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 n1 21 ( p, q)
In this paper I also proved that f(z) Cr [ p,q] where f(z) r and satisfying the inequality given by
nn ( p, q) a n 1
n
r n1 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 n0 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 fz is in the class r
( p, q ) for such functions. A function fz) 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 n0 z n0
By applying the condition of the theorem, we have
(k 1) (n k ) an r n1 (k 1 2 ( p, q)) 2 ( p, q) n k an r n1 n 0
2 ( p, q) 1
0
n 0
(n k 2 ( p, q) n k ) an r n 1
n0
Which shows that
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International Educational Scientific Research Journal [IESRJ]
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 21 ( 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 fz 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
nn ( 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 fz*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 n0 n zf ( z ) Re f ( z) n 1 1 an z n0
For all z Dr. n+1
Letting z= r e we have that an z which implies that
1
p,q
n0
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
n0 *
Putting p=1 & q=0, we have
an an e
< 1.
n ( p, q)an r n 1 1 ( p, q)
n0
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 n0 n0
which is equivalent to
( n ) a n r n 1 1
n0
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 fzCr[ p,q] if and only if
then fz *r [ p,q ] for 0 r r0 with
nn ( 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 fz 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 fz
and
2
6
n 1
for 0
1
6
r1 1
n 1
f ( z)
1.282550
6
f ( z)
corollary : 6 A function fz 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 fz 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.
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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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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.
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H.Silverman, “Partial sums of starlike and convex functions”, J.Math. Anal, 209(1997), 221-227.
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