The Number of Zeros of a Polynomial in a Disk
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 restrictions and give bounds for the number of zeros in a specific 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
1. Introduction
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 states [6] that if the coefficients of the polynomial
j n
j jz
a z
P
∑
= =
0
)
( satisfy
0≤a0 ≤a1 ≤...≤an−1 ≤an,
then all the zeros of P(z) lie in the closed disk z ≤1. By putting a restriction on
the coefficients of a polynomial similar to that of the Enestrom-Kakeya Theorem, Mohammad [7] proved the following result:
Theorem A: Let j
n
j jz
a z
P
∑
= =
0
)
( be a polynomial such that
0<a0 ≤a1 ≤...≤an−1 ≤an. Then the number of zeros of P(z) in
2 1
≤
0
log 2 log
1 1
a an
+ .
For polynomials with complex coefficients, Dewan [1] proved the following results:
Theorem B: Let j
n
j jz
a z
P
∑
= =
0
)
( be a polynomial such that
, 0,1,2,..., ,
2
argaj − ≤ ≤ j = n
π α β
for some real αand β and
0< a0 ≤ a1 ≤...≤ an−1 ≤ an . Then the number of zeros of P(z) in
2 1
≤
z does not exceed
0
1
0
sin 2 ) 1 sin (cos
log 2 log
1
a
a a
n
j j
n
∑
− = +
+
+ α α
α
.
Theorem C: Let j
n
j jz
a z
P
∑
= =
0
)
( b e a polynomial of degree n with Re(aj)=αj,
n j
aj) j, 0,1,2,...,
Im( =β = such that
0<α0 ≤α1 ≤...≤αn−1 ≤αn. Then the number of zeros of P(z) in
2 1
≤
z does not exceed
0 0
log 2 log
1 1
a
n
j j
n
∑
= + +
β α
.
Regarding the number of zeros of P(z) in z ≤δ,0<δ <1, Gulzar [4,5] proved the
following generalizations of the above results:
Theorem D: Let j
n
j jz
a z
P
∑
= =
0
)
( b e a polynomial of degree n with Re(aj)=αj,
n j
aj) j, 0,1,2,...,
Im( =β = such that for some 0<k ≤1,0<τ ≤1 and 0≤l ≤n,
kαn ≤αn−1 ≤...≤αl+1 ≤αl ≥αl−1 ≥...≥α1 ≥τα0.
0
log 1 log
1
a M
δ
,
where
M
∑
= + + −
+ − +
= n
j j n
n
l k
0 0
0
0 ( ) 2
2 ) (
2α α α α τ α α β .
Theorem E: : Let j
n
j jz
a z
P
∑
= =
0
)
( b e a polynomial of degree n with Re(aj)=αj,
n j
aj) j, 0,1,2,...,
Im( =β = such that for some 0<k1,k2 ≤1,0<τ1,τ2 ≤1 ,0≤l ≤n
and 0≤m≤n,
k1αn ≤αn−1 ≤...≤αl+1 ≤αl ≥αl−1 ≥...≥α1 ≥τ1α0
k2βn ≤βn−1 ≤...≤βm+1 ≤βm ≥βm−1 ≥...≥β1 ≥τ2β0.
Then for 0<δ <1 the number of zeros of P(z) in z ≤δ does not exceed
0
log 1 log
1
a M′
δ
,
where
M′ = an +(1−k1)αn +(1−k2)βn +2(αl +βm)−(k1αn +k2βn)+(1−τ1)α0
−τ1α0 +(1−τ2)β0 −τ2β0.
Recently, Gardner and Sheilds [2] proved the following results:
Theorem F: Let j
n
j jz
a z
P
∑
= =
0
)
( be a polynomial of degree n such that for some t>0,
n l≤ ≤
0 and for some real numbers α,β ,
, 0,1,2,..., ,
2
argaj −β ≤α ≤π j= n
and
tn an ≤tn−1an−1 ≤...≤tl+1al+1 ≤tl al ≥tl−1al−1 ≥...≥ta1 ≥ a0 >0.
Then the number of zeros of P(z) in z ≤δt,0<δ <1 is less than
0 *
log 1 log
1
a M
δ
, where
∑
− =
+ +
+ + + − +
+ −
−
= 1
1 1 1
1 0
*
sin 2 ) cos sin
1 ( cos
2 ) sin cos
1 (
n
j j j n
n l
l t a t a t
a t
a
Theorem G: Let j n
j jz
a z
P
∑
= =
0
)
( b e a polynomial of degree n with Re(aj)=αj,
n j
aj) j, 0,1,2,...,
Im( =β = such that for some t>0, and some 0≤l≤n,
tnαn ≤tn−1αn−1 ≤...≤tl+1αl+1 ≤tlαl ≥tl−1αl−1 ≥...≥tα1 ≥α0 ≠0.
Then for 0<δ <1 the number of zeros of P(z) in z ≤δt is less than
0 * *
log 1 log
1
a M
δ
,
where
* *
M
∑
=
+ +
+ + − +
+ −
= n
j
j j n
n n l
lt t t
t
0
1 1
1 0
0 ) 2 ( ) 2
(α α α α α β .
Theorem H: Let j
n
j jz
a z
P
∑
= =
0
)
( b e a polynomial of degree n with Re(aj)=αj,
n j
aj) j, 0,1,2,...,
Im( =β = such that for some t>0, ,0≤l ≤nand 0≤m≤n,
tnαn ≤tn−1αn−1 ≤...≤tl+1αl+1 ≤tlαl ≥tl−1αl−1 ≥...≥tα1 ≥α0 ≠0
1 1 0
1 1
1 1
1
...
... β β β β β
β
β ≤ ≤ ≤ ≤ ≥ − − ≥ ≥ ≥
+ + −
−
t t
t t
t
t m m
m m l m n
n n
n .
Then for 0<δ <1 the number of zeros of P(z) in z ≤δt is less than
0 * * *
log 1 log
1
a M
δ
,
where
* * *
M =(α0 −α0)t+2αltl+1+1(αn −αn)tn+1 +(β0 −β0)t+2βmtm+1 +(βn −βn)tn+1.
In this paper, we prove the following results:
Theorem 1: Let j
n
j jz
a z
P
∑
= =
0
)
( be a polynomial of degree n such that for some t>0,
n l≤ ≤
0 ,0<k ≤1,0<τ ≤1 and for some real numbers α,β ,
, ,..., 2 , 1 , 0 , 2
argaj −β ≤α ≤π j= n
and
ktn an ≤tn−1an−1 ≤...≤tl+1al+1 ≤tl al ≥tl−1al−1 ≥...≥ta1 ≥τ a0 >0.
Then the number of zeros of P(z) in z ≤δt,0<δ <1 is less than
0 1
log 1 log
1
a M
δ
, where
∑
− =
+ +
−
+ 1
1 1
sin 2 )
1 (
n
j j j n
nt a t
a
k α .
Remark 1: For different values of the parameters, we get many interesting results.
For example for k=1,τ =1, Theorem 1 reduces to Theorem F . For t=1, it reduces
to Theorem D .
Theorem 2: Let j
n
j jz
a z
P
∑
= =
0
)
( b e a polynomial of degree n with Re(aj)=αj,
n j
aj) j, 0,1,2,...,
Im( =β = such that for some t>0,0<k ≤1,0<τ ≤1 and 0≤l≤n,
ktnαn ≤tn−1αn−1 ≤...≤tl+1αl+1 ≤tlαl ≥tl−1αl−1 ≥...≥tα1 ≥τα0 ≠0.
Then for 0<δ <1 the number of zeros of P(z) in z ≤δt is less than
0 2
log 1 log
1
a M
δ
,
where
2
M
∑
=
+ +
+
+ + − + +
+ + −
= n
j
j j n
n n n
n l
lt t k t t
t t
0
1 1
1 1
0 0
0 ( ) 2 2 ( ) 2
2α τ α α α α α α β .
Remark 2: For different values of the parameters, we get many interesting results.
For example for k=1,τ =1, Theorem 2 reduces to Theorem G . For t=1, it reduces
to the following result:
Corollary 1: Let j
n
j jz
a z
P
∑
= =
0
)
( b e a polynomial of degree n with Re(aj)=αj,
n j
aj) j, 0,1,2,...,
Im( =β = such that for some 0<k ≤1,0<τ ≤1 and 0≤l ≤n,
kαn ≤αn−1 ≤...≤αl+1 ≤αl ≥αl−1 ≥...≥α1 ≥τα0 ≠0.
Then for 0<δ <1 the number of zeros of P(z) in z ≤δ does not exceed
0 2
* log 1 log
1
a M
δ
,
where
* 2
M
∑
= + + −
+ + + −
= n
j j n
n n
l k
0 0
0
0 ( ) 2 2 ( ) 2
2α τ α α α α α α β .
Applying Theorem 2 to the polynomial –iP(z) , we get the following result:
Corollary 2: Let j
n
j jz
a z
P
∑
= =
0
)
n j
aj) j, 0,1,2,...,
Im( =β = such that for some t>0,0<k ≤1,0<τ ≤1 and 0≤l≤n,
1 1 0
1 1
1 1
1
...
... β β β β τβ
β
β ≤ ≤ ≤ ≤ ≥ − − ≥ ≥ ≥
+ + −
−
t t
t t
t
kt l
l l l l l n
n n n
.
Then for 0<δ <1 the number of zeros of P(z) in z ≤δt is less than
0 * * 2
log 1 log
1
a M
δ
,
where
* * 2
M
∑
=
+ +
+
+ + − + +
+ + −
= n
j
j j n
n n n
n l
lt t k t t
t t
0
1 1
1 1
0 0
0 ( ) 2 2 ( ) 2
2β τ β β β β β β α .
If aj is real i.e. βj =0,∀j, Theorem 2 gives the following result:
Corollary 3: Let j
n
j jz
a z
P
∑
= =
0
)
( b e a polynomial of degree n such that for some
t>0,0<k ≤1,0<τ ≤1 and 0≤l ≤n,
... 1 1 ... 1 0 0
1 1 1
1 ≤ ≤ ≤ ≥ ≥ ≥ ≥ ≠
≤ − −
+ + −
−
a ta a
t a t a t a
t a
kt l l
l l l l n
n n
n τ .
Then for 0<δ <1 the number of zeros of P(z) in z ≤δt is less than
0 * * * 2
log 1 log
1
a M
δ
,
where
* * * 2
M =2a0 t−τ(a0 +a0)t+2altl+1+2antn+1−k(an +an)tn+1.
Theorem 3: Let j
n
j jz
a z
P
∑
= =
0
)
( b e a polynomial of degree n with Re(aj)=αj,
n j
aj) j, 0,1,2,...,
Im( =β = such that for some t>0,0<k1,k2 ≤1,0<τ1,τ2 ≤1 ,0≤l≤n
and 0≤m≤n,
k1tnαn ≤tn−1αn−1 ≤...≤tl+1αl+1 ≤tlαl ≥tl−1αl−1 ≥...≥tα1 ≥τ1α0 ≠0
1 1 2 0
1 1
1 1
1
2 β ≤ β ≤...≤ β ≤ β ≥ β − ≥...≥ β ≥τ β
− +
+ −
−
t t
t t
t t
k n n n n l l l m l m .
Then for 0<δ <1 the number of zeros of P(z) in z ≤δt is less than
0 3
log 1 log
1
a M
δ
,
where
3
M 1 1
1 1
0 0 1
0 ( ) 2 ( )
2 − + + + + + − + +
= n
n n n
n l
lt t k t
t
t τ α α α α α α
α
1
2 1 1
0 0 1
0 ( ) 2 ( )
2 − + + + + + − + +
+ n
n n n
n m
mt t k t
t
t τ β β β β β β
Remark 3: For different values of the parameters, we get many interesting results.
For example for k=1,τ =1, Theorem 3 reduces to Theorem H . For t=1, it reduces
to Theorem E .
2. Lemmas
For the proofs of the above results, we need the following results:
Lemma 1: Let z1,z2 ∈Cwith z1 ≥ z2 and , 1,2
2
argzj −β ≤α ≤π j = for some real
numbers α and β. Then
z1 −z2 ≤(z1 − z2)cosα +(z1 + z2)sinα .
The above lemma is due to Govil and Rahman [3].
Lemma 2: Let F(z) be analytic in z ≤R, F(z) ≤Mfor z ≤Rand F(0)≠0. Then
for 0<δ <1 the number of zeros of F(z) in the disk z ≤δt is less than
0
log 1 log
1
a M
δ
.
For the proof of this lemma see [8, P.171].
3. Proofs of Theorems Proof of Theorem 1: Consider the polynomial
n
n n n l
l l l l
l z a z a z a z a z
a z
a z a a z t z P z t z
F( )=( − ) ( )=( − )( 0 + 1 )+ 2 2 +...+ −1 −1 + + +1 +1+... −1 −1+
1
1 1
2 1 2 0
0 0
1 0
1 1
1 1
1 0
1 0
) (
) (
... ) (
)] (
) [(
) (
... )
( ) (
... ) (
+ +
−
+ −
+ +
−
− +
− + + −
+ −
+ − + =
− −
+ + −
+ −
+ + − + =
l l l l l l
n n n n n l
l l l l l
z a t a z a t a z
a t a z a a a
t a t a
z a z a t a z
a t a z a t a z
a t a t a
τ
τ
1
1) ( )]
[(
...+ − − − − − +
+ n
n n n n n
nt a ka t a t z a z
ka
For z =t, we have by using the hypothesis and Lemma 1,
1 1
1 2
1 2 0 1 0 0
0 ...
)
( ≤ + − + − + − + + − − + l+ − l l+
l l
lt a t a t a t
a t
a t a t a t a t a a t a z
F τ τ
1
1
...+ − − + − + +
+ n
n n n n n n
nt a t ka t a t a t
ka
1
1 1
2 1 2 0 1 0
0 (1 ) ...
+ +
− + −
− + + − + − + −
+
= l
l l l l
lt a t a t a t
a t
a t a t a t a t a t
a τ
1 1 (1 )
...+ − − + − + +
+ n
n n n n
n
nt a t k a t a t
≤ a0 t+(1−τ)a0t+[(a1t−τa0)cosα+(a1t+τ a0 )sinα]t 1 1 1 1 1 1 1 1 2 1 2 1 2 ) 1 ( ] sin ) ( cos ) [( ... ] sin ) cos ) [( ] sin ) ( cos ) [( ... ] sin ) ( cos ) [( + − − + + + − − + − + + + − + + + + − + − + − + + + + − + n n n n n n n n n l l l l l l l l l l t a t a k t t a k a t a k a t t a a t a a t a t a a t a t a t a a t a α α α α α α α α
2a0t τ a0 (1 cosα sinα) 2a t 1cosα a t 1(1 ksinα kcosα) n
n l
l + + −
+ − + − = + +
∑
− = + + − + 1 1 1 sin 2 ) 1 ( n j j j nnt a t
a
k α
=M1.
Since F(z) is analytic for z ≤t and F(z) ≤M1 for z =t, it follows by Lemma 2 and
Maximum Modulus Theorem that the number of zeros of F(z) and hence of P(z) in
t
z ≤δ is less than or equal to
0 1 log 1 log 1 a M δ
and the theorem follows.
Proof of Theorem 2: Consider the polynomial
n n n n l l l l l
l z a z a z a z a z
a z a z a a z t z P z t z
F = − = − + + + + − − + + + + + −1 −1+
1 1 1 1 2 2 1
0 ) ... ...
)( ( ) ( ) ( ) ( 1 1 1 2 1 2 0 0 0 1 0 1 1 1 1 1 0 1 0 ) ( ) ( ... ) ( )] ( ) [( ) ( ... ) ( ) ( ... ) ( + + − + − + + − − + − + + − + − + − + = − − + + − + − + + − + = l l l l l l n n n n n l l l l l l z a t a z a t a z a t a z a a a t a t a z a z a t a z a t a z a t a z a t a t a τ τ 1
1) ( )]
[( ...+ − − − − − + + n n n n n n
nt a ka t a t z a z
ka
=(α0 +iβ0)t+[{(α1 +iβ1)t−τ(α0 +iβ0)}+{τ(α0 +iβ0)−(α0 +iβ0)}]z
1 1 1 1 1 1 1 1 ) ( )}] ( ) ( { )} ( ) ( [{ ... )] ( ) [( )] ( ) [( ... + − − + + + − − + − + − + − + − + + + + − + + + − + + + n n n n n n n n n n n n l l l l l l l l l l z i z i t i k i t i k z i t i z i t i β α β α β α β α β α β α β α β α β α 1 1 1 0 0 0 1
0 [( ) ( )] ... ( ) ( )
+ + − + − − + + − + − + = l l l l l
lt z t z
z t
t α τα τα α α α α α
α ]. ) ( ... ) ( [ )] ( ) [( ... 1 1 0 1 0 1 1 + − + − − − + + − + + − − − − + + n n n n n n n n n n n n z z t z t t i z z t t k t k β β β β β β α α α α α
1 1 1 2 1 2 0 1 0 0 0 ... )
( ≤ + − + − + − + + − − + l+ − l l+
l l
lt t t t
t t t t t t z
F α τα α α τα α α α α α α
1 1 0 1 0 1 1 ) ( ... ) ( ... + − + − + + + + + + + + − + − + + n n n n n n n n n n n n n t t t t t t t t t t k t t k β β β β β β α α α α α l l
lt t
t t t t t
t (1 ) 0 ( 1 0) ( 2 1) 2 ... ( 1)
0 + − + − + − + + − +
≤ α τ α α α α α α α
∑
= + + + − + + + + − + − + + − + n j j j n n n n n n n l ll t t k t t k t t t
0 1 1 1 1 1
1 ) ... ( ) (1 ) 2
(α α α α α α β
∑
= + + + + + − + + + − − = n j j j n n n n n llt t k t t
t t 0 1 1 1 1 0 0
0 ) 2 2 ( ) 2
2
( α α τα α α α α β
=M2.
The theorem now follows as in the proof of Theorem 1.
Proof of Theorem 3: As in the proof of Theorem 2,
1 1 1 0 0 1 0 1 1
0 [( ) ( )] ... ( ) ( )
)
(z = t+ t− + − z+ + lt− l− zl + l+t− l zl+
F α α τ α τ α α α α α α
]. )} ( ) {( ... ) ( ) ( ... )} ( ) {( [ )] ( ) [( ... 1 2 1 2 1 1 1 0 0 2 0 2 1 0 1 1 1 1 + − + + − + − − − − − + + − + − + + − + − + + − − − − + + n n n n n n n m m m m m m n n n n n n n z z t t k t k z t z t z t t i z z t t k t k β β β β β β β β β β β τ β τ β β α α α α α
For z =t, we have by using the hypothesis,
1 1 1 2 1 2 0 1 0 0 0 ... )
( ≤ + − + − + − + + − − + l+ − l l+
l l
lt t t t
t t t t t t z
F α τα α α τα α α α α α α
1 2 1 2 1 1 1 0 0 2 0 2 1 0 1 1 1 1 ... ... ... + − + + − + − + − + − + + − + − + + − + − + + + − + − + + n n n n n n n n m m m m m m n n n n n n n n t t t t k t t k t t t t t t t t t t t t k t t k β β β β β β β β β β β τ β τ β β α α α α α l l
lt t
t t t t
t
t (1 1) 0 ( 1 1 0) ( 2 1) 2 ... ( 1)
0 + − + − + − + + − −
1 1
1 1
0 0 1
0 ( ) 2 ( )
2 − + + + + + − + +
= n
n n n
n l
lt t k t
t
t τ α α α α α α
α
1
2 1 1
0 0 1
0 ( ) 2 ( )
2 − + + + + + − + +
+ n
n n n
n m
mt t k t
t
t τ β β β β β β
β .
=M3.
The result now follows as in the proof of Theorem 1.
References
[1] K.K. Dewan, Extremal Properties and Coefficient Estimates for Polynomials with Restricted Coefficients and on Location of Zeros of Polynomials,Ph.D Thesis, IIT Delhi, 1980.
[2] R. Gardner and B. Sheilds, The Number of Zeros of a Polynomial in a Disk, Journal of Classical Analysis, Vol.3, No.2(2013), 167-176.
[3] N. K. Govil and Q. I. Rahman, On the Enestrom-Kakeya Theorem, Tohoku Mathematics Journal, 20(1968), 126-136.
[4] M. H. Gulzar, On the Zeros of Polynomials with Restricted Coefficients ISST Journal of Mathematics &Computing System, Vol.3 No.2 (July- December 2012), 33-40.
[5] M. H. Gulzar, On the Zeros of a Polynomial, Int. Journal of Scientific and Research Publications, Vol. 2 Issue 7,July 2012 , 1-7.
[6] M. Marden, , Geometry of Polynomials,Math. Surveys No. 3, Amer. Math
Society(R.I. Providence) (1966).
[7] Q. G. Mohammad, On the Zeros of Polynomials, American Mathematical Monthly, 72 (6) (1965), 631-633.
[8] E. C. Titchmarsh, The Theory of Functions, 2nd Edition, Oxford University Press
London, 1939.