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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 1k ,
......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
tkz ,
where
10
sin2)sin(cos
j
j
j taa
kM
.
Theorem B: Let 0)(0
j
j
j zazf be analytic in tz . If for some 1k 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
tkz ,
where
100
* sin2sincos)
2(
j
j
j
kkta
akkt
a
aM
.
The aim of this paper is to generalize the above - mentioned results. In fact , we are going to prove the
following results:
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
taz ,
where
100
0 sin2)sin)(cos(
j
j
j taaa
aM
.
Remark1: Taking 0)1( ak , 1k , 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 1k ,
......2
2
10 attaka .
Then f(z) does not vanish in
1212
)1(
k
kt
k
tkz .
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 0ja 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
taz ,
where
10000
* sin2sin1cos)12(
j
j
j taaaa
ta
aM
.
Remark2: Taking 0)1( ak , 1k , Theorem 2 reduces to Theorem B.
The result of Aziz and Mohammad (Theorem 6 of [1]) follows from Theorem 2 by taking 0 .
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
taaa
atj
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 Mztta
za
0
if
Mztta
z
0
i.e. if
ta
zMzt
0
.
Since the region defined by
ta
zMzt
0
is precisely the disk
On The Zeros Of Analytic Functions…
9
222
0
222
0
0
Ma
Mt
Ma
taz ,
we conclude that F(z) and therefore f(z) does not vanish in the disk
222
0
222
0
0
Ma
Mt
Ma
taz .
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
sin2sin1cos)12(
j
j
j taaaa
ta
aat
*
0 Mat ,
where
10000
* sin2sin1cos)12(
j
j
j taaaa
ta
aM
.
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 Mztta
za
0
On The Zeros Of Analytic Functions…
10
if
*
0
Mztta
z
i.e. if
ta
zMzt
0
* .
Since the region defined by
ta
zMzt
0
*
is precisely the disk
22*2
0
*
22*2
0
0
Ma
tM
Ma
taz ,
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
taz .
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.
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