# p-Valently Meromorphic Functions with Fixed First Coefficient DOI : http://dx.doi.org/10.17577/IJERTV9IS010273 Text Only Version

#### 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.

1. 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* = {zÂ¢ : 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  the following result :

Result 1 : Let the function f(z)Mp analytic in Dr = {zÂ¢ : 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* = {zÂ¢ : 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 , Uralegaddi , Owa and Srivastava  and M.K. Aouf and

H.E. Darwish .

2. 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).

3. 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.

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

5. THE CLASS S*

(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

. S. Owa, and H. Srivastava A class of analytic functions with fixed finitely many coefficients. J. Fac. Tech. Kinki Univ. 23(1987), 110. . H. Silverman and F.M. Silvia Fixed coefficients for subclasses of starlike functions, Houston J. Math. 7(1981), 129136.

. B.A. Uralegaddi Meromorphically starlike functions with positive and fixed second coefficients, Kyungpook Math J. 29(1989) no. 1, 6468.

. 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.

. Poonam Sharma and Deepali Chowdhary Coefficient properties of meromorphic function belonging to pvalently starlike and convex class of order

.