1. Research Inventy: International Journal Of Engineering And Science
Vol.2, Issue 8 (March 2013), Pp 06-10
Issn(e): 2278-4721, Issn(p):2319-6483, Www.Researchinventy.Com
6
On The Zeros of Analytic Functions
M. H. Gulzar
P. G. Department of Mathematics University of Kashmir, Srinagar 190006
Abstract : In this paper we consider a certain class of analytic functions whose coefficients are restricted to
certain conditions, and find some interesting zero-free regions for them. Our results generalize a number of
already known results in this direction.
Mathematics Subject Classification: 30C10, 30C15
Key-words and phrases: Analytic Function , Coefficients, Zeros
I. INTRODUCTION AND STATEMENT OF RESULTS
Regarding the zeros of a class of analytic functions , whose coefficients are restricted to certain
conditions, W. M. Shah and A. Liman [4] have proved the following results:
Theorem A: Let 0)(
0
 

j
j
j zazf be analytic in tz  . If for some 1k ,
......2
2
10  atatak .,
and for some  ,
,......,2,1,0,
2
arg  ja j


then f(z ) does not vanish in
2222
)1()1(
)1(





kM
Mt
kM
tk
z ,
where




10
sin
2)sin(cos
j
j
j ta
a
kM

 .
Theorem B: Let 0)(
0
 

j
j
j zazf be analytic in tz  . If for some 1k and 0 ,
............ 1
2
2
10  k
k
k
tatatatak .,
and for some  ,
,......,2,1,0,
2
arg  ja j


then f(z ) does not vanish in
22*
*
22
)1()1(*
)1(





kM
tM
kM
tk
z ,
where




100
* sin
2sincos)
2
(
j
j
j
kk
ta
a
kkt
a
a
M

 .
The aim of this paper is to generalize the above - mentioned results. In fact , we are going to prove the
following results:

2. On The Zeros Of Analytic Functions…
7
Theorem 1: Let 0)(
0
 

j
j
j zazf be analytic for tz  . If for some 0 ,
......2
2
10  atata .,
and for some real  and  ,
,......,2,1,0,
2
arg  ja j


then f(z ) does not vanish in
222
0
2
0
222
0
0






Ma
aMt
Ma
ta
z ,
where






100
0 sin
2)sin)(cos(
j
j
j ta
aa
a
M



.
Remark1: Taking 0)1( ak  , 1k , Theorem 1 reduces to Theorem A.
Also taking 0  , we get the following result, proved earlier by Aziz and Shah [2] :
Corollary 1: Let 0)(
0
 

j
j
j zazf be analytic for tz  such that for some 1k ,
......2
2
10  attaka .
Then f(z) does not vanish in
1212
)1(





k
kt
k
tk
z .
Taking 0  and 0 , we get the following result proved earlier by Aziz and Mohammad [1] :
Corollary 2: Let 0)(
0
 

j
j
j zazf be analytic for tz  such that 0ja and
,......3,2,1,01  jtaa jj .Then f(z) does not vanish in tz  .
Theorem 2: Let 0)(
0
 

j
j
j zazf be analytic for tz  . If for some 0 ,
0 ,
............ 1
1
2
2
10  





 atatatata .,
and for some real  and ,
,......,2,1,0,
2
arg  ja j


then f(z ) does not vanish in
22*2
0
2
0
*
22*2
0
0






Ma
atM
Ma
ta
z ,
where









10000
* sin2
sin1cos)12(
j
j
j ta
aaa
t
a
a
M




 
.
Remark2: Taking 0)1( ak  , 1k , Theorem 2 reduces to Theorem B.
The result of Aziz and Mohammad (Theorem 6 of [1]) follows from Theorem 2 by taking 0 .

3. On The Zeros Of Analytic Functions…
8
II. LEMMA
For the proofs of the above theorems , we need the following lemma, which is due to Govil and
Rahman [3]:
Lemma : If
2
arg

 ja and for some t>0, ,,......,1,0,1 njata jj   , then
]sin)(cos)[( 111    jjjjjj ataataata .
III. Proofs Of The Theorems
Proof of Theorem 1: Consider the function
)()()( zftzzF 
......)()(
......)()(
......))((
2
21100
2
21100
2
210



ztaaztaazta
ztaaztaata
zazaatz

)(0 zGzta   ,
where



 
2
110 )()()(
j
j
jj ztaaztaazG  .
Since ,......,2,1,0,
2
arg  ja j

 by using the lemma and the hypothesis, we have for tz  ,
 sin}){(cos})[{()( 1010 taataatzG 
......]}sin)(cos){( 2121   taataat
.
]
sin2
)sin)(cos1[(
0
100
0
Mat
ta
aa
at
j
j
j

 





Since G(z) is analytic for tz  , G(0)=0, it follows by Schwarz’s lemma that
zMatzG 0)(  for tz  .
Hence it follows that
)()( 0 zGztazF  
][
0
0 Mztt
a
z
a 

0
if
Mztt
a
z

0

i.e. if
t
a
z
Mzt 
0

.
Since the region defined by
t
a
z
Mzt 
0

is precisely the disk

4. On The Zeros Of Analytic Functions…
9
222
0
222
0
0






Ma
Mt
Ma
ta
z ,
we conclude that F(z) and therefore f(z) does not vanish in the disk
222
0
222
0
0






Ma
Mt
Ma
ta
z .
That completes the proof of Theorem 1.
Proof of Theorem 2: Consider the function
)()()( zftzzF 
1
1
2
21100
1
2
21100
2
210
)(......)()(
......)(......)()(
......))((






k
kk
n
nn
ztaaztaaztaazta
ztaaztaaztaata
zazaatz

......)(...... 1  
n
nn zaa
)(0 zGzta   .
Since ,......,2,1,0,
2
arg  ja j

 by using the lemma and the hypothesis, we have for tz  ,
1
1
2
2110 )(......)()()( 
 
 ztaaztaaztaazG
......)(...... 1
1  

n
nn zaa
 sin)(cos)[( 0101 ataatat 
......]
sin)(cos)(......
sin)(cos)(
sin)(cos)(
......sin)(cos)(
1
1
1
1
1
1
1
1
1
1
1
1
12
2
12
2

















n
n
n
n
n
n
n
n
n
n
at
atatatat
atatatat
atatatat
taattaat





























10000
0
sin2
sin1cos)12(
j
j
j ta
aaa
t
a
a
at




 
*
0 Mat ,
where









10000
* sin2
sin1cos)12(
j
j
j ta
aaa
t
a
a
M




 
.
Since G(z) is analytic for tz  , G(0)=0, it follows by Schwarz’s lemma that
zMatzG *
0)(  for tz  .
Hence it follows that
)()( 0 zGztazF  
][ *
0
0 Mztt
a
z
a 

0

5. On The Zeros Of Analytic Functions…
10
if
*
0
Mztt
a
z


i.e. if
t
a
z
Mzt 
0
* 
.
Since the region defined by
t
a
z
Mzt 
0
* 
is precisely the disk
22*2
0
*
22*2
0
0






Ma
tM
Ma
ta
z ,
we conclude that F(z) and therefore f(z) does not vanish in the disk
22*2
0
*
22*2
0
0






Ma
tM
Ma
ta
z .
That completes the proof of Theorem 2.
REFERENCES
[1] A. Aziz and Q. G. Mohammad, On the zeros of a certain class of polynomials and related analytic functions, J. Math. Anal.
ppl.75(1980), 495-502.
[2] A. Aziz and W. M. Shah, On the location of zeros of polynomials and related analytic functions, onlinear Studies 6 (1999), 91-
101.
[3] N. K. Govil and Q.I. Rahman, On the Enestrom-Kakeya Theorem II, Tohoku Math.J. 20 (1968), 126-136.
[4] W. M. Shah and A. Liman, On Enestrom-Kakeya Theorem and related analytic functions, Proc. Indian Acad. Sci. (Math. Sci.), Vol.
117, No.3, August 2007359-370.