J02055460

International Journal of Engineering Inventions e-ISSN: 2278-7461, p-ISSN: 2319-6491 Volume 2, Issue 5 (March 2013) PP: 54-60

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 n

A well-known result due to Enestrom and Kakeya [5] states that a polynomial

P( z )   a j z j of degree n j 0

with

a n  a n1  ......  a1  a0  0

z  1.

has all its zeros in

Q. G. Mohammad [6] initiated the problem of finding an upper bound for the number of zeros of P(z) satisfying

1 . Many generalizations and refinements were later given by researchers on the 2 bounds for the number of zeros of P(z) in z   ,0    1 (for reference see [1]),[2], [4]etc.). the above conditions in

z

In this paper we consider the same problem for analytic functions and prove the following results: 

Theorem 1: Let

f ( z )   a j z j  0 be analytic for z  1 , where a j   j  i j , j  0,1,......, n . If for j 0

some

  0,

   0   1   2  ...... , then the number of zeros of f(z) in

a0

 z   ,0    1 , does not exceed

M



2    0   0  2  j

1 log

1

j 0

log

a0



,

where 

M  2    0   0  2  j j 1

Taking 

 0 , the following result immediately follows from Theorem 1: 

Corollary 1: Let

f ( z )   a j z j  0 be analytic for z  1 , where a j   j  i j , j  0,1,......, n . If j 0

 0   1   2  ...... , then the number of zeros of f(z) in

a0 M

 z   ,0    1 , does not exceed

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On the Zeros of Analytic Functions inside the Unit Disk 

 0   0  2  j

1 log

1

j 0

log

a0



,

where 

M   0   0  2  j . j 1

If the coefficients

a j are real i.e.  j  0, j  0,1,...., n , we get the following result from Theorem 1:

Corollary 2 : Let

f ( z )   a j z j  0 be analytic for z  1 , where



j 0

  a0  a1  a 2  ...... ,

1 log where

1

log

a0

 z   ,0    1 , does not exceed M 2  a0  a0

then the number of zeros of f(z) in

,

a0



M  2  a0 .

Taking

  (k  1) 0 , k  1 , we get the following result from Theorem 1: 

Corollary 3 : Let

f ( z )   a j z j  0 be analytic for z  1 , where j 0

a j   j  i j , j  0,1,......, n . If for some k  1 ,

k 0   1   2  ...... , then the number of zeros of f(z) in

a0 M

 z   ,0    1 , does not exceed 

(2k  1) 0   0  2  j

1 log

1

j 0

log

a0



,

where 

M  (2k  1) 0   0  2  j j 1

Applying Theorem 1 to the function –if(z), we get the following result: 

Theorem 2 : Let

f ( z )   a j z j  0 be analytic for z  1 , where a j   j  i j , j  0,1,......, n . If for j 0

some

  0,

   0  1   2  ...... , then the number of zeros of f(z) in

a0 M

 z   ,0    1 , does not exceed

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On the Zeros of Analytic Functions inside the Unit Disk 

2    0   0  2  j

1 log

1

j 0

log

a0



,

where 

M  2    0   0  2  j . j 1



Theorem 3 : Let

f ( z )   a j z j  0 be analytic for z  1 . If for some   0 , j 0

  a0  a1  a 2  ...... , and for real some

,

arg a j      then the number of zeros of f(z) in

 2 a0 M

, j  0,1,......., n ,  z   ,0    1 , does not exceed 

(   a 0 )(cos   sin   1)  sin   a j

1 log

1

j 1

log

a0



,

where 

M    (    0 )(cos   sin  )  sin   a j . j 1

Taking

  0 , Theorem 3 reduces to the following result: 

Corollary 4:Let

f ( z )   a j z j  0 be analytic for z  1 . If, j 0

a0  a1  a 2  ...... , and for some real

,

arg a j      then the number of zeros of f(z) in

 2 a0

M

, j  0,1,......., n ,

 z   ,0    1 , does not exceed 

a 0 (cos   sin   1)  sin   a j

1 log

1

j 1

log

a0



,

where 

M  a 0 (cos   sin  )  sin   a j . j 1

Taking

  (k  1) a 0 , k  1 , we get the following result from Theorem 3:

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On the Zeros of Analytic Functions inside the Unit Disk 

Corollary 5:Let

f ( z )   a j z j  0 be analytic for z  1 . If, j 0

k a0  a1  a 2  ...... , and for some real

,



arg a j     

, j  0,1,......., n , 2 a0 then the number of zeros of f(z) in  z   ,0    1 does not exceed M 

k a 0 (cos   sin   1)  sin   a j

1 log

1

j 1

log

a0



,

where 

M  2k  0 (cos   sin  )  a 0  sin   a j . j 1

2. Lemmas For the proofs of the above results we need the following results: Lemma 1 : Let f(z) be analytic for

z  1 , f (0)  0 and f ( z )  M for z  1 ,

Then the number of zeros of f(z) in

z   ,0    1 , does not exceed

1 log

1

log

M (see [7] , page 171 ). f (0)



ta j  a j 1

ta j  a j 1 and arg a j     



, j  0,1,2,... , for some real  , then 2  ( ta j  a j 1 ) cos   ( ta j  a j 1 ) sin  .

Lemma 2 : If for some t>0,

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

F ( z)  (1  z) f ( z)

 (1  z )(a0  a1 z  a 2 z 2  ......)  a0  (a0  a1 ) z  (a1  a2 ) z 2  ......   0  z  (    0   1 ) z  ( 1   2 ) z 2  ......

 i 0  i{( 0  1 ) z  (1   2 ) z  ......} .

For

z 1, F ( z)     0     0   1   1   2   2   3  ......   0   0  1  1   2  ...... 

 2    0   0  2  j . j 0

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On the Zeros of Analytic Functions inside the Unit Disk Since F(z) is analytic for F(z) in

z  1 , F (0)  a 0  0 , it follows , by using Lemma 1, that the number of zeros of

z   ,0    1 , does not exceed 

2    0   0  2  j

1 log

1

j 0

log

.

a0



On the other hand, consider

F ( z)  (1  z) f ( z)  (1  z )(a0  a1 z  a 2 z 2  ......)  a0  (a0  a1 ) z  (a1  a2 ) z 2  ......

 a 0  q( z ) , where

q( z)  (a0  a1 ) z  (a1  a 2 ) z 2  ......  z  (    0   1 ) z  ( 1   2 ) z 2  ......  i{( 0  1 ) z  (1   2 ) z 2  ......} For

z  1, q( z )       0   1   1   2  ......   0  1  1   2  ...... 

 2    0   0  2  j  M . j 1

Since q(z ) is analytic for

z  1 , q(0)=0, it follows , by Schwarz’s lemma , that

q( z )  M z for z  1 .

Hence for

z 1 ,

F ( z )  a 0  q( z )  a 0  q( z )  a0  M z 0 if

z

a0 M

.

This shows that all the zeros of F(z) lie in

z

follows that all the zeros of f(z) lie

in

a0 M

a0 M

. Since the zeros of f(z) are also the zeros of F(z), it

z

a0 M

. Thus, the number of zeros of f(z) in

 z   ,0    1 , does not exceed

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On the Zeros of Analytic Functions inside the Unit Disk 

2    0   0  2  j

1 log

1

j 0

log

a0



.

Proof of Theorem 2: Consider the function

F ( z)  (1  z) f ( z)

 (1  z )(a0  a1 z  a 2 z 2  ......)  a0  (a0  a1 ) z  (a1  a 2 ) z 2  ......  a0  z  (   a0  a1 ) z  (a1  a2 ) z 2  ......

For

z  1 , we have, by using the hypothesis and Lemma 2, F ( z)    a0  [(   a0  a1 ) cos   (   a0  a1 ) sin   ( a1  a 2 ) cos   ( a1  a 2 ) sin   ...... 

 (   a 0 )(cos   sin   1)  sin   a j . j 1

Since F(z) is analytic for

z  1 , F (0)  a 0  0 , it follows , by using the lemma 1, that the number of zeros

z   ,0    1 , does not exceed

of F(z) in



(   a 0 )(cos   sin   1)  sin   a j

1 log

1

j 1

log

a0



.

Again , Consider the function

F ( z)  (1  z) f ( z)  (1  z )(a0  a1 z  a 2 z 2  ......)

 a0  (a0  a1 ) z  (a1  a2 ) z 2  ......  a0  z  (   a0  a1 ) z  (a1  a2 ) z 2  ......

 a 0  q( z ) , where

q( z)  (a0  a1 ) z  (a1  a 2 ) z 2  ......  z  (   a0  a1 ) z  (a1  a 2 ) z 2  ......

For

z  1 , by using lemma 2, we have, q( z )    (    0  a1 ) cos   (    0  a1 ) sin 

 ( a1  a 2 ) cos   ( a1  a 2 ) sin   ...... 

   (   a 0 )(cos   sin  )  sin   a j  M . j 1

Since q(z ) is analytic for

z  1 , q(0)=0, it follows , by Schwarz’s lemma , that

q( z )  M z for z  1 .

Hence for

z 1 ,

F ( z )  a 0  q( z ) www.ijeijournal.com

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On the Zeros of Analytic Functions inside the Unit Disk

 a 0  q( z )  a0  M z 0 if

z

a0 M

.

This shows that all the zeros of F(z) lie in

z

follows that all the zeros of f(z) lie

in

a0 M

a0 M

. Since the zeros of f(z) are also the zeros of F(z), it

z

a0 M

. Thus, the number of zeros of f(z) in

 z   ,0    1 , does not exceed 

(   a 0 )(cos   sin   1)  sin   a j

1 log

1



log

j 1

a0

.

References [1] [2] [3] [4] [5] [6] [7]

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.

K. K.Dewan, Theory of Polynomials and Applications,Deep & Deep Publications Pvt. Ltd., New Delhi, 2007.

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

M. H. Gulzar, On the number of zeros of a polynomial in a prescribed region, Research Journal of Pure Algebra-2(2), Feb. 2012, 3546.

M. Marden, Geometry of Polynomials, IInd. Ed. Math. Surveys, No.3, Amer. Math. Soc. Providence, R.I. 1966.

Q. G. Mohammad, On the Zeros of Polynomials, Amer. Math. Monthly, 72 (1965), 631-633.

E. C Titchmarsh, The Theory of Functions, 2nd edition, Oxford University Press, London, 1939.

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