A NEW SUB CLASS OF MEROMORPHICALLY CONVEX FUNCTIONS WITH NEGATIVE AND
Department of Mathematics, S.J.N.P.G. College, Lucknow
ARTICLE INFO ABSTRACT
In this paper, we introduce and study a subclass
We obtain coefficients inequalities, extreme points, distortion and growth bounds, radii of meromorphically starlikeness and meromorphically convexity for this class. Further it is shown that this class is closed under convex linear combination.
Copyright©2017, Dr. Jitendra Awasthi.This is an open access article distributed under the Creative Commons Att use, distribution, and reproduction in any medium, provided the original work is properly cited.
INTRODUCTION
Let
denote the class of functions of the form
1
1
)
(
n n n
z
a
z
z
f
………(Which are analytic in the punctured open unit disk
}
0
{
\
}
1
0
,
:
{
*
D
z
C
z
z
D
with a simple pole at the origin and residue 1 there.
Let
kdenote the subclass of
consisting of functions f(z) which are convex with respect to the origin, i.e. satisfying the condition:)
.(
0
)
(
)
(
"
1
z
D
*z
f
z
zf
R
……….(Let
k(
)
denote the subclass of
consisting of functions f(z) which are convex of order*Corresponding author: Dr. Jitendra Awasthi,
Department of Mathematics, S.J.N.P.G. College, Lucknow
ISSN: 0975-833X
Article History:
Received 24th February, 2017
Received in revised form 21st March, 2017
Accepted 04th April, 2017
Published online 31st May,2017
Citation: Dr. Jitendra Awasthi. 2017. “A new sub class of meromorphically convex functions with negative and fixed second coefficients Journal of Current Research, 9, (05), 51141-51148.
Key words:
Meromorphic, Univalent, Convex, Analytic, Linear combinations.
RESEARCH ARTICLE
A NEW SUB CLASS OF MEROMORPHICALLY CONVEX FUNCTIONS WITH NEGATIVE AND
FIXED SECOND COEFFICIENTS
*Dr. Jitendra Awasthi
Department of Mathematics, S.J.N.P.G. College, Lucknow-226001
ABSTRACT
In this paper, we introduce and study a subclass (,,A,B,)
k
of meromorphic univalent functions.
We obtain coefficients inequalities, extreme points, distortion and growth bounds, radii of meromorphically starlikeness and meromorphically convexity for this class. Further it is shown that this class is closed under convex linear combination.
is an open access article distributed under the Creative Commons Attribution License, which use, distribution, and reproduction in any medium, provided the original work is properly cited.
denote the class of functions of the form
………(1.1)
Which are analytic in the punctured open unit disk
a simple pole at the origin and residue 1 there.
consisting of functions f(z) which are convex with respect to the origin, i.e. satisfying the
……….(1.2)
consisting of functions f(z) which are convex of order
,i.e. satisfying the conditionDepartment of Mathematics, S.J.N.P.G. College, Lucknow-226001
International Journal of Current Research
Vol. 9, Issue, 05, pp.51141-51148, May, 2017
A new sub class of meromorphically convex functions with negative and fixed second coefficients
A NEW SUB CLASS OF MEROMORPHICALLY CONVEX FUNCTIONS WITH NEGATIVE AND
226001
of meromorphic univalent functions. We obtain coefficients inequalities, extreme points, distortion and growth bounds, radii of meromorphically starlikeness and meromorphically convexity for this class. Further it is shown that
ribution License, which permits unrestricted
consisting of functions f(z) which are convex with respect to the origin, i.e. satisfying the
,i.e. satisfying the condition
)
1
0
;
.(
)
(
)
(
"
1
*
z
D
z
f
z
zf
R
……….(1.3)and similar other classes of meromorphically univalent functions have been defined and studied by Altintas et al.[1], Aouf[2,3], Ganigi and Uralegaddi[6],Uralegaddi[9],Uralegaddi and Ganigi[10] and others.
Let
k(
,
A
,
B
)
denote the class of functions f(z) in
which satisfy the condition
)
4
.
1
(
...
...
...
1
)
1
)(
(
)
(
'
)
(
"
1
2
)
(
'
)
(
"
B
A
B
z
f
z
zf
B
z
f
z
zf
)
1
0
;
1
1
;
1
0
*,
(
z
D
A
B
B
We note that
k(
,
1
,
1
)
K(
)
.Let
denote the subclass of
consisting of functions of the form..(1.5)
...
...
...
1
)
(
1
n
n n
z
a
z
z
f
Now in the following definition, we define a subclass
k(
,
,
A
,
B
)
for functions in the class
.Definition 1.1: A function f(z) defined by (1.5) is in the class
k(
,
,
A
,
B
)
if it satisfies the condition
)
1
)(
(
)
(
'
)
(
"
1
2
)
(
'
)
(
"
B
A
B
z
f
z
zf
B
z
f
z
zf
……..………(1.6)
)
1
0
;
1
1
;
1
0
;
1
0
*,
(
z
D
A
B
B
For the class
k(
,
,
A
,
B
)
,the following characterization was given bySrivastava et al. [8].Theorem 1.1: Let the function f(z) defined by (1.5) be analytic in D* .Then
f
(
z
)
K(
,
,
A
,
B
)
if and only if
1
1
1
n
n
B
A
a
A
A
B
Bn
n
n
. ………(1.7)For a function f(z) defined by (1.5) and in the class
k(
,
,
A
,
B
)
,Theorem 1.1 yields]
)
(
[
2
)
1
(
)
(
1A
A
B
B
A
B
a
………...(1.8)
Hence we may take
)
1
0
(
,
]
)
(
[
2
)
1
(
)
(
1
A
A
B
B
A
B
class of functions and use the similar techniques to prove our results.
Let class
k(
,
,
A
,
B
,
)
be the subclass of
k(
,
,
A
,
B
)
consisting of functions of the form
2
]}
)
(
[
)
1
{(
}
)
(
[
2
)
1
(
)
(
1
)
(
n
n n
z
a
A
A
B
Bn
n
n
z
A
A
B
B
A
B
z
z
f
...………(1.10)
where
0
1
.
In this paper, we obtain coefficient inequalities, extreme points, distortion and growth bounds, radii of meromorphically starlikeness and meromorphically convexity for the class
k(
,
,
A
,
B
,
)
by fixing the second coefficient. Further it is shownthat the class
k(
,
,
A
,
B
,
)
is closed under convex linear combinations.2.Cofficients Inequalities
Theorem 2.1:Let the function f(z) is defined by (1.10). then
f
(
z
)
k(
,
,
A
,
B
,
)
iff
1
1
(
1
)
2
n
n
B
A
a
A
A
B
Bn
n
n
.………(2.1)
The result is sharp.
Proof: By putting
)
1
0
(
,
]
)
(
[
2
)
1
(
)
(
1
A
A
B
B
A
B
a
.………(2.2)in (1.7),the result is easily derived. The result is sharp for the function
.
2
,
]}
)
(
[
)
1
{(
)
1
)(
1
(
)
(
}
)
(
[
2
)
1
(
)
(
1
)
(
z
n
A
A
B
Bn
n
n
A
B
z
A
A
B
B
A
B
z
z
f
n
…...……….(2.3)
Corollary 2.2: If the function f(z) defined by (1.10) is in the class
k(
,
,
A
,
B
,
)
then
.
2
,
]}
)
(
[
)
1
{(
)
1
)(
1
(
)
(
n
A
A
B
Bn
n
n
A
B
a
n
………...(2.4)
The result is sharp for the function f(z) given by (2.3).
3. DISTORTION THEOREMS
Theorem 3.1: If the function f(z) defined by (1.10) is in the class
k(
,
,
A
,
B
,
).
Then for
0
z
r
1
, we have2 2
]}
)
(
2
[
3
{
2
)
1
)(
1
(
)
(
]
)
(
[
2
)
1
(
)
(
1
f(z)
]}
)
(
2
[
3
{
2
)
1
)(
1
(
)
(
]
)
(
[
2
)
1
(
)
(
1
r
A
A
B
B
A
B
r
A
A
B
B
A
B
r
r
A
A
B
B
A
B
r
A
A
B
B
A
B
r
………(3.1)
2
]}
)
(
2
[
3
{
2
)
1
)(
1
(
)
(
]
)
(
[
2
)
1
(
)
(
1
f(z)
z
A
A
B
B
A
B
z
A
A
B
B
A
B
z
.……….(3.2) andr
A
A
B
B
A
B
A
A
B
B
A
B
r
r
A
A
B
B
A
B
A
A
B
B
A
B
r
]}
)
(
2
[
3
{
)
1
)(
1
(
)
(
]
)
(
[
2
)
1
(
)
(
1
(z)
f
]}
)
(
2
[
3
{
)
1
)(
1
(
)
(
]
)
(
[
2
)
1
(
)
(
1
2 2
………(3.3)Proof: Since
f
(
z
)
K(
,
,
A
,
B
,
)
,then from theorem (2.1).
2
,
]}
)
(
[
)
1
{(
)
1
)(
1
(
)
(
n
A
A
B
Bn
n
n
A
B
a
n
…….……….(3.4)Then for
0
z
r
1
2]
)
(
[
2
)
1
(
1
)
(
n n nz
a
z
A
A
B
B
A
B
z
z
f
2 2]
)
(
[
2
)
1
(
1
n n
a
r
r
A
A
B
B
A
B
r
..(3.5)
...
...
...
]}
)
(
2
[
3
{
2
)
1
)(
1
(
)
(
]
)
(
[
2
)
1
(
1
2
r
A
A
B
B
A
B
r
A
A
B
B
A
B
r
and
2]
)
(
[
2
)
1
(
1
)
(
n n nz
a
z
A
A
B
B
A
B
z
z
f
2 2]
)
(
[
2
)
1
(
1
n n
a
r
r
A
A
B
B
A
B
r
.(3.6)
...
...
...
]}
)
(
2
[
3
{
2
)
1
)(
1
(
)
(
]
)
(
[
2
)
1
(
1
2
r
A
A
B
B
A
B
r
A
A
B
B
A
B
r
Thus (3.5) and (3.6) together yield (3.1).
Further more,from theorem 2.1,it follows that
7)
...(3.
...
...
...
...
.
2
,
]}
)
(
2
[
3
{
)
1
)(
1
(
)
(
n
A
A
B
B
A
B
na
n
22
2
[
(
)
]
)
1
(
1
)
(
n nz
na
A
A
B
B
A
B
z
z
f
...(3.8)
...
...
]}
)
(
2
[
3
{
)
1
)(
1
(
)
(
]
)
(
[
2
)
1
(
1
2
r
A
A
B
B
A
B
A
A
B
B
A
B
r
2 12
2
[
(
)
]
)
1
(
1
)
(
n n n
z
na
A
A
B
B
A
B
z
z
f
and
9)
...(3.
...
...
]}
)
(
2
[
3
{
)
1
)(
1
(
)
(
]
)
(
[
2
)
1
(
1
2
r
A
A
B
B
A
B
A
A
B
B
A
B
r
Thus (3.8) and (3.9) together yield (3.3).
4. CLOSURE THEREMS
Theorem 4.1: If
....(4.1)
...
...
...
]
)
(
[
2
)
1
(
)
(
1
)
(
1
z
A
A
B
B
A
B
z
z
f
(4.2)
...
.
2
,
]}
)
(
[
)
1
{(
)
1
)(
1
(
)
(
]
)
(
[
2
)
1
(
)
(
1
)
(
2
n
z
A
A
B
nB
n
n
A
B
z
A
A
B
B
A
B
z
z
f
n n n
Then
f
(
z
)
K(
,
,
A
,
B
,
)
if and only if it can be expressed in the form3)
...(4.
...
...
...
...
)
(
)
(
1
n n nf
z
z
f
1 nn
0
and
n1
.
where
Proof: From (4.2) and (4.3), we have
1 2 11
(
)
(
)
)
(
)
(
n n n n nn
f
z
f
z
f
z
z
f
.
]}
)
(
[
)
1
{(
)
1
)(
1
(
)
(
]
)
(
[
2
)
1
(
)
(
1
2
n n nz
A
A
B
nB
n
n
A
B
z
A
A
B
B
A
B
z
)
,
,
,
,
(
)
(
that
follows
it
(2.1)
theorem
from
f
z
A
B
So
K).
,
,
,
,
(
f(z)
let
K
A
B
Conversely
Since
.
2
,
]}
)
(
[
)
1
{(
)
1
)(
1
(
)
(
n
A
A
B
Bn
n
n
A
B
a
n
Setting
and
2 1
1
)
1
)(
1
(
)
(
]}
)
(
[
)
1
{(
n n
n
n
a
A
B
A
A
B
Bn
n
n
It follows that
.
)
(
)
(
1
n n n
f
z
z
f
This complete the proof.
Theorem 4.2: The class
f
(
z
)
K(
,
,
A
,
B
,
)
is closed under convex linear combinations.Proof: Suppose the function f(z) be given by (1.10) and let the function g(z) be given by
2
.
2
,
]
)
(
[
2
)
1
(
1
)
(
n
n n
z
n
b
z
A
A
B
B
A
B
z
z
g
Assuming that f(z) and g(z) are in the class
K(
,
,
A
,
B
,
)
, it is enough to prove that the function h(z) defined by.
1
0
),
(
)
1
(
)
(
)
(
z
f
z
g
z
h
is also in the class
K(
,
,
A
.,
B
,
)
.Since
2
.
)
1
(
]
)
(
[
2
)
1
(
1
)
(
n
n n n
b
z
a
z
A
A
B
B
A
B
z
z
h
We observe that
2
)
1
)(
1
(
)
(
)
1
(
]}
)
(
[
)
1
{(
n
n
n
b
B
A
a
A
A
B
Bn
n
n
with the aid of theorem 2.1.Thus
h
(
z
)
k(
,
,
A
,
B
,
)
.5.RADIUS OF STARLIKENESS AND CONVEXITY
Theorem 5.1: Let the function f(z) defined by (1.10) be in the class
K(
,
,
A
,
B
,
)
,then we have(i) f is meromorphically starlike of order
(0
1)
in the diskz
r
1(
,
,
A
,
B
,
,
)
where)
,
,
,
,
,
(
1
A
B
..(5.1)
...
.
2
,
1
]}
)
(
[
)
1
{(
A]
A)
-(B
[B
2
r
n
n
Bn
B
A
A
r
n
(ii)f is meromorphically convex of order
(0
1)
in the diskz
r
2(
,
,
A
,
B
,
,
)
wherer
2(
,
,
A
,
B
,
,
)
is the largest value for which...(5.2)
...
.
2
,
1
]}
)
(
[
)
1
{(
)
1
)(
1
(
)
)(
2
(
A]
A)
-(B
[B
2
)
-(1
A)
-)(B
-(3
2 1
n
r
A
A
B
Bn
n
A
B
n
r
n
Proof: It is enough to show that
.
z
,
1
1
)
(
)
(
r
1z
f
z
f
z
Thus we have
.3)
...(5
...
...
)
(
[
2
)
1
(
)
(
1
)
1
(
)
(
[
2
)
1
(
)
(
2
1
)
(
)
(
2 2
n n n n n nz
a
z
A
A
B
B
A
B
z
z
a
n
z
A
A
B
B
A
B
z
f
z
f
z
Hence (5.3) holds ture if
,
)
(
[
2
)
1
(
)
(
1
)
1
(
)
1
(
)
(
[
2
)
1
(
)
(
2
2 1 2 2 1 2
n n n n n nr
a
r
A
A
B
B
A
B
r
a
n
r
A
A
B
B
A
B
or,
2 1 2).
1
(
)
2
(
]
)
(
[
2
)
1
(
)
)(
3
(
n n nr
a
n
r
A
A
B
B
A
B
and it follows that from (2.1),we may take
.
2
,
]}
)
(
[
)
1
{(
)
1
)(
1
(
)
(
n
A
A
B
Bn
n
n
A
B
a
n
For each fixed r, we choose the positive integer
n
1
n
1(
r
1)
for which]}
)
(
[
)
1
{(
)
2
(
1
nr
A
A
B
Bn
n
n
n
is maximal. Then it follows that
2 1 1 1 1 1 1 1]}
)
(
[
)
1
{(
)
1
)(
1
(
)
)(
2
(
)
2
(
n n n nr
A
A
B
Bn
n
n
A
B
n
r
a
n
Then f(z) is starlike of order δ in
0
z
r
1(
,
,
A
,
B
,
,
)
provided that.
1
]}
)
(
[
)
1
{(
)
1
)(
1
(
)
)(
2
(
A]
A)
-(B
[B
2
)
-(1
A)
-)(B
-(3
1 1 1 1 12 1
We find the value
r
1
r
1(
,
,
A
,
B
,
,
)
and the corresponding integern
1
n
1(
r
1)
so that.
1
]}
)
(
[
)
1
{(
)
1
)(
1
(
)
)(
2
(
A]
A)
-(B
[B
2
)
-(1
A)
-)(B
-(3
11 1
1 1
2 1
n
r
A
A
B
Bn
n
n
A
B
n
r
It is the value for which the functionf(z) is starlike in
0
z
r
1.(ii) In a similar manner, we can prove our result providing the radius of meromorphically convexity of order
(0
1)
for the function
K(
,
,
A
,
B
,
)
.REFERENCES
[1]. Altintas, O., Irmak, H. and Srivastava, H.M. 1995. A family of meromorphically univalent functions with positive coefficients, Panamer. Math. J., 5, 75-81.
[2]. Aouf, M. K. 1989. A certain subclass of meromorphically starlike functions with positive coefficients, Rend. Mat. Appl. (7) 9, 225-235.
[3]. Aouf, M. K. 1991. On a certain class of meromorphic univalent functions with positive coefficients, Rend. Mat. Appl. (7) 11, 209-219.
[4]. Aouf, M. K. 1997. and H. E. Darwish, Certain meromorphically starlike functions with positive and fixed second coefficients, Turkish j. Math., 21, 311-316.
[3]. Aouf, M. K. and S. B. Joshi. 1998. On certain subclasses of meromorphically starlike functions with positive coefficients, Soochow J. Math. 24.79-90.
[4]. Ganigi, M. R. and B. A.Uralegaddi, 1989. New criteria for meromorphic univalent functions, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 33(81), 9-13.
[5]. Sivasubramanian, S., N. Magesh and MaslinaDarus, 2013. A new subclass of meromorphic functions with positive and fixed second coefficients,TamkangJournal of Mathematics, vol. 44, Number 3, 271-278.
[6]. Srivastava, H. M., Hossan, H. M. and Aouf, M. K. 1988. A certain subclass of meromorphic convex functions with negative coefficients, Math. J., Ibaraki Uni., vol. 30, 33-51.
[7]. Uralegaddi, B. A. 1989. Meromorphically starlike functions with positive and fixed second coefficients, Kyungpook Math. J., 29, 64-68.
[8]. Uralegaddi, B. A. and M. D. Ganigi, 1987. A certain class of meromorphically starlike functions with positive coefficients,Pure Appl. Math. Sci., 26, 75-81.
