B02806010

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

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 f ( z ) 



a j 0

j

z j  0 be analytic in z  t . If for some k  1 ,

k a0  t a1  t a 2  ...... ., 2

and for some  ,

arg a j     

 2

, j  0,1,2,......,

then f(z ) does not vanish in

z

(k  1)t Mt ,  2 2 M  (k  1) M  (k  1) 2 2

where

M  k (cos   sin  )  2 Theorem B: Let f ( z ) 

sin  a0



a j 0

j



a j 1

j

tj.

z j  0 be analytic in z  t . If for some k  1 and   0 ,

k a0  t a1  t a 2  ......  t k a k  t k 1  ...... ., 2

and for some  ,

arg a j     

 2

, j  0,1,2,......,

then f(z ) does not vanish in

z

(k  1)t M *t  , M *2 (k  1) 2 M * 2  (k  1) 2

where

M* (

2 ak a0

t k  k ) cos   k sin   2

sin  a0



a j 1

j

tj .

The aim of this paper is to generalize the above - mentioned results. In fact , we are going to prove the following results:

6

On The Zeros Of Analytic Functions…

Theorem 1: Let f ( z ) 



a j 0

j

z j  0 be analytic for z  t . If for some   0 ,

  a0  t a1  t a 2  ...... ., and for some real  and  ,  arg a j      , j  0,1,2,......, 2

2

then f(z ) does not vanish in

z

 a0 t

Mt a 0



a0 M 2   2 2

2

a0 M 2   2 2

,

where

M (

  a0 a0

)(cos   sin  )  2

Remark1: Taking Also taking

sin  a0



a j 1

j

tj .

  (k  1) a 0 , k  1 , Theorem 1 reduces to Theorem A.

    0 , we get the following result, proved earlier by Aziz and Shah [2] :

Corollary 1: Let f ( z ) 



a j 0

j

z j  0 be analytic for z  t such that for some k  1 ,

ka0  ta1  t a 2  ...... .

2

Then f(z) does not vanish in

(k  1)t kt  .

2k  1 2k  1 Taking     0 and   0 , we get the following result proved earlier by Aziz and Mohammad [1] : z

Corollary 2: Let f ( z ) 



a j 0

j

z j  0 be analytic for z  t such that a j  0 and

a j 1  ta j  0, j  1,2,3,...... .Then f(z) does not vanish in z  t .

Theorem 2: Let f ( z ) 



a j 0

j

z j  0 be analytic for z  t . If for some   0 ,

  0,   a0  t a1  t 2 a 2  ......  t  a  t  1 a 1  ...... ., and for some real  and  ,  arg a j      , j  0,1,2,......, 2

then f(z ) does not vanish in

z

 a0 t a0 M *   2 2

2



M *t a0

2

a0 M *   2 2

2

,

where

 a    2 sin  M *  (2 t  1 ) cos   1  sin   a0 a0 a0  a 0 Remark2: Taking



a j 1

j

t j .

  (k  1) a 0 , k  1 , Theorem 2 reduces to Theorem B.

The result of Aziz and Mohammad (Theorem 6 of [1]) follows from Theorem 2 by taking

7

 0.

On The Zeros Of Analytic Functions…

II.

LEMMA

For the proofs of the above theorems , we need the following lemma, which is due to Govil and Rahman [3]: Lemma : If arg a j     



and for some t>0, ta j  a j 1 , j  0,1,......, n, , then 2  [( ta j  a j 1 ) cos   ( ta j  a j 1 ) sin  ] .

ta j  a j 1

III.

Proofs Of The Theorems

Proof of Theorem 1: Consider the function

F ( z)  ( z  t ) f ( z)  ( z  t )(a 0  a1 z  a 2 z 2  ......)

 ta 0  (a 0  ta1 ) z  (a1  ta 2 ) z 2  ......  ta 0  z  (   a 0  ta1 ) z  (a1  ta 2 ) z 2  ......  ta 0  z  G( z ) , where 

G( z )  (   a 0  ta1 ) z   (a j 1  ta j ) z j . j 2

Since arg a j     



, j  0,1,2,......, by using the lemma and the hypothesis, we have for z  t , 2 G( z )  t[{(  a0 )  ta1 } cos   {(  a0 )  ta1 } sin 

 t{( a1  ta 2 ) cos   ( a1  ta 2 ) sin }  ......]

 t a 0 [(1 

 a0

)(cos   sin  ) 

2 sin  a0



a j 1

j

t j]

 t a0 M .

Since G(z) is analytic for z  t , G(0)=0, it follows by Schwarz’s lemma that

G( z )  t a 0 M z for z  t .

Hence it follows that

F ( z )  ta 0  z  G( z )  a0 [

z a0

 t t z M]

0 if

z a0

t t zM

i.e. if

tzM 

z a0

t .

Since the region defined by

tzM 

z a0

t

is precisely the disk

8

On The Zeros Of Analytic Functions…

z

 a0 t a0 M   2

2

2



Mt a0 M 2   2 2

,

we conclude that F(z) and therefore f(z) does not vanish in the disk

z

 a0 t a0 M 2   2 2



Mt a0 M 2   2 2

.

That completes the proof of Theorem 1.

Proof of Theorem 2: Consider the function

F ( z)  ( z  t ) f ( z)

 ( z  t )(a 0  a1 z  a 2 z 2  ......)  ta 0  (a 0  ta1 ) z  (a1  ta 2 ) z 2  ......  (a n 1  ta n ) z n  ......  ta 0  z  (   a 0  ta1 ) z  (a1  ta 2 ) z 2  ......  (a k  ta k 1 ) z k 1  ......  (a n1  a n ) z n  ......

 ta 0  z  G( z ) .

Since arg a j     



, j  0,1,2,......, by using the lemma and the hypothesis, we have for z  t , 2 G( z )  (   a0  ta1 ) z  (a1  ta 2 ) z 2  ......  (a  ta  1 ) z  1

 ......  (a n1  a n ) z n1  ......  t[( ta1    a0 ) cos   ( ta1    a0 ) sin 

 ( t 2 a 2  ta1 ) cos   ( t 2 a 2  ta1 ) sin   ......  ( t  a   t  1 a  1 ) cos   ( t  a   t  1 a  1 ) sin   ( t  a   t  1 a  1 ) cos   ( t  a   t  1 a  1 ) sin   ......  ( t n 1 a n 1  t n a n ) cos   ( t n 1 a n 1  t n a n ) sin   t n a n  ......]

 a    2 sin   t a 0 (2 t  1 ) cos   1  sin   a0 a0 a0  a 0 *  t a0 M ,



a j 1



j

tj

j

t j .

where

 a    2 sin  M *  (2 t  1 ) cos   1  sin   a0 a0 a0  a 0



a j 1

Since G(z) is analytic for z  t , G(0)=0, it follows by Schwarz’s lemma that

G( z )  t a0 M * z for z  t .

Hence it follows that

F ( z )  ta 0  z  G( z )

 a0 [

z a0

 t  t z M *]

0

9

On The Zeros Of Analytic Functions… if

z a0

t t zM*

i.e. if

t zM* 

z a0

t .

Since the region defined by

t zM* 

z a0

t

is precisely the disk

z

 a0 t a 0 M *2   2 2



M *t a 0 M *2   2 2

,

we conclude that F(z) and therefore f(z) does not vanish in the disk

z

 a0 t a 0 M *2   2 2



M *t a 0 M *2   2 2

.

That completes the proof of Theorem 2.

REFERENCES [1] [2] [3] [4]

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.

A. Aziz and W. M. Shah, On the location of zeros of polynomials and related analytic functions, onlinear Studies 6 (1999), 91101.

N. K. Govil and Q.I. Rahman, On the Enestrom-Kakeya Theorem II, Tohoku Math.J. 20 (1968), 126-136.

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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