http://www.ijeijournal.com/pages/v2i5.html

1. International Journal of Engineering Inventions
e-ISSN: 2278-7461, p-ISSN: 2319-6491
Volume 2, Issue 5 (March 2013) PP: 54-60
www.ijeijournal.com Page | 54
On the Zeros of Analytic Functions inside the Unit Disk
M. H. Gulzar
Department of Mathematics, University of Kashmir, Srinagar 190006
Abstract: In this paper we find an upper bound for the number of zeros of an analytic function inside the unit
disk by restricting the coefficients of the function to certain conditions.
AMS Mathematic Subject Classification 2010: 30C 10, 30C15
Keywords and Phrases: Bound, Analytic Function, Zeros
I. Introduction and Statement of Results
A well-known result due to Enestrom and Kakeya [5] states that a polynomial 

n
j
j
j zazP
0
)( of degree n
with
0...... 011   aaaa nn
has all its zeros in .1z
Q. G. Mohammad [6] initiated the problem of finding an upper bound for the number of zeros of P(z) satisfying
the above conditions in
2
1
z . Many generalizations and refinements were later given by researchers on the
bounds for the number of zeros of P(z) in 10,  z (for reference see [1]),[2], [4]etc.).
In this paper we consider the same problem for analytic functions and prove the following results:
Theorem 1: Let 0)(
0
 

j
j
j zazf be analytic for 1z , where njia jjj ,......,1,0,   . If for
some 0 ,
......210   ,
then the number of zeros of f(z) in 10,
0
 z
M
a
, does not exceed
0
0
00 22
log
1
log
1
a
j
j


 

,
where




1
00 22
j
jM 
Taking 0 , the following result immediately follows from Theorem 1:
Corollary 1: Let 0)(
0
 

j
j
j zazf be analytic for 1z , where njia jjj ,......,1,0,   . If
......210   ,
then the number of zeros of f(z) in 10,
0
 z
M
a
, does not exceed

2. On the Zeros of Analytic Functions inside the Unit Disk
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0
0
00 2
log
1
log
1
a
j
j


 

,
where




1
00 2
j
jM  .
If the coefficients ja are real i.e. njj ,....,1,0,0  , we get the following result from Theorem 1:
Corollary 2 : Let 0)(
0
 

j
j
j zazf be analytic for 1z , where
......210  aaa ,
then the number of zeros of f(z) in 10,
0
 z
M
a
, does not exceed
0
002
log
1
log
1
a
aa 

,
where
02 aM   .
Taking 0)1(   k , 1k , we get the following result from Theorem 1:
Corollary 3 : Let 0)(
0
 

j
j
j zazf be analytic for 1z , where
njia jjj ,......,1,0,   . If for some 1k ,
......210  k ,
then the number of zeros of f(z) in 10,
0
 z
M
a
, does not exceed
0
0
00 2)12(
log
1
log
1
a
k
j
j


 

,
where




1
00 2)12(
j
jkM 
Applying Theorem 1 to the function –if(z), we get the following result:
Theorem 2 : Let 0)(
0
 

j
j
j zazf be analytic for 1z , where njia jjj ,......,1,0,   . If for
some 0 ,
......210   ,
then the number of zeros of f(z) in 10,
0
 z
M
a
, does not exceed

3. On the Zeros of Analytic Functions inside the Unit Disk
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0
0
00 22
log
1
log
1
a
j
j


 

,
where




1
00 22
j
jM  .
Theorem 3 : Let 0)(
0
 

j
j
j zazf be analytic for 1z . If for some 0 ,
......210  aaa ,
and for real some  ,
nja j ,.......,1,0,
2
arg 

 ,
then the number of zeros of f(z) in 10,
0
 z
M
a
, does not exceed
0
1
0 sin)1sin)(cos(
log
1
log
1
a
aa
j
j


 

,
where




1
0 sin)sin)(cos(
j
jaM  .
Taking 0 , Theorem 3 reduces to the following result:
Corollary 4:Let 0)(
0
 

j
j
j zazf be analytic for 1z . If,
......210  aaa ,
and for some real  ,
nja j ,.......,1,0,
2
arg 

 ,
then the number of zeros of f(z) in 10,
0
 z
M
a
, does not exceed
0
1
0 sin)1sin(cos
log
1
log
1
a
aa
j
j


 

,
where




1
0 sin)sin(cos
j
jaaM  .
Taking 0)1( ak  , 1k , we get the following result from Theorem 3:

4. On the Zeros of Analytic Functions inside the Unit Disk
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Corollary 5:Let 0)(
0
 

j
j
j zazf be analytic for 1z . If,
......210  aaak ,
and for some real  ,
nja j ,.......,1,0,
2
arg 

 ,
then the number of zeros of f(z) in 10,0
 z
M
a
does not exceed
0
1
0 sin)1sin(cos
log
1
log
1
a
aak
j
j


 

,
where




1
00 sin)sin(cos2
j
jaakM  .
2. Lemmas
For the proofs of the above results we need the following results:
Lemma 1 : Let f(z) be analytic for 1z , 0)0( f and Mzf )( for 1z ,
Then the number of zeros of f(z) in 10,  z , does not exceed
)0(
log
1
log
1
f
M

(see [7] , page 171 ).
Lemma 2 : If for some t>0, 1 jj ata and ,...2,1,0,
2
arg  ja j

 , for some real  , then
 sin)(cos)( 111   jjjjjj ataataata .
The proof of Lemma 2 follows from a lemma of Govil and Rahman [3].
3. Proofs of Theorems:
Proof of Theorem 1: Consider the function
)()1()( zfzzF 
......))(1( 2
210  zazaaz
......)()( 2
21100  zaazaaa
......)()( 2
21100  zzz 
......})(){( 21100  zzii  .
For 1z ,
03221100 ......)(  zF
......2110  




0
00 22
j
j .

5. On the Zeros of Analytic Functions inside the Unit Disk
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Since F(z) is analytic for 1z , 0)0( 0  aF , it follows , by using Lemma 1, that the number of zeros of
F(z) in 10,  z , does not exceed
0
0
00 22
log
1
log
1
a
j
j


 

.
On the other hand, consider
)()1()( zfzzF 
......))(1( 2
210  zazaaz
......)()( 2
21100  zaazaaa
)(0 zqa  ,
where
......)()()( 2
2110  zaazaazq
......)()( 2
2110  zzz 
......})(){( 2
2110  zzi 
For 1z ,
............)( 21102110  zq
M
j
j  

1
00 22  .
Since q(z ) is analytic for 1z , q(0)=0, it follows , by Schwarz’s lemma , that
zMzq )( for 1z .
Hence for 1z ,
)()( 0 zqazF 
)(0 zqa 
zMa  0
0
if
M
a
z
0
 .
This shows that all the zeros of F(z) lie in
M
a
z
0
 . Since the zeros of f(z) are also the zeros of F(z), it
follows that all the zeros of f(z) lie in
M
a
z
0
 . Thus, the number of zeros of f(z) in
10,
0
 z
M
a
, does not exceed

6. On the Zeros of Analytic Functions inside the Unit Disk
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0
0
00 22
log
1
log
1
a
j
j


 

.
Proof of Theorem 2: Consider the function
)()1()( zfzzF 
......))(1( 2
210  zazaaz
......)()( 2
21100  zaazaaa
......)()( 2
21100  zaazaaza 
For 1z , we have, by using the hypothesis and Lemma 2,
 sin)(cos)[()( 10100 aaaaazF 
......sin)(cos)( 2121   aaaa




1
0 sin)1sin)(cos(
j
jaa  .
Since F(z) is analytic for 1z , 0)0( 0  aF , it follows , by using the lemma 1, that the number of zeros
of F(z) in 10,  z , does not exceed
0
1
0 sin)1sin)(cos(
log
1
log
1
a
aa
j
j


 

.
Again , Consider the function
)()1()( zfzzF 
......))(1( 2
210  zazaaz
......)()( 2
21100  zaazaaa
......)()( 2
21100  zaazaaza 
)(0 zqa  ,
where
......)()()( 2
2110  zaazaazq
......)()( 2
2110  zaazaaz 
For 1z , by using lemma 2, we have,
 sin)(cos)()( 1010 aazq 
......sin)(cos)( 2121   aaaa
Maa
j
j  

1
0 sin)sin)(cos(  .
Since q(z ) is analytic for 1z , q(0)=0, it follows , by Schwarz’s lemma , that
zMzq )( for 1z .
Hence for 1z ,
)()( 0 zqazF 

7. On the Zeros of Analytic Functions inside the Unit Disk
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)(0 zqa 
zMa  0
0
if
M
a
z
0
 .
This shows that all the zeros of F(z) lie in
M
a
z
0
 . Since the zeros of f(z) are also the zeros of F(z), it
follows that all the zeros of f(z) lie in
M
a
z
0
 . Thus, the number of zeros of f(z) in
10,
0
 z
M
a
, does not exceed
0
1
0 sin)1sin)(cos(
log
1
log
1
a
aa
j
j


 

.
References
[1] K. K. Dewan, Extremal Properties and Coefficient estimates for polynomials with restricted zeros and on location of zeros of
polynomials, Ph.D Thesis,IIT Delhi, !980.
[2] K. K.Dewan, Theory of Polynomials and Applications,Deep & Deep Publications Pvt. Ltd., New Delhi, 2007.
[3] N. K. Govil and Q. I. Rahman, On the Enestrom-Kakeya Theorem, Tohoku Math. J .20(1968), 126-136.
[4] M. H. Gulzar, On the number of zeros of a polynomial in a prescribed region, Research Journal of Pure Algebra-2(2), Feb. 2012, 35-
46.
[5] M. Marden, Geometry of Polynomials, IInd. Ed. Math. Surveys, No.3, Amer. Math. Soc. Providence, R.I. 1966.
[6] Q. G. Mohammad, On the Zeros of Polynomials, Amer. Math. Monthly, 72 (1965), 631-633.
[7] E. C Titchmarsh, The Theory of Functions, 2nd
edition, Oxford University Press, London, 1939.