DOI : 10.17577/IJERTV9IS010273
- Open Access
- Authors : Dr. Deepaly Nigam
- Paper ID : IJERTV9IS010273
- Volume & Issue : Volume 09, Issue 01 (January 2020)
- Published (First Online): 08-02-2020
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License:
This work is licensed under a Creative Commons Attribution 4.0 International License
p-Valently Meromorphic Functions with Fixed First Coefficient
Dr. Deepaly Nigam
Dr. Akhilesh Das Gupta Institute of Technology & Management
New Delhi-110053, (INDIA).
Abstract : In this paper, we have considered the class of the functions of the form
1 p i
np
a 0 and
p p. Coefficient inequalities, closure
f (z)
e
zp
np1
anp z
, (p N)
np
2
theorems and radius of convexity for this class are determined.
-
- INTRODUCTION Let Mp denote the class of functions of the form
(1.1)
f (z)
1
zp np
anpz
np ,
p N
which are analytic in the punctured disk U* = {z¢ : 0 < |z| < 1}. A function f(z)Mp is said to be starlike of order if it
satisfies the inequality
zf ‘(z)
(1.2)
Re
f (z)
,
(z U U * U{0})
for (0 < p). We say that f(z) is in the class S*(p, ) for such functions.
We have obtained [5] the following result :
Result 1 : Let the function f(z)Mp analytic in Dr = {z¢ : 0 < |z| < r 1} be given by (1.1) with anp 0, then f(z)S*(p,r; ) if and only if
(1.3)
np
(n p)anprn p
for some p p .
2
2
Letting r 1 in above result 1, we get the following result 2.
Result 2 : If f(z)Mp defined on U* = {z¢ : 0 < |z| < 1} satisfies
(1.4)
np
(n p)anp p
anp 0
for some p p , then f(z)S*(p, )
2
2
In view of result 2, a function of the form (1.1) belonging to the class S*(p, ) must satisfy the coefficient inequality
p
(1.5)
hence, we write
anp n p
(n p)
a p ei
0
by fixing the first coefficient a0, we introduce a new subclass S*(p,; ) of S*(p, ) consisting of functions of the form
1 p i
np
(1.6)
and
p p .
2
f (z)
zp
e
np1
anpz
, p N
In this paper, we obtain coefficient inequality, closure theorems and radius of convexity for the class S*(p,;). Techniques used are similar to those of Silverman and Silvia [2], Uralegaddi [3], Owa and Srivastava [1] and M.K. Aouf and
H.E. Darwish [4].
- COEFFICIENT INEQUALITY :
Lemma 2.1 : Let function f(z) defined by (1.6) is in the class S*(p,;) if and only if
(2.1)
np1
(n p)anp (p )(1 ei ),
(anp 0)
for
p p . The result is sharp.
2
Proof : Putting a
p ei
in (1.4), we get
0
(p )ei
np1
(n p)anp (p )
np1
(n p)anp (p )(1 ei )
Further, by taking the function f(z) of the form
1 p i
(p )(1 ei )
np
(2.2)
f (z)
zp
e
(n p) z
for n p + 1, we can see that results (2.1) is sharp.
Corollary 2.2 : Let the function defined by (1.6) be in the class S*(p,;), then
(2.3) anp (p ) (1 ei) (n + p)1 (n p + 1) The result (2.3) is sharp for the function f(z) is given by (2.2).
- CLOSURE THEOREMS :
Theorem 3.1 : Let the function
1
p i
np
(3.1)
f j (z) z e
np1
anp, jz z
be in the class S*(p,;) for j = 1, 2, m, then the function F(z) defined by
m
(3.2)
F(z)
j1
d jFj (z)
(d j 0)
is also in the same class S*(p,;) where
m
(3.3)
j1
d j 1
Proof : Combining (3.1) and (3.2), we get
m
F(z)
d 1 p ei a
znp
(3.4)
j1
j zp
np1
n-p, j
1 p i
m
np
F(z)
p e
- d ja np, j z
(using 3.3)
z
np1 j1
Since fj(z) S*(p,;) for every j = 1, 2, , m, therefore, theorem 2.1 yields
np1
(n p)anp, j (p )(1 ei )
for j = 1, 2, , m. Thus, we obtain
m
m
(n p)
d ja np, j
d j
(n p)a np, j
np1
j1
j1
np1
which implies F(z) S*(p,;).
(p )(1 ei )
Theorem 3.2 : Let the function fj(z) be defined by (3.1). If fj(z) S*(p,;) for every j = 1, 2, , m, then the function
1 p i
np
(3.5)
g(z) e
zp
np1
bnpz
is in the same class S*(p,;), where
1 m
(3.6)
bnp
m
m
j1
anp, j
Proof : Since fj(z) S*(p,;) it follows from theorem 2.1 that
hence
np1
(n p)anp, j (p )(1 ei )
m j1
m j1
1 m
np1
(n p)bnp, j
np1
(n p)
a np, j
m
m
1
(n p)a np, j
m j1
1
1
m
m j1
np1
(p )(1 ei )
(p )(1 ei )
which (in view of theorem 2.1) implies g(z) S*(p,;). This completes the proof of theorem.
Theorem 3.3 : The class S*(p,;) is closed under convex linear combination.
Proof : Let the function fj(j = 1, 2) defined by (3.1) be in the class S*(p,;). It is sufficient to prove that the function H(z)
defined by
H(z) = f1(z) + (1 ) f2(z) (0 1)
is also in the class S*(p,;).
1 p i
np
H(z) e
zp
np1
{anp,1 (1 )an-p,2}z
p
p
Since f1(z) and f2(z) belong to the class S* ( ; ).
Therefore,
and
np1
np1
| anp,1 | (n p) (p )(1 ei )
| anp,2 | (n p) (p )(1 ei )
Now, we observe that
np1
| anp,1 (1 )anp,2 | (n p) (p )(1 ei )
p
p
hence, in view of theorem 2.1, we get H(z) S* ( ; )
Theorem 3.4 : Let
(3.7)
f (z)
1 p i
and
p zp
e
1 p i
(p )(1 ei )
np
(3.8)
fn (z)
zp
e
(n p) z
(n p 1)
Then f(z) is in the class S*(p,;) if and only if it can be expressed in the form
(3.9)
f (z)
np
nfn (z)
wheren 0
and (3.10)
np
n 1.
Proof : We supose that f(z) can be expressed in the form (3.9) then it follows from (3.8), (3.9) and (3.10) that
1 p i
(p )(1 ei )
np
Note that
f (z) e
zp
np1
(n p)
n z
np1
(p )(1 ei )
(n p)
(n p)
n (p )(1 ei )
n np1
= 1 p 1.
hence f(z) S*(p,;)
for the converse assume that the function f(z) of the form (1.6) belongs to the class S*(p,;). Since f(z) satisfies (2.3) for n p
+ 1, we may set
and
n
(n p) (p )(1 ei )
anp
, n p 1
Then
p 1
n np1
1 p i
(p )(1 ei )
np
f (z) e
zp
np1
(n p)
n z
1 p
i
1 p
i
e
zp
np1
n fn (z)
zp
e
1 p ei
(z) 1
zp
1 n
np1
np1
n fn
zp
1 p ei
f (z)
1
np1
n zp
p n n np1
1
p
i
p zp
e
np1
nfn (z)
pfp (z)
np1
nfn (z)
np
nfn (z)
This complete the proof of the theorem.
- RADIUS OF CONVEXITY :
Theorem 4.1 : Let the function f(z) defined by (1.6) be in the class S*(p,;), then f(z) is pvalent meromorphically convex in
0 < |z| < r = r(,; ) where
p2 (n p)
1/ n
(4.1)
r(p, , ) inf
np (n 2
- p2
)(p )(1 ei
)
)
Proof : If is suffices to show that
(4.1)
Consider
zf “(z) 1 p p f ‘(z)
zf “(z) (p 1)f ‘(z)
np1
(n p)na np znp1
f ‘(z)
p zp1
np1
(n p)a
np
znp1
np1
n(n p)anp zn
Thus, the result follows if
- p
np1
(n p)anp zn
np1
or
n(n p)a
np
| z |n pp
np1
(n p)a
np
r n
(4.3)
np1
(n 2 p2 )anp | z |n p2
But by theorem 2.1, we have
(4.4)
np1
(n p)anp (p )(1 ei )
Hence, (4.3) holds if and only if
n
n
(n 2 p2 )
| z |
p2
(n p) (p )(1 ei )
or
p2 (n p)
1/ n
2 2 i
2 2 i
| z |
(n p )(p )(1 e )
(n p, p N)
This proves the theorem.
m
m
- p2
- THE CLASS S*
- INTRODUCTION Let Mp denote the class of functions of the form
(p, ; ) :
S
S
Instead of fixing just the first coefficient, we can fix finitely many coefficients. Let
* (p,; ) denote the class of
m
m
functions of the form
1 m (p )(1 eik )
kp
np
(5.1)
f (z)
zp
(k p) z
- anpz
, (m p)
where 0 eik
1 and
m
m
0
kp
eik
kp
ei 1
nm1
p
p
Note that S* (p,; ) S*(p, ; ).
m
m
Theorem 5.1 : The extreme points of the class S*
(p,; ) are
m
m
1 m (p )(1 eik )
kp
(5.2)
and
f (z)
zp kp
(k p) z
m i
m ik
(p )1 e k
(5.3)
f (z) 1
(p )(1 e ) zkp kp znp
(n m 1)
n zp
kp
(k p)
(n p)
m
m
Theorem 5.2 : Let the function f(z) defined by (5.1) be in the class S*
(p,; ), then f(z) is pvalently meromorphic convex
function in 0 < |z| < rm(p, ; ) where rm(p, ; ) in the largest value for which
2 2 m i
m ik 2 2
(n p )(p )1 e k
(p )(1 e )(k p ) r k kp r n p2
kp
(k p)
(n p)
for n m + 1, the result is sharp for the function given by (5.3).
REFERENCES
[1]. S. Owa, and H. Srivastava A class of analytic functions with fixed finitely many coefficients. J. Fac. Tech. Kinki Univ. 23(1987), 110. [2]. H. Silverman and F.M. Silvia Fixed coefficients for subclasses of starlike functions, Houston J. Math. 7(1981), 129136. [3]. B.A. Uralegaddi Meromorphically starlike functions with positive and fixed second coefficients, Kyungpook Math J. 29(1989) no. 1, 6468. [4]. M.K. Aouf and H.E. Darwish Meromorphically starlike functions with positive and fixed many coefficients, Analete Schntifice Ale Universitath. AL.I.CUZA IASI Tomul XLI, S.I.a. Matematica, 1995, 109116. [5]. Poonam Sharma and Deepali Chowdhary Coefficient properties of meromorphic function belonging to pvalently starlike and convex class of order.