n. If for some

then P(z) has all its zeros in n

z  k 1 

Theorem E: Let P ( z ) 

k n   0   0  2  j j 0

n n

a

z j be a polynomial of degree n with Re( a j )   j and Im( a j )   j ,

j

j 0

.

k  1, k n   n 1  ......  1   0 ,

j=0,1,2,……,n. If for some

then P(z) has all its zeros in n

k n   0   0  2  j

j 0 . n M. H. Gulzar [4] proved the following generalizations of Theorems D and E:

z  k 1 

Theorem F: Let P ( z ) 

n

a j 0

j

z j be a polynomial of degree n with Re( a j )   j and Im( a j )   j ,

  0 , 0    1,  ......   1   0 ,

j=0,1,2,……,n. If for some real numbers

   n   n 1

then P(z) has all its zeros in the disk n

z

  n

   n  2  0   ( 0   0 )  2  j j 0

n

Theorem G: Let P ( z ) 

n

a j 0

j

.

z j be a polynomial of degree n with Re( a j )   j and Im( a j )   j ,

j=0,1,2,……,n. If for some real number   0 ,

   n   n 1  ......  1   0 , then P(z) has all its zeros in the disk n

 z  n

   n  2  0   (  0   0 )  2  j j 0

n

.

In this paper we give generalization of Theorems F and G. In fact, we prove the following: Theorem 1: Let P ( z ) 

n

a j 0

j

z j be a polynomial of degree n with Re( a j )   j and Im( a j )   j ,

j=0,1,2,……,n. If for some real numbers  ,   0 , 1  k  n, a n  k  0,0    1,

and

 n  k 1

   n   n 1  ... n  k 1   n  k   n  k 1  ...   1   0 ,   n  k , then P(z) has all its zeros in the disk n

z and if

 an

 n  k   n  k 1 ,

   n  (  1) n  k    1  n  k  2  0   ( 0   0  2  j j 0

an

,

then P(z) has all its zeros in the disk

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Further Generalizations of Enestrom-Kakeya Theorem n

z

 an

Remark 1: Taking If a j are real i.e.

   n  (1   ) n  k  1    n  k  2  0   ( 0   0  2  j j 0

an

  1 , Theorem 1 reduces to Theorem F.

 j  0 for all j, we get the following result:

Corollary 1: Let P ( z ) 

n

a j 0

j

z j be a polynomial of degree n. If for some real numbers  ,   0 ,

1  k  n, a n k  0, 0    1,

  a n  a n 1  ...  a n  k 1  a n  k  a n  k 1  ...  a1  a 0 , and a n  k 1  a n  k , then P(z) has all its zeros in the disk

z

 an

  a n  (  1)a n  k    1 a n  k  2 a 0   (a 0  a 0 ) an

,,

and if a n  k  a n  k 1 , then P(z) has all its zeros in the disk

z

 an

  a n  (1   )a n  k  1   a n  k  2 a 0   (a 0  a 0 ) an

If we apply Theorem 1 to the polynomial –iP(z) , we easily get the following result: Theorem 2: Let P ( z ) 

n

a j 0

j

z j be a polynomial of degree n with Re( a j )   j and Im( a j )   j ,

 ,   0 , 1  k  n, a n k  0, 0    1,    n   n 1  ....   n  k 1   n  k   n  k 1  ..   1   0 ,   n  k , then P(z) has all its zeros in the disk

j=0,1, 2,……,n. If for some real numbers

and

 n  k 1

n

z and if

 an

 n  k   n  k 1 ,

   n  (  1)  n  k    1  n  k  2  0   (  0   0 )  2  j j 0

,

an then P(z) has all its zeros in the disk n

z

 an

Remark 2: Taking

   n  (1   )  n  k  1    n  k  2  0   (  0   0 )  2  j j 0

an

  1 , Theorem 2 reduces to Theorem G.

Theorem 3: Let P ( z ) 

n

a j 0

j

z j be a polynomial of degree n such that for some real numbers  ,   0 ,

1  k  n, a n  k  0,  ,0    1, ,

  an  an1  ...  ank 1   ank  ank 1  ...  a1  a0 and

arga j      If

 2

, j  0,1,......,n.

ank 1  ank (i.e.   1 ) , then P(z) has all its zeros in the disk

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Further Generalizations of Enestrom-Kakeya Theorem

[   a n (cos  sin  )  a n  k (cos  sin    cos   sin     1) z If

  a 0 (sin   cos  1)  2 a 0  2 sin 

an

n 1

a

j 1, j  n  k

j

]

an

ank  ank 1 (i.e.   1 ), then P(z) has all its zeros in the disk [   a n (cos  sin  )  a n  k (cos  sin    cos   sin   1   )

z

  a 0 (sin   cos  1)  2 a 0  2 sin 

an

n 1

a

j 1, j  n  k

j

]

an

Remark 3: Taking

  1 in Theorem 3 , we get the following result:

Corollary 2: Let P ( z ) 

n

a j 0

j

z j be a polynomial of degree n. If for some real numbers   0 , 0    1,

  an  an1  ...... a1   a0

,

then P(z) has all its zeros in the disk n 1

z

 an

[   a n (cos  sin  )   a 0 (sin   cos  1)  2 a 0  2 sin   a j ] j 1,

.

an

Remark 4: Taking

  (k  1)a n , k  1 in Cor.2, we get the following result:

Corollary 3: Let P ( z ) 

n

a j 0

j

z j be a polynomial of degree n. If for some real numbers   0 , 0    1,

k an  an1  ...... a1   a0 , then P(z) has all its zeros in the disk n 1

z Taking

 an

[k a n (cos  sin  )   a 0 (sin   cos  1)  2 a 0  2 sin   a j ] j 1,

an

  (k  1)a n , k  1 , and   1 in Cor. 3, we get a result of Shah and Liman [7,Theorem 1]. II.

LEMMAS

For the proofs of the above results , we need the following results: n

Lemma 1: .

Let P(z)=

a z j 0

arg a j     

ta j  a j 1

j

j

be a polynomial of degree n with complex coefficients such that

, j  0,1,...n, for some real  . Then for some t>0, 2  t a j  a j 1 cos  t a j  a j1 sin  .

The proof of lemma 1 follows from a lemma due to Govil and Rahman [2].

p( z)  M in z  1, then the number of zeros of p(z)in 1 M (see[8],p171). z   ,0    1, does not exceed log 1 p ( 0 ) log 

Lemma 2.If p(z) is regular ,p(0)  0 and

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Further Generalizations of Enestrom-Kakeya Theorem III.

PROOFS OF THEOREMS

Proof of Theorem 1: Consider the polynomial

F ( z )  (1  z ) P( z )

 (1  z )(a n z n  a n1 z n1  ...... a1 z  a0 )  a n z n1  (a n  a n1 ) z n  ...... (a nk 1  a nk ) z nk 1  (a nk  a n k 1 ) z nk  (a n  k 1  a n  k  2 ) z n  k 1  ......  (a1  a 0 ) z  a 0  ( n  i n ) z n 1  ( n   n 1 ) z n  ......  ( n  k 1   n  k ) z n  k 1

 ( n  k   n k 1 ) z n  k  ( n  k 1   n k  2 ) z n  k 1  ...... (1   0 ) z n

 (   1) 0 z   0  i  (  j   j 1 ) z j  i 0 j 1

If

 n  k 1   n  k ,

then

 n  k 1   n  k , and we have

F ( z )  ( n  i n ) z n 1  z n  (    n   n 1 ) z n  ......  ( n  k 1   n  k ) z n  k 1

 ( n  k   n  k 1 ) z n  k  (  1)an  k z n  k  ( n  k 1   n  k  2 ) z n  k 1  ...... n

 ( 1   0 ) z  (   1) 0 z   0  i  (  j   j 1 ) z j  i 0 . j 1

For

z  1,

F ( z)  an z n1  z n  (    n   n1 ) z n  ( n1   n2 ) z n1  ...... ( nk 1   nk ) z nk 1  ( n  k   n  k 1 ) z n  k  (  1)a n  k z n  k  ( n  k 1   n  k  2 ) z n  k 1  ...... n

 ( 1   0 ) z  (   1) 0 z   0  i  (  j   j 1 ) z j  i 0 j 1

1 1 n  z  an z    (    n   n1 )  ( n1   n2 )  ...... ( nk 1   nk ) k 1 z z  1 1 1  ( n  k   n  k 1 ) k  (  1) n  k k  ( n  k 1   n  k  2 )  .... k 1 z z n    1 1  ( 1   0 ) n 1  (   1) n 01  n0  i  (  j   j 1 ) n  j i n0  z z z z z j 1  z

n

a z     n

   n   n1   n1   n2  ......  nk 1   nk

  n  k   n  k 1    1  n  k   n  k 1   n  k  2  ......  1   0 n

   1  0   0    j   j 1   0

 z

n

a z       n



j 1

n

  n1   n1   n2  ...... nk 1   nk   nn k   nk 1

   1  n  k   n  k 1   n  k  2  ......  1   0  (1   )  0 n

  0  2  j



j 0

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Further Generalizations of Enestrom-Kakeya Theorem

 z

a

n

n

z    [{   n  (  1) n  k    1  n  k   ( 0   0 )  2  0

n

 2  j

j 0

>0 if n

a n z       n  (  1) n  k    1  n  k   ( 0   0 )  2  0  2  j j 0

This shows that the zeros of F(z) whose modulus is greater than 1 lie in n

z

 an

   n  (  1) n  k    1  n  k  2  0   ( 0   0 )  2  j j 0

an

.

But the zeros of F(z) whose modulus is less than or equal to 1 already satisfy the above inequality. Hence all the zeros of F(z) lie in n

z

 an

   n  (  1) n  k    1  n  k  2  0   ( 0   0 )  2  j j 0

an

.

Since the zeros of P(z) are also the zeros of F(z), it follows that all the zeros of P(z) lie in n

z If

 an

   n  (  1) n  k    1  n  k  2  0   ( 0   0 )  2  j j 0

an

.

 n  k   n  k 1 , then  n  k   n  k 1 , and we have F ( z )  ( n  i n ) z n 1  z n  (    n   n 1 ) z n  ......  ( n  k 1   n  k ) z n  k 1

 ( n  k   n  k 1 ) z n  k  (1   ) n  k z n  k 1  ( n  k 1   n  k  2 ) z n  k 1  ...... n

 ( 1   0 ) z  (   1) 0 z   0  i  (  j   j 1 ) z j  i 0 . j 1

For

z  1,

F ( z)  an z n1  z n  (    n   n1 ) z n  ( n1   n2 ) z n1  ...... ( nk 1   nk ) z nk 1  ( n  k   n  k 1 ) z n  k  (1   ) n  k z n  k 1  ( n  k 1   n  k  2 ) z n  k 1  ...... n

 ( 1   0 ) z  (   1) 0 z   0  i  (  j   j 1 ) z j  i 0 j 1

1 1 n  z  an z    (    n   n1 )  ( n1   n2 )  ...... ( nk 1   nk ) k 1 z z  1 1 1  ( n  k   n  k 1 ) k  (1   ) n  k k 1  ( n  k 1   n  k  2 ) k 1  .... z z z n    1 1  ( 1   0 ) n 1  (   1) n 01  n0  i  (  j   j 1 ) n  j i n0  z z z z z j 1  z

n

a z     n

   n   n1   n1   n2  ......  nk 1   nk

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Further Generalizations of Enestrom-Kakeya Theorem

  n  k   n  k 1  1    n  k   n  k 1   n  k  2  ......  1   0 n

   1  0   0    j   j 1   0

 z

n

a z       n



j 1

n

  n1   n1   n2  ...... nk 1   nk   nn k   nk 1

 1    n  k   n  k 1   n  k  2  ......  1   0  (1   )  0 n

  0  2  j



j 0

 a n z    {   n  (1   ) n  k  1    n  k   ( 0   0 )     n  z   2   2  j }  0  j 0   n

>0 if n

a n z       n  (1   ) n  k  1    n  k   ( 0   0 )  2  0  2  j j 0

This shows that the zeros of F(z) whose modulus is greater than 1 lie in n

z

 an

   n  (1   ) n  k  1    n  k  2  0   ( 0   0 )  2  j j 0

an

.

But the zeros of F(z) whose modulus is less than or equal to 1 already satisfy the above inequality. Hence all the zeros of F(z) lie in n

z

 an

   n  (1   ) n  k  1    n  k  2  0   ( 0   0 )  2  j j 0

an

.

Since the zeros of P(z) are also the zeros of F(z), it follows that all the zeros of P(z) lie in n

z

 an

   n  (1   ) n  k  1    n  k  2  0   ( 0   0 )  2  j j 0

an

.

That proves Theorem 1. Proof of Theorem 3: Consider the polynomial

F ( z )  (1  z ) P( z )  (1  z )(a n z n  a n1 z n1  ...... a1 z  a0 )

 a n z n1  (a n  a n1 ) z n  ...... (a nk 1  a nk ) z nk 1  (a nk  a n k 1 ) z nk  (a n  k 1  a n  k  2 ) z n  k 1  ......  (a1  a 0 ) z  a 0 . If

ank 1  ank , then ank 1  ank ,   1 and we have, for z  1, by using Lemma1,

F ( z)  an z n1  z n  (   an  an1 ) z n  (an1  an2 ) z n1  ...... (ank 1  ank ) z nk 1  (a n  k  a n  k 1 ) z n  k  (  1)a n  k z n  k  (a n  k 1  a n  k  2 ) z n  k 1  ......  (a1  a 0 ) z  (   1)a 0 z  a 0

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Further Generalizations of Enestrom-Kakeya Theorem

1 1 n  z  an z    (   an  an1 )  (an1  an2 )  ...... (ank 1  ank ) k 1 z z  1 1 1  (a n  k  a n  k 1 ) k  (  1)a n  k k  (a n  k 1  a n  k  2 ) k 1  .... z z z a a 1  (a1  a 0 ) n 1  (   1) n01  0n  z z z n  z  an z       an  an1  an1  an2  ...... ank 1  ank  a n  k  a n  k 1    1 a n  k  a n  k 1  a n  k  2  ...... a1  a 0    1 a0  a0

 z

n



 a z    {(   a n

n

 an1 ) cos  (   an  an1 ) sin 

 ( an1  an2 ) cos  ( an1  an2 ) sin   ...... ( ank 1  ank ) cos  ( ank 1  ank ) sin   ( ank  ank 1 ) cos  ( ank  ank 1 ) sin 

   1 ank  ( ank 1  ank 2 ) cos  ( ank 1  ank 2 ) sin   ...... ( a1   a0 ) cos  ( a1   a0 ) sin   (1   ) a0  a0 }

 z

n

a z    {   a n

n

(cos  sin  )  ank (cos  sin    cos   sin 

   1)   a 0 (sin   cos   1)  2 a 0  2 sin 

n 1

a

j 1, j  n  k

j

}

0 if

a z      a n

n

(cos  sin  )  ank (cos  sin    cos   sin 

   1)   a 0 (sin   cos   1)  2 a 0  2 sin 

n 1

a

j 1, j  n  k

j

This shows that the zeros of F(z) whose modulus is greater than 1 lie in

[   a n (cos  sin  )  a n  k (cos  sin    cos   sin     1)

z

 an

  a 0 (sin   cos  1)  2 a 0  2 sin  

n 1

a

j 1, j  n  k

j

] ..

an

But the zeros of F(z) whose modulus is less than or equal to 1 already satisfy the above inequality. Hence all the zeros of F(z) lie in

[   a n (cos  sin  )  a n  k (cos  sin    cos   sin     1)

z

 an

  a 0 (sin   cos  1)  2 a 0  2 sin  

n 1

a

j 1, j  n  k

j

] .

an

Since the zeros of P(z) are also the zeros of F(z), it follows that all the zeros of P(z) lie in

[   a n (cos  sin  )  a n  k (cos  sin    cos   sin     1)

z

 an

  a 0 (sin   cos  1)  2 a 0  2 sin  

n 1

a

j 1, j  n  k

j

]

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Further Generalizations of Enestrom-Kakeya Theorem If

ank  ank 1 , then ank  ank 1 ,   1 and we have, for z  1, by using Lemma 1,

F ( z)  an z n1  z n  (   an  an1 ) z n  (an1  an2 ) z n1  ...... (ank 1  ank ) z nk 1  (a n  k  a n  k 1 ) z n  k  (1   )a n  k z n  k 1  (a n  k 1  a n  k  2 ) z n  k 1  ......  (a1  a 0 ) z  (   1)a 0 z  a 0

1 1 n  z  an z    (   an  an1 )  (an1  an2 )  ...... (ank 1  ank ) k 1 z z  1 1 1  (a n  k  a n  k 1 ) k  (1   )a n  k k 1  (a n  k 1  a n  k  2 ) k 1  .... z z z a a 1  (a1  a 0 ) n 1  (   1) n01  0n  z z z n  z  an z       an  an1  an1  an2  ...... ank 1  ank  a n  k  a n  k 1  1   a n  k  a n  k 1  a n  k  2  ...... a1  a 0

  1 a0  a0  z

n



 a z    {(   a n

n

 an1 ) cos  (   an  an1 ) sin 

 ( an1  an2 ) cos  ( an1  an2 ) sin   ...... ( ank 1  ank ) cos  ( ank 1  ank ) sin   ( ank  ank 1 ) cos  ( ank  ank 1 ) sin 

 1   ank  ( ank 1  ank 2 ) cos  ( ank 1  ank 2 ) sin   ...... ( a1   a0 ) cos  ( a1   a0 ) sin   (1   ) a0  a0 }

 z

n

a z    {   a n

n

(cos  sin  )  ank (cos  sin    cos   sin 

 1   )   a 0 (sin   cos   1)  2 a 0  2 sin 

n 1

a

j 1, j  n  k

j

}

0 if

a z      a n

n

(cos  sin  )  ank (cos  sin    cos   sin 

 1   )   a 0 (sin   cos   1)  2 a 0  2 sin 

n 1

a

j 1, j  n  k

j

This shows that the zeros of F(z) whose modulus is greater than 1 lie in

[   a n (cos  sin  )  a n  k (cos  sin    cos   sin   1   )

z

 an

  a 0 (sin   cos  1)  2 a 0  2 sin  

n 1

a

j 1, j  n  k

j

]

an

.

But the zeros of F(z) whose modulus is less than or equal to 1 already satisfy the above inequality. Hence all the zeros of F(z) and therefore P(z) lie in

[   a n (cos  sin  )  a n  k (cos  sin    cos   sin   1   )

z

 an

  a 0 (sin   cos  1)  2 a 0  2 sin  

n 1

a

j 1, j  n  k

j

]

an

That proves Theorem 3.

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Further Generalizations of Enestrom-Kakeya Theorem REFERENCES 1) 2) 3) 4) 5) 6) 7) 8)

A.Aziz and B.A.Zargar, Some extensions of the Enestrom-Kakeya Theorem , Glasnik Mathematiki, 31(1996) , 239-244. N.K.Govil and Q.I.Rehman, On the Enestrom-Kakeya Theorem, Tohoku Math. J.,20(1968) , 126-136. N.K.Govil and G.N.Mc-tume, Some extensions of the Enestrom- Kakeya Theorem , Int.J.Appl.Math. Vol.11,No.3,2002, 246-253. M. H. Gulzar, On the Zeros of polynomials, International Journal of Modern Engineering Research, Vol.2,Issue 6, Nov-Dec. 2012, 4318- 4322. A. Joyal, G. Labelle, Q.I. Rahman, On the location of zeros of polynomials, Canadian Math. Bull.,10(1967) , 55-63. M. Marden , Geometry of Polynomials, IInd Ed.Math. Surveys, No. 3, Amer. Math. Soc. Providence,R. I,1996. W. M. Shah and A.Liman, On Enestrom-Kakeya Theorem and related Analytic Functions, Proc. Indian Acad. Sci. (Math. Sci.), 117(2007), 359-370. E C. Titchmarsh,The Theory of Functions,2 nd edition,Oxford University Press,London,1939.

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