A Monthly Double-Blind Peer Reviewed Refereed Open Access International e-Journal - Included in the International Serial Directories International Journal in Physical & Applied Sciences
http://www.ijmr.net.in email id- [email protected] Page 31
REFINEMENT OF SOME RESUTS FOR ZEROS OF POLYNOMIAL AND ANALYTIC FUNCTION
Roshan Lal Asstt. Prof. Mathematics Govt. Degree College Chaubattakhal Pauri Garhwal, Uttarakhand-246162
Abstract: If
p
(
z
)
is a polynomial of degreen
. In this paper, we have obtained ring shaped regions containing zeros of complex valued polynomials as well as analytic functions in terms of coefficients of function. Our results improve upon the results earlier proved.AMS Subject Classification: 30C10, 30C15, 30C26. 1. Introduction and Statement of Results:
The following elegant result is commonly known as Enestrom-Kakeya Theorem, firstly proved by Enestrom [3] and later independently by Kakeya [7] and Hurwitz [5].
Theorem A. If
n a z zp
0
) (
is a polynomial of degree
n
such thatan an1a1a00 (1.1.) then p(z) does not vanish in z 1.
Joyal, Labelle and Rahman [6] extended Theorem Ato the polynomials with coefficients not necessarily non-negative. More precisely, they proved the following
Theorem B. Let
n a z z
p
0
) (
is a polynomial of degree
n
such thatanan1a1a0 . (1.2) Then p(z) has all its zeros in
n n
a a a a
A Monthly Double-Blind Peer Reviewed Refereed Open Access International e-Journal - Included in the International Serial Directories International Journal in Physical & Applied Sciences
http://www.ijmr.net.in email id- [email protected] Page 32
The following result is recently proved by Rather and Ahmad [8]
Theroem C. If
n j
j j z
a z
p
0
)
( be a polynomial of degree n with complex coefficients. Let
j j
j i
a
forj
0
,
1
,
2
,
,
n
and for some K,1K
n
n1
1
0 ,and
K
n
n1
1
0 , (1.4) then p(z) has all its zeros in
n n n
a
a
K
K
z
1
0
0
0 (1.5)Aziz and Mohammad [1] extended Enestrom-Kakeya Theorem to the class of analytic function
0
)
(
j
j j
z
a
z
f
(not identically zero), with its coefficients a satisfying a relation analogous to(1.1) and proved the following theorem.
Theorem D. Let
0
)
(
j
j j
z
a
z
f
(not identically zero) be analytic in z t. If aj 0 and
, 3 , 2 , 1 , 0
1
ta j
aj j then
f
(
z
)
does not vanish in z t.Aziz and Shah [2] relaxed the hypothesis of Theorem D and proved the following theorem.
Theroem E. Let
0
)
(
j
j j
z
a
z
f
(not identically zero) be analytic in z t, such that for some1
K
Ka0ta1t2a2 (1.6)
then
f
(
z
)
does not vanish in2 1 2 1
1
K t K t K K
A Monthly Double-Blind Peer Reviewed Refereed Open Access International e-Journal - Included in the International Serial Directories International Journal in Physical & Applied Sciences
http://www.ijmr.net.in email id- [email protected] Page 33
Shah and Liman [9] proved the following result concerning the location of zeros of analytic function.
Theroem F. Let
0
)
(
j
j j
z
a
z
f
(not identically zero) be analytic in z t. If for someK1,
K
a
0
t
a
1
t
2a
2
(1.8) and for some real
0,1, 2,
2
arg aj j
then
f
(
z
)
does not vanish in2 ( 1)2 2 ( 1)2
) 1 (
K M
t M K
M
t K z
, (1.9)
where
j
j
j t a a
K
M
1 0
sin 2 ) sin
(cos
(1.10)
Firstly we prove the following result which gives maximum number of zeros that can lie in a prescribed region and also a zero-free region thereby improving Theorem C.
Theroem 1.1. If
n j
j j z
a z
p
0
)
( be a polynomial of degree
n
with complex coefficients. Letj j
j i
a and for some K,L1
K
n
n1
1
1
0and
A Monthly Double-Blind Peer Reviewed Refereed Open Access International e-Journal - Included in the International Serial Directories International Journal in Physical & Applied Sciences
http://www.ijmr.net.in email id- [email protected] Page 34
where 0 and 0 are not zero simultaneously, then the maximum number of zeros in
1 0 ,
1
0 z
M a
does not exceed
0 0 0 0
0
) (
) (
) (
2 ) (
) (
log ) / 1 log(
1
a
L K
n n
n n
,
where
M1(
0
0)2(
)K(
n
n)L(
n
n). (1.12)For0, Theorem 1.1 reduces to following
Corollary 1.1. If
n j
j jz
a z
p
0
)
( be a polynomial of degree n with complex coefficients. Let
j j
j i
a for
j
0
,
1
,
2
,
,
n
and for some K,L1
K
n
n1
1
0 ,and
L
n
n1
1
0 , (1.13)with 0 and 0 are not simultaneously zero, then the maximum number of zeros in
1
0
,
20
z
M
a
does not exceed
0 0 0 0
0 ) ( ) ( ) ( )
( log ) / 1 log(
1
a
L
K
n
n
n
n
A Monthly Double-Blind Peer Reviewed Refereed Open Access International e-Journal - Included in the International Serial Directories International Journal in Physical & Applied Sciences
http://www.ijmr.net.in email id- [email protected] Page 35
M2 K(
n
n)L(
n
n)(
0
0). (1.14)If j 0,j 0, for j0,1,2,,n; in (1.13)then we get the following
Corollary1.2.If
n j
j j z
a z
p
0
)
( ; aj jij,j0,1,2,,n; be a polynomial of degree
n
such that for some K,L1
K
n
n1
1
00and
L
n
n1
1
0 0 (1.5)then maximum number of zeros in
,
0
1
30
z
M
a
does not exceed
0
) (
2 log ) / 1 log(
1
a L K
n
n
,where
M32(K
nL
n)(
0
0). (1.16)For the location of zeros of analytic functions we prove the following result which not only generalizes Theorem F, but in particular cases reduce to Theorem E and Theorem D also. More precisely we prove
Theroem 1.2. If
0
) (
j
j j z
a z
f (not identically zero) be analytic in z t. If for some K1,
A Monthly Double-Blind Peer Reviewed Refereed Open Access International e-Journal - Included in the International Serial Directories International Journal in Physical & Applied Sciences
http://www.ijmr.net.in email id- [email protected] Page 36
and for some
0,1,2,
2
argaj j ,
then
f
(
z
)
does not vanish in2 2 4
4 2
2
4 ( 1) ( 1)
) 1 (
K M
t M K
M
t K z
, (1.18)
where
j
j j t a a
K a
a t K
M
1 0 0
4
sin 2 sin cos
2
. (1.19)
Remark 1.1.If we let
in Theorem 1.2 we get Theorem F. For
,
0
,
Theorem 1.2 reduces to Theorem E and
,
0
,
K
1
, Theorem 1.2 reduces to Theorem D.2. Lemmas:
We need the following lemma for the proof of the above theorems.
Lemma 2.1. If
n a z z
p
0
) (
is a polynomial of degree
n
such that for some real ,aj , j 0,1,2, ,n 2
arg ,
then for some t 0,
taj aj1 taj aj1 cos
(taj aj1)sin
. (2.1)A Monthly Double-Blind Peer Reviewed Refereed Open Access International e-Journal - Included in the International Serial Directories International Journal in Physical & Applied Sciences
http://www.ijmr.net.in email id- [email protected] Page 37
3. Proof of the Theorems:
Proof of Theorem 1.1. Consider the polynomial 1 1 2 1 2 0 1 0 2 2 1 0
)
(
)
(
)
(
)
)(
1
(
)
(
)
1
(
)
(
n n n n n n nz
a
z
a
a
z
a
a
z
a
a
a
z
a
z
a
z
a
a
z
z
p
z
z
F
1 1 10
(
)
n n n j j jj
a
z
a
z
a
a
Now for z 1
n j j j n j j j n n n n j jj a a
A Monthly Double-Blind Peer Reviewed Refereed Open Access International e-Journal - Included in the International Serial Directories International Journal in Physical & Applied Sciences
http://www.ijmr.net.in email id- [email protected] Page 38
) ( ) ( ) ( 2 ) ( ) ( 2 ) 1 ( 2 ) 1 ( ) 1 ( ) 1 ( ) ( 0 0 0 0 1 0 1 1 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 n n n n n n n n n n n n n n n j j j j j j n n n n j j j j j j n n n n n L K L L K K L L K K z F
(Let)Thus
F
(
z
)
M
, forz
1
.Also F(0) p(0) a0 0, as
0 and
0 are not zero simultaneously.Now it is known (see [10; page 171] that if
f
(
z
)
is regular,f
(
0
)
0
and F(z) M in z 1; thenthe number of zeros of
f
(
z
)
inz
1
does not exceed ) 0 ( log ) / 1 ( log 1 f M
. Applying this
result to F(z), we get the number of zeros of F(z) and hence of p(z) in z does not exceed
0 0 0 0 0 ) ( ) ( ) ( 2 ) ( ) ( log ) / 1 log( 1 a L K n n n n .
This proves one part of the theorem.
Now to show that no zeros lie in
1 0
M a
z .
M
A Monthly Double-Blind Peer Reviewed Refereed Open Access International e-Journal - Included in the International Serial Directories International Journal in Physical & Applied Sciences
http://www.ijmr.net.in email id- [email protected] Page 39
For this we have
F(z)a0(a1a0)z(a2a1)z2(anan1)znanzn1
F(z)a0h(z) (3.1)
where
1 1 1 1 1 1 1 2 1 2 0 1 ) ( ) ( ) ( ) ( ) ( ) ( ) ( n j j j j n n n n n n n n n n z a a z a a z a z h z a z a a z a a z a a zh
For
z
1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ) ( ) ( ) ( m ax n j j j j j j n j j j j j j n n n n n n n n n n n j j j n j j j n n n n n n n j j j n n n z L L K K a a a a a z h ) ( ) ( ) ( 2 ) ( 2 ) 1 ( 2 ) 1 ( ) ( 0 0 1 0 1 1 0 1 n n n n n n n n n n n n n n L K L L K K M1 (Let)
Thus 1 1
) (
maxh z M
z
Therefore by Schwarz’s lemma
A Monthly Double-Blind Peer Reviewed Refereed Open Access International e-Journal - Included in the International Serial Directories International Journal in Physical & Applied Sciences
http://www.ijmr.net.in email id- [email protected] Page 40
Now from inequality (3.1)
F(z)a0h(z)
0
) ( )
(
1 0 0
z M a
z h a z F
for
z
1
if
1 0
M a
z .
Therefore no zeros of
F
(
z
)
and hence ofp
(
z
)
lie in
1 0
M a
z .
This completes the proof of Theorem 1.1.
Proof of Theorem 1.2. Since
f
(
z
)
is analytic in,z
t
Therefore lim 0
j j
j a z . Now consider the function
2 1 1
0 0
0 0
2 2 1 1 0 0
2 2 1 0
) (
) (
) (
) (
) )(
(
) ( ) ( ) (
j
j j j ta z a
z a t Ka z a K z a a t
z a t a z a t a a t
z a z
a z a a t z
z f t z z F
or
A Monthly Double-Blind Peer Reviewed Refereed Open Access International e-Journal - Included in the International Serial Directories International Journal in Physical & Applied Sciences
http://www.ijmr.net.in email id- [email protected] Page 41
where
2 1 1 10 ) ( )
( ) ( j j j
j ta z
a z z a t Ka z G .
Clearly
G
(
z
)
is analytic;G
(
0
)
0
and forz
t
sin
)
(
cos
)
(
sin
)
(
cos
)
(
sin
)
(
cos
)
[(
)
(
3 3 2 2 3 3 2 2 2 2 1 2 2 1 1 0 1 0 2 1 1 1 0a
t
a
t
a
t
a
t
a
t
a
t
a
t
a
t
a
t
a
K
a
t
a
K
t
t
a
t
a
t
a
t
Ka
t
z
G
j j j j
1 0 0 0 1 0 0 1 1 1 1 1 1 1sin
2
sin
cos
)
2
(
sin
2
sin
cos
)
2
(
]
sin
)
(
cos
)
(
sin
)
(
cos
)
(
j j j j j jt
a
a
K
a
a
t
K
a
t
t
a
a
K
a
t
a
K
t
a
t
a
t
a
t
a
t
a
t
a
t
a
t
a
t
=t a0 M4, (Let)
where j j j t a a K a a t K M
1 0 0 4 sin 2 sin cos2
(3.3)
This implies
A Monthly Double-Blind Peer Reviewed Refereed Open Access International e-Journal - Included in the International Serial Directories International Journal in Physical & Applied Sciences
http://www.ijmr.net.in email id- [email protected] Page 42
Hence from (3.2), we have
0
) 1 (
) ( )
(
4 0 0
0 0 0
z M a t z K a
z G z Ka z a a t z F
if
M4z (K1)zt (3.4)
Since it is easy to verify that the region defined by (3.4) is precisely the disk
2 2
4 4 2
2
4 ( 1) ( 1)
) 1 ( :
K M
t M
K M
t K z
z . (3.5)
It follows from (3.4) that F(z) and hence f(z) does not vanish in the disk defined by (3.5). This completes the proof of Theorem 1.2.
A Monthly Double-Blind Peer Reviewed Refereed Open Access International e-Journal - Included in the International Serial Directories International Journal in Physical & Applied Sciences
http://www.ijmr.net.in email id- [email protected] Page 43
References:
[1] A. Aziz and Q.G. Mohammad, On the zeros of a certain class of polynomials and related analytic functions, J. Math. Anal. Appl. 75(1980) 495-502.
[2] A. Aziz and W.M. Shah, On the location of zeros of polynomials and related analytic functions, Nonlinear Studies, 6(1999) 91-101.
[3] G.Enestrom, Remarquee sur un theorem relatif aux racines de I’ equation
0
0 1
1
x a
a x
an n n n ou tous les coefficients sont reels et possitifs, Tohoku Math.J.,18(1920) 34-36.
[4] N.K.Govil and Q.I.Rahman, On the Enestrom-Kakeya Theorem, Tohoku Math. J., 20(1968) 126-136.
[5] A.Hurwitz,
U
ber einen Satz des Herrn Kakeya, Tohoku Math. J., 4(1913-14), 29-93; Math.Werke, 2,626-631.[6] A. Joyal, G.Labelle and Q.I.Rahman, On the location of zeros of polynomials, Canad. Math. Bull., 10(1967) 53-63.
[7] S.Kakeya, On the limits of the roots of an algebraic equation with positive coefficients, Tohoku Math. J., 2(1912-13) 140-142.
[8] N.A.Rather and S.S.Ahmad, A remark on a generalization of Enestro m- Kakeya theorem, Journal of Analysis and Computation, vol. 3, no. 1(2007), 33-41.
[9] W.M.Shah, and A. Liman, On Enestrom-Kakeya Theorem and related analytic functions, Proc. Indian Acad. Sci.(Math. Sci), Vol.-117,no.3, 2007, 359-370.
