International Journal of Engineering Research and Development e-ISSN: 2278-067X, p-ISSN: 2278-800X, www.ijerd.com Volume 10, Issue 6 (June 2014), PP.60-69
Zeros of Polynomials in Ring-shaped Regions M. H. Gulzar Department of Mathematics University of Kashmir, Srinagar 190006 Abstract:- In this paper we subject the coefficients of a polynomial and their real and imaginary parts to certain conditions and give bounds for the number of zeros in a ring-shaped region. Our results generalize many previously known results and imply a number of new results as well. Mathematics Subject Classification: 30 C 10, 30 C 15 Keywords and Phrases:- Coefficient, Polynomial, Zero
I.
INTRODUCTION AND STATEMENT OF RESULTS
A large number of research papers have been published so far on the location in the complex plane of some or all of the zeros of a polynomial in terms of the coefficients of the polynomial or their real and imaginary parts. The famous Enestrom-Kakeya Theorem [5] states that if the coefficients of the polynomial n
P( z ) a j z j satisfy 0 a0 a1 ...... an1 an , then all the zeros of P(z) lie in the closed disk j 0
z 1. By putting a restriction on the coefficients of a polynomial similar to that of the Enestrom-Kakeya Theorem, Mohammad [6] proved the following result: n
Theorem A: Let
P( z ) a j z j be a polynomial such that 0 a0 a1 ...... an1 an . Then the j 0
number of zeros of P(z) in
1
z
1 does not exceed 2
a 1 log n . log 2 a0
For polynomials with complex coefficients, Dewan [1] proved the following results: n
Theorem B: Let
P( z ) a j z j be a polynomial such that j 0
arg a j for some real
and
2
, j 0,1,2,....., n,
and
0 a0 a1 ...... an1 an . Then the number of zeros of P(z) in
z
1 does not exceed 2 n 1
1 log log 2
a n (cos sin 1) 2 sin a j j 0
a0 n
Theorem C: Let
.
P( z ) a j z j b e a polynomial of degree n with Re( a j ) j , j 0
Im(a j ) j , j 0,1,2,......, n such that 0 0 1 ...... n1 n . 60
Zeros of Polynomials in Ring-shaped Regions
z
Then the number of zeros of P(z) in
1 does not exceed 2
n
1
n j
1 log log 2
j 0
a0
.
Recently Gulzar [3,4] proved the following results: n
Theorem D: Let
P( z ) a j z j be a polynomial of degree n such that for some k 1, o 1 and for j 0
some integer
,0 n 1 , n n1 ...... 1 k 1 ...... 1 0 .
Then P(z) has no zero in
z
a0
, where
M1
n
M 1 a n n 2(k 1) ( 0 0 ) 0 0 n 2 j . j 1
n
Theorem E: Let
P( z ) a j z j be a polynomial of degree n such that for some j 0
real
, ; arg a j
2
,0 n 1 ,
, j 0,1,2,......, n and for some k 1, o 1 and some integer
a n a n1 ...... a 1 k a a 1 ...... a1 a0 . Then P(z) has no zeros in
z
a0
, where
M2
M 2 a n (cos sin 1) 2 a (k k sin 1) a0 (cos sin 1) a0 2 sin
n 1
a
j 1, j
j
.
n
Theorem F: Let
P( z ) a j z j be a polynomial of degree n with Re( a j ) j , j 0
Im(a j ) j , j 0,1,......, n such that for some k1 1, k 2 1,0 1 ,
k1 n k 2 n1 n2 ......1 0 Then P(z) has no zero in z
a0 M3
, where
M 3 a n (k1 n k 2 n1 ) (k1 n k 2 n1 ) n1 ( n n1 ) n 1
( 0 0 ) 0 0 n 2 j . j 1
n
Theorem G: Let
P( z ) a j z j be a polynomial of degree n such that for some k1 1, k 2 1,0 1 , j 0
k1 a n k 2 a n1 a n2 ...... a1 a0 .
61
Zeros of Polynomials in Ring-shaped Regions
Then P(z) has no zero in
z
a0 M4
, where
M 4 k1 a n (cos sin 1) k 2 a n1 (sin cos 1) a n1 (1 cos ) n2
a0 (cos sin 1) a0 2 sin a j . j 1
In this paper, we prove the following results: n
Theorem 1: Let
P( z ) a j z j be a polynomial of degree n such that for some k 1, o 1 and for j 0
some integer
,0 n 1 , n n1 ...... 1 k 1 ...... 1 0 .
Then the number of zeros of P(z) in
z
R ( R 0, c 1) does not exceed c n
a n R n 1 a 0 R n [ n 2(k 1) ( 0 0 ) 0 0 n 2 j ]
1 log log c for R 1 and 1 log log c for R 1 .
j 1
a0 n 1
an R
n
a 0 R[ n 2(k 1) ( 0 0 ) 0 0 n 2 j ] j 1
a0
Combining Theorem 1 and Theorem D, we get a bound for the number of zeros of P(z) in a ring-shaped region as follows: n
P( z ) a j z j be a polynomial of degree n such that for some k 1, o 1 and for
Corollary 1: Let
j 0
some integer
,0 n 1 , n n1 ...... 1 k 1 ...... 1 0 .
Then the number of zeros of P(z) in
a0 M1
z
R ( R 0, c 1) does not exceed c n
a n R n 1 a 0 R n [ n 2(k 1) ( 0 0 ) 0 0 n 2 j ]
1 log log c for R 1 and
j 1
a0 n
a n R n 1 a 0 R[ n 2(k 1) ( 0 0 ) 0 0 n 2 j ]
1 log log c for R 1 , where
j 1
a0 n
M 1 a n n 2(k 1) ( 0 0 ) 0 0 n 2 j . j 1
n
Theorem 2: Let
P( z ) a j z j be a polynomial of degree n such that for some j 0
62
Zeros of Polynomials in Ring-shaped Regions
real
, ; arg a j
,0 n 1 ,
2
, j 0,1,2,......, n and for some k 1, o 1 and some integer
a n a n1 ...... a 1 k a a 1 ...... a1 a0 . R ( R 0, c 1) does not exceed c a n R n 1 a 0 R n { a n (cos sin ) 2 a (k k sin 1) 1 1 n 1 log for R 1 log c a 0 a 0 (cos sin 1) a 0 2 sin a j } j 1, j z
Then the number of zeros of P(z) in
and
a n R n 1 a 0 R n { a n (cos sin ) 2 a (k k sin 1) 1 1 n 1 log for R 1 . log c a 0 a 0 (cos sin 1) a 0 2 sin a j } j 1, j Combining Theorem 2 and Theorem E, we get a bound for the number of zeros of P(z) in a ring-shaped region as follows: n
Corollary 2: Let
P( z ) a j z j be a polynomial of degree n such that for some j 0
real
, ; arg a j
,0 n 1 ,
2
, j 0,1,2,......, n and for some k 1, o 1 and some integer
a n a n1 ...... a 1 k a a 1 ...... a1 a0 .
a0
Then the number of zeros of P(z) in
M2
z
R ( R 0, c 1) does not exceed c
a n R n 1 a 0 R n { a n (cos sin ) 2 a (k k sin 1) 1 1 n 1 log for R 1 log c a 0 a 0 (cos sin 1) a 0 2 sin a j } j 1, j and
a n R n 1 a 0 R n { a n (cos sin ) 2 a (k k sin 1) 1 1 n 1 log for R 1 , log c a 0 a 0 (cos sin 1) a 0 2 sin a j } j 1, j where
M 2 a n (cos sin 1) 2 a (k k sin 1) a0 (cos sin 1) a0 n 1
a
2 sin
j 1, j
j
.
n
Theorem 3: Let
P( z ) a j z j be a polynomial of degree n with Re( a j ) j , j 0
Im(a j ) j , j 0,1,......, n such that for some k1 1, k 2 1,0 1 ,
k1 n k 2 n1 n2 ......1 0 R Then the number of zeros of P(z) in z ( R 0, c 1) does not exceed c 63
Zeros of Polynomials in Ring-shaped Regions
a n R n 1 R n {(k1 n k 2 n 1 ) (k1 n k 2 n 1 ) n 1 ( n n 1 ) 1 1 n 1 log log c a 0 ( 0 0 ) 0 0 n 2 j } j 1 for R 1 and a n R n 1 R{(k1 n k 2 n 1 ) (k1 n k 2 n 1 ) n 1 ( n n 1 ) 1 1 n 1 log log c a 0 ( 0 0 ) 0 0 n 2 j } j 1 for R 1 . Combining Theorem 3 and Theorem F, we get a bound for the number of zeros of P(z) in a ring-shaped region as follows: n
Corollary 3: Let
P( z ) a j z j be a polynomial of degree n with Re( a j ) j , j 0
Im(a j ) j , j 0,1,......, n such that for some k1 1, k 2 1,0 1 ,
k1 n k 2 n1 n2 ......1 0 Then the number of zeros of P(z) in
a0 M3
z
R ( R 0, c 1) does not exceed c
a n R n 1 R n {(k1 n k 2 n 1 ) (k1 n k 2 n 1 ) n 1 ( n n 1 ) 1 1 n 1 log log c a 0 ( 0 0 ) 0 0 n 2 j } j 1 for R 1 and a n R n 1 R{(k1 n k 2 n 1 ) (k1 n k 2 n 1 ) n 1 ( n n 1 ) 1 1 n 1 log log c a 0 ( 0 0 ) 0 0 n 2 j } j 1 for R 1 , where M 3 a n (k1 n k 2 n1 ) (k1 n k 2 n1 ) n1 ( n n1 ) n 1
( 0 0 ) 0 0 n 2 j . j 1
n
Theorem 4: Let
P( z ) a j z j be a polynomial of degree n such that for some k1 1, k 2 1,0 1 , j 0
k1 a n k 2 a n1 a n2 ...... a1 a0 . R Then the number of zeros of P(z) in z ( R 0, c 1) does not exceed c n 1 n a n R a 0 R {k1 a n (cos sin 1) k 2 a n 1 (sin cos 1) a n 1 1 n2 log log c a 0 a n 1 (1 cos sin ) a 0 (cos sin 1) 2 sin a j } j 1 for R 1 and
64
Zeros of Polynomials in Ring-shaped Regions
a n R n 1 a 0 R{k1 a n (cos sin 1) k 2 a n 1 (sin cos 1) a n 1 1 n2 log log c a 0 a n 1 (1 cos sin ) a 0 (cos sin 1) 2 sin a j } j 1 for R 1 . Combining Theorem 4 and Theorem G, we get a bound for the number of zeros of P(z) in a ring-shaped region as follows: n
Corollary 4: Let
P( z ) a j z j be a polynomial of degree n such that for some k1 1, k 2 1,0 1 , j 0
k1 a n k 2 a n1 a n2 ...... a1 a0 . Then the number of zeros of P(z) in
a0 M4
z
R ( R 0, c 1) does not exceed c
a n R n 1 a 0 R n {k1 a n (cos sin 1) k 2 a n 1 (sin cos 1) a n 1 1 n2 log log c a 0 a n 1 (1 cos sin ) a 0 (cos sin 1) 2 sin a j } j 1 for R 1 and a n R n 1 a 0 R{k1 a n (cos sin 1) k 2 a n 1 (sin cos 1) a n 1 1 n2 log log c a 0 a n 1 (1 cos sin ) a 0 (cos sin 1) 2 sin a j } j 1 for R 1 , where M 4 k1 a n (cos sin 1) k 2 a n1 (sin cos 1) a n1 (1 cos )
a0 (cos sin 1) a0 . For different values of the parameters in the above results, we get many interesting results including generalizations of some well-known results.
II.
LEMMAS
For the proofs of the above results, we need the following results: Lemma 1: Let
z1 , z 2 C with z1 z 2 and arg z j
and . Then
2
, j 1,2 for some real numbers
z1 z 2 ( z1 z 2 ) cos ( z1 z 2 ) sin . The above lemma is due to Govil and Rahman [2]. Lemma 2: Let F(z) be analytic in
z R , F ( z ) M for z R and F (0) 0 . Then
for c>1, the number of zeros of F(z) in the disk
z
R does not exceed c
1 M log . log c a0 For the proof of this lemma see [7].
III.
PROOFS OF THEOREMS
Proof of Theorem 1: Consider the polynomial F(z)=(1-z)P(z)
(1 z)(an z n an1 z n1 ...... a 1 z 1 a z a 1 z 1 ...... an1 z n1 an z n 65
Zeros of Polynomials in Ring-shaped Regions
an z n1 (an an1 ) z n (an1 an2 ) z n1 ...... (a 1 a ) z 1 (a a 1 ) z (a 1 a 2 ) z 1 ...... (a1 a0 ) z a0 a n z n1 ( n n1 ) z n ( n1 n2 ) z n1 ...... ...... {( 1 k ) (k )}z 1 {(k 1 ) (k )}z ( 1 2 ) z 1 ...... {( 1 0 ) ( 0 0 )}z n
i ( j j 1 )z j a 0 j 1
For
z R , we have, by using the hypothesis,
F ( z ) a n R n1 n n1 R n n 1 n2 R n1 ....... 1 k R 1 (k 1) R k 1 R (k 1) R 1 2 R 1 ...... 1 0 R n
(1 ) 0 R ( j j 1 ).R j a 0 j 1
an R
n 1
a0 R [ n n1 n1 n2 ....... 1 k (k 1) n
k 1 (k 1) 1 2 ...... 1 0 n
(1 ) 0 ( j j 1 )] j 1
n
a n R n 1 a 0 R n [ n 2(k 1) ( 0 0 ) 0 0 n 2 j ] j 1
for
R 1
and
F ( z ) a n R n1 a0 R[ n n1 n1 n2 ....... 1 k (k 1) k 1 (k 1) 1 2 ...... 1 0 n
(1 ) 0 ( j j 1 )] j 1
n
a n R n 1 a 0 R[ n 2(k 1) ( 0 0 ) 0 0 n 2 j ] j 1
for
R 1.
Hence, by Lemma 2, it follows that the number of zeros of F(z) in
z
R ,c 1 c
does not exceed n
a n R n 1 a 0 R n [ n 2(k 1) ( 0 0 ) 0 0 n 2 j ]
1 log log c for R 1 and
j 1
a0 n
1 log log c
a n R n 1 a 0 R[ n 2(k 1) ( 0 0 ) 0 0 n 2 j ] j 1
a0 66
1
Zeros of Polynomials in Ring-shaped Regions for R 1 . Since the zeros of P(z) are also the zeros of F(z), the proof of Theorem 1is complete. Proof of Theorem 2: Consider the polynomial F(z)=(1-z)P(z)
(1 z)(an z n an1 z n1 ...... a 1 z 1 a z a 1 z 1 ...... an1 z n1 an z n
an z n1 (an an1 ) z n (an1 an2 ) z n1 ...... (a 1 a ) z 1 (a a 1 ) z (a 1 a 2 ) z 1 ...... (a1 a0 ) z a0 For
z R , we have, by using the hypothesis and Lemma 1,
F ( z ) a n R n1 a n a n1 R n a n1 a n2 R n1 ....... a 1 ka R 1 (k 1) a R 1 ka a 1 R (k 1) a R a 1 a 2 R 1 ...... a1 a 0 R (1 ) a 0 R
a n R n1 a0 R n [( a n a n1 ) cos ( a n a n1 ) sin ( a n1 a n2 ) cos ( a n 1 a n 2 ) sin ...... ( a 1 k a ) cos ( a 1 k a ) sin (k 1) (k 1 ) cos (k 1 ) sin (k 1) ( a 1 a 2 ) cos ( a 1 a 2 ) sin ...... ( a1 a 0 ) cos ( a1 a 0 ) sin (1 ) a 0 ]
a n R n1 a0 R n [ a n (cos sin ) 2 a (k k sin 1) a0 (cos sin 1) a0 2 sin for
n 1
a
j 1, j
j
]
R 1
and
F ( z ) a n R n1 a0 R[ a n (cos sin ) 2 a (k k sin 1) a0 (cos sin 1) a0 2 sin for
n 1
a
j 1, j
j
]
R 1 .
Hence, by Lemma 2, it follows that the number of zeros of F(z) in
z
R ,c 1 c
does not exceed
a n R n 1 a 0 R n { a n (cos sin ) 2 a (k k sin 1) 1 1 n 1 log for R 1 log c a 0 a 0 (cos sin 1) a 0 2 sin a j } j 1, j and
a n R n 1 a 0 R n { a n (cos sin ) 2 a (k k sin 1) 1 1 n 1 log for R 1 . log c a 0 a 0 (cos sin 1) a 0 2 sin a j } j 1, j Since the zeros of P(z) are also the zeros of F(z), Theorem 2 follows. Proof of Theorem 3: Consider the polynomial
F ( z ) (1 z ) P( z ) (1 z)(an z n an1 z n1 ...... a1 z a0 ) an z n1 (an an1 ) z n (an1 an2 ) z n1 ...... (a1 a0 ) z a0 67
Zeros of Polynomials in Ring-shaped Regions
an z n1 {(k1 n k 2 n1 ) (k1 n n ) (k 2 n1 n1 )}z n ( n1 n2 ) z n1
...... {( 1 0 ) ( 0 0 )}z i{( n n 1 ) z n ...... ( 1 0 ) z} a 0 For
z R , we have, by using the hypothesis,
F ( z) a n R n1 k1 n k 2 n1 R n (k1 1) n R n (k 2 1) n1 R n n1 n2 R n1 ...... n
1 0 R (1 ) 0 R j j 1 R j a 0 j 1
an R
n 1
R [k1 n k 2 n1 (k1 1) n (k 2 1) n1 n1 n2 ..... n
n
1 0 (1 ) 0 ( j j 1 )] j 1
an R
n 1
R [(k1 n k 2 n1 ) (k1 n k 2 n1 ) n1 ( n n1 ) n
n 1
( 0 0 ) 0 0 n 2 j ] j 1
for
R 1
and
F ( z) a n R n1 R[(k1 n k 2 n1 ) (k1 n k 2 n1 ) n1 ( n n1 ) n 1
( 0 0 ) 0 0 n 2 j ] j 1
for
R 1 .
Hence, by Lemma 2, it follows that the number of zeros of F(z) in
z
R ,c 1 c
does not exceed
a n R n 1 R n {(k1 n k 2 n 1 ) (k1 n k 2 n 1 ) n 1 ( n n 1 ) 1 1 n 1 log log c a 0 ( 0 0 ) 0 0 n 2 j } j 1 for R 1 and a n R n 1 R{(k1 n k 2 n 1 ) (k1 n k 2 n 1 ) n 1 ( n n 1 ) 1 1 n 1 log log c a 0 ( 0 0 ) 0 0 n 2 j } j 1 for R 1 . Since the zeros of P(z) are also the zeros of F(z), Theorem 3 follows. Proof of Theorem 4: Consider the polynomial
F ( z ) (1 z ) P( z ) (1 z)(an z n an1 z n1 ...... a1 z a0 )
an z n1 (an an1 ) z n (an1 an2 ) z n1 ...... (a1 a0 ) z a0 a n z n1 {(k1 a n k 2 a n1 ) (k1 a n a n ) (k 2 a n1 a n1 )}z n (a n1 a n2 ) z n1
...... {(a1 a0 ) (a0 a0 )}z a0 For
z R , we have, by using the hypothesis and Lemma 1, 68
Zeros of Polynomials in Ring-shaped Regions
F ( z) a n R n1 k1 a n k 2 a n1 R n (k1 1) a n R n (k 2 1) a n1 R n a n1 a n2 R n1 ...... a1 a0 R (1 ) a0 R a0 a n R n1 a0 R n [(k1 a n k 2 a n1 ) cos (k1 a n k 2 a n1 ) sin (k1 1) a n (k 2 1) a n 1 ( a n 1 a n 2 ) cos ( a n 1 a n 2 ) sin ( a1 a 0 ) cos ( a1 a 0 ) sin (1 ) a 0
a n R n1 a0 R n [k1 a n (cos sin 1) k 2 a n1 (sin cos 1) a n n2
a n 1 (1 cos sin ) a 0 (cos sin 1) 2 sin a j ] j 1
for
R 1
and
F ( z ) a n R n1 a0 R[k1 a n (cos sin 1) k 2 a n1 (sin cos 1) a n n2
a n 1 (1 cos sin ) a 0 (cos sin 1) 2 sin a j ] j 1
R 1 . R Hence, by Lemma 2, it follows that the number of zeros of F(z) in z , c 1 c for
does not exceed
a n R n 1 a 0 R n {k1 a n (cos sin 1) k 2 a n 1 (sin cos 1) a n 1 1 n2 log log c a 0 a n 1 (1 cos sin ) a 0 (cos sin 1) 2 sin a j } j 1 for R 1 and a n R n 1 a 0 R{k1 a n (cos sin 1) k 2 a n 1 (sin cos 1) a n 1 1 n2 log log c a 0 a n 1 (1 cos sin ) a 0 (cos sin 1) 2 sin a j } j 1
for R 1 . Since the zeros of P(z) are also the zeros of F(z), Theorem 4 follows.
REFERENCES [1].
K.K. Dewan, Extremal Properties and Coefficient Estimates for Polynomials with Restricted Coefficients and on Location of Zeros of Polynomials,Ph. Thesis, IIT Delhi, 1980. [2]. N. K. Govil and Q. I. Rahman, On the Enestrom-Kakeya Theorem, Tohoku Mathematics Journal, 20(1968), 136. [3]. M. H. Gulzar, The Number of Zeros of a Polynomial in a Disk,International Journal of Innovative Science, Engineering & Technology, Vol.1, Issue 2,April 2014, 37-46. [4]. M. H. Gulzar, Zeros of Polynomials under Certain Coefficient Conditions, Int. Journal of Advance Foundation and Research in Science and Technology (to appear) . [5] M. Marden, , Geometry of Polynomials,Math. Surveys No. 3, Amer. Math Society(R.I. Providence) (1966). [6] Q. G. Mohammad, On the Zeros of Polynomials, American Mathematical Monthly, 72 (6) (1965), 631633. [7] E. C. Titchmarsh, The Theory of Functions, 2 nd Edition, Oxford University Press,London, 1939.
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