PROPERTIES OF MEROMORPHICALLY STARLIKE AND CONVEX FUNCTIONS

Research Paper

Mathematics

E-ISSN No : 2454-9916 | Volume : 2 | Issue : 11 | Nov 2016

PROPERTIES OF MEROMORPHICALLY STARLIKE AND CONVEX FUNCTIONS Dr. M. Aparna Sr.Asst.Prof in Mathematics, G.Narayanamma Institute of Technology & Science, Shaikpet, Hyderabad.

ABSTRACT In this paper i studied some properties of meromorphically starlike and meromorphically convex functions. We have proved f(z)  r *

 ( p, q)

where

f(z  r satisfying an inequality given by 

 n  k  2 ( p, q)  n  k  a n 0

n

r n1  21   ( p, q)

In this paper I also proved that f(z)  Cr [  p,q] where f(z)  r and satisfying the inequality given by 

 nn   ( p, q) a n 1

n

r n1  1   ( p, q) .

KEY WORDS: Univalent Function, Starlike Function, convex Function, Analytic Function.

1. INTRODUCTION: Let r denote the class of functions f(z) of the form

1  f ( z )    an z n z n0 which are analytic in the disk function

Ozaki has shown that the necessary and sufficient condition that f(z)r with an  0 (n=1,2,3……) is meromorphic and univalent in Dr is that there should exist the relation



 nan r n 1  1

Dr  z  c : 0  z  r  1 .

n 1 A

f ( z )   r is said to be strlike of order  ( p, q ) if it satisfies the

Cofficient Inequalities for functions Theoram 1 : If f(z  r satisfies for some  ( p, q ) (0 

inequality

 zf ( z )  Re     ( p, q )  f ( z )( p  q) 

( z  Dr )

For some  ( p, q ) (0  ( p, q ) <1. We say that fz is in the class r

 ( p, q )  for such functions. A function fz)  r is said to be convex of order  ( p, q ) if it satisfies the inequality   zf ( z )     ( p, q) Re  1  ( z  Dr )  f ( z )( p  q )     ( p , q )  ( p , q ) < 1). We say that f(z) is in the class Cr[ For some (0  ( p, q ) ] if it is convex of order  ( p, q ) in Dr. We note that f(z)  Cr

 ( p, q) if

between its coefficients.

and only if –z f(z)  

*

r

 ( p, q)

.

There are many papers

discussing various properties of classes consisting of univalent, starlike, convex, multivalent, and meromorphic functions in the book by Srivastava and Owa.

f(z)  r *

 ( p, q)

 ( p, q ) < 1 ) and

k[  ( p, q ) < k  1]

then

.

Proof: For f(z)  r we know that

zf ( z )  kf ( z )( p  q)  zf ( z )  2 ( p, q)  k  f ( z )( p  q) 1  1   (k  1)   (n  k )a n z n  2 ( p, q)  k  1   2 ( p, q)  n  k a n z n z n0 z n0

By applying the condition of the theorem, we have 



 (k  1)   (n  k ) an r n1  (k  1  2 ( p, q))   2 ( p, q)  n  k an r n1 n 0

 2 ( p, q)  1 

0

n 0



 (n  k  2 ( p, q)  n  k ) an r n 1

n0

Which shows that

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72

Research Paper

E-ISSN No : 2454-9916 | Volume : 2 | Issue : 11 | Nov 2016

 n  k  2 ( p, q)  n  k a 

n 0



n 1  21   ( p, q) n r

 1   ( p, q )   ( p, q ) a 1  0 f ( z)   a 0  z ( p  q) n   ( p , q )  

It follows that

zf ( z )  kf ( z )( p  q ) 1 zf ( z )  2 ( p, q)  k  f ( z )( p  q )

So that

 zf ( z )     ( p, q) Re   f ( z )( p  q) 

belongs to the class *r   for some real ‘’ with ½ 

( z  Dr )

By Putting k=0, p=1 and q=0 in the above theoram. We have Corollary –1 : If f(z) r satisfies



,then fz r(  ).

Corollary –2 : Let the function f(z)r be given in the introduction with  n 1 2

i

f(z)  *r [  p,q] if and only if

for some  p,q,

Remark 1: If f(z)  r with a0=0, then corollalry-2 holds true for some p,q [0   p,q < 1].

 ( n   ) a n r n 1  1  

 nn   ( p, q) an r n 1  1   ( p, q)

n 1

 1 / 2    1

Proof: In view of the above theorem, we see that if the coefficient inequality

 1 / 2    1 , then fz*r(  ).

Conversely, let f(z) be in the class *r(  ), then

for some  p,q [0   p,q < 1], then f(z) belongs to the class Cr[  p,q].

Proof : Note that f(z)  Cr [  p,q] if and only if

holds true for some

    1   na z n 1   n0 n  zf ( z )    Re    f ( z)   n 1   1   an z  n0  

For all z  Dr. n+1

Letting z= r e we have that an z which implies that

1

 p,q



n0

1/2 i,



Theoram 2 : If f(z)  r satisfies



 Re  

1/ 2   ( p, q)  1 .

Remark 2: If f(z)  r with a0 = 0, then corollary-3 holds true for 0  < 1.

,

then f(z)   *r (  ) if and only if

For some

Corollary 3 : Let the function f(z)  r be given by introduction with an  0, then

n0 *

Putting p=1 & q=0, we have

an  an e

< 1.

 n   ( p, q)an r n 1  1   ( p, q)

n0

 1 / 2    1

  i n e z    p,q  1 1  a0



 ( n   ) a n r n 1  1  

For some

Example : The function f(z) given by

= | an  r

n+1



   n 1 n 1   n a r   1  a r .  n  n   n0  n0 

which is equivalent to



 ( n   ) a n r n 1  1  

n0

International Educational Scientific Research Journal [IESRJ]

–z

f  (z)  *r [  p,q] , and

 zf ( z ) 

 1   nan z n , z ( p  q) n 1

With the help of theoram-1, we complete the proof of the theorem.

Example 2 : The function f(z) given by

f ( z) 

 1   ( p, q)  i n 1  a0   e z z ( p  q)  n(n   ( p, q)) 

belongs to the class Cr [  p,q] for some real  with 0 

 p,q < 1.

Corollary 4 : Let the function f(z)  r be given by introduction with

an  an e

 n 1 2

i ,

then f(z)  Cr  p,q if and only if the inequality (remark -1 holds true for some  (p,q , [0   p,q<1].

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Research Paper

E-ISSN No : 2454-9916 | Volume : 2 | Issue : 11 | Nov 2016

Corollary 5 : Let the function f(z)r be given by introduction with an  0, then fzCr[  p,q] if and only if



then fz  *r [  p,q ] for 0  r  r0 with







 nn   ( p, q)an r n 1  1   ( p, q)

n 1n

n 1

for some

= 

Theorem:3 A function f(z)r belongs to the class *r[  p,q] for 0  r  r0 where r0 is the smallest positive root of the equation.

By Putting

 ( p, q ) a0 r 3    1   ( p, q) r 2   ( p, q) a0 r  1   ( p, q)  0

 n an

2



  ( p, q )

n 1

1

 n an

 

2 and

n 1





n 1

n 1

 n   ( p, q) an r n 1   ( p, q) a0 r   an r n 1 

 n an

2

n 1





n 1

n 1

1

 nr 2n  2   ( p, q)  n an

2



  ( p, q ) a 0 r 

 1   ( p, q )

r4

1 r

2



Where



*r

Where

 n an





1

 n an

   1   ( p, q ) 

 n 3 an



2

  ( p, q )



 n an

2

n 1

i z n 1  1  e n z n 1n 2 n

( n is real)

see that fz  Cr[  p,q] for 0  r  r0 with  Taking



we get

 

 1         6 3 10 

= 0,



 1.282550

6

r0 

6 6 

 0.661896.

2 REFERENCES:

n 1

Example 3: If we consider the function f(z) given by

 1 1 i n n  e z z ( p  q) n 1n n

 0.661896.

Example : Let us consider the function fz

and

2

6 

n 1

   for 0 

 1 

6

r1  1 



n 1

f ( z) 

 1.282550

6

f ( z) 



corollary : 6 A function fz  r with a0=0 belongs to the class r < r0.

0

 =0 we have

r0 

n 1

By corollary - 1 we get f(z)  *r [  p,q] for 0  r r0 Letting a0=0, p=1, q=0 in the above theorem.

r  1

4





r2

n 1n

 nr 2n  2

   2 2  1  n a   ( p , q ) a n n  n 2  n 1 1  r 2  n 1 

  ( p, q ) a 0 r 

1

Theorem 4 : A function fz  r belongs to the class Cr[  p,q] for 0  r  r1 where

Proof: Using the Cauchy inequality, we have

  ( p, q ) a 0 r 



  ( p, q )

2

 1  ( p, q)     3 10   6

And





  (2)   ( p, q)  (1)

 p,q [0   p,q < 1].

Starlikeness and convexity of functions Consider the radius problems for starlikeness and convexity of functions f(z) belonging to the class r.



1

( n is real)

International Educational Scientific Research Journal [IESRJ]

1.

Duren P.L, Univalent Functions, Springer-Verlag Berlin Heidelberg and Tokyo, 1983.

2.

I.S.Jack, “Functions starlike and convex of order”, J.London Math Soc. 3,(1971),469 - 474.

3.

V.Ravichandran, C.Selvaraj and R.Rajalaksmi, ”Sufficient conditions for starlike functions of order ”, J.pure and Appl.Math, 1443-5756.

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5.

E-ISSN No : 2454-9916 | Volume : 2 | Issue : 11 | Nov 2016

Herb Silverman, “Convex and starlike criteria”, Internat.J.Math. & Math.

Sci,Vol. 22, No.1(1999) ,75-79.

O.P.Juneja and T.R.Reddy, “Meromorphic starlike and univalent functions with positive coefficients, Ann. Univ. Mariae Curie-Sklodowsku, 39(1985), 65-76.

6.

H.Silverman, “Partial sums of starlike and convex functions”, J.Math. Anal, 209(1997), 221-227.

7.

M.S.Robertson, “On the theory of univalent functions”, coefficients”, Proc.

Amer. Math. Soc. 51(1975), 109-116.

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