p-Valently Meromorphic Functions with Fixed First Coefficient

Download Full-Text PDF Cite this Publication

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* = {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].

    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.

    4. 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

       

    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

[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

.

Leave a Reply

Your email address will not be published. Required fields are marked *