A New Sub Class of Univalent Analytic Functions Associated with a Multiplier Linear Operator

International Research Journal of Engineering and Technology (IRJET)

e-ISSN: 2395-0056

Volume: 04 Issue: 07 | July -2017

p-ISSN: 2395-0072

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A NEW SUB CLASS OF UNIVALENT ANALYTIC FUNCTIONS ASSOCIATED WITH A MULTIPLIER LINEAR OPERATOR DR. JITENDRA AWASTHI DEPARTMENT OF MATHEMATICS, S.J.N.P.G.COLLEGE, LUCKNOW-226001 ---------------------------------------------------------------------***--------------------------------------------------------------------ABSTRACT: This paper deals with a new class

km, ( , A, B)

which is a subclass of uniformly starlikefunctions involving a

multiplier linear operator k ,  . Coefficients inequality, Distortion theorem, Extreme points, Radius of starlikeness and radius of m

convexity for functions belonging to this class are obtained. Key words and phrases: Univalent, starlike, subordination, convex, analytic, linear operator. 2010 Mathematics Subject Classification: 30C45. 1. INTRODUCTION Let S denote the class of functions of the form 

(1.1) f ( z )  z   a n z n , n2

which are analytic and univalent in the unit open disk ∆={z: |z|<1}. Silverman[9] had introduced and studied a subclass T of S consisting of functions of the form 

(1.2) f ( z )  z   a n z n , (a n  0, n  2) n2

Let f and g be analytic in ∆.Then g is said to be subordinate to f, written as

g  f or g(z)  f(z) , if there exists a Schwartz function ω, which is analytic in ∆ with ω(0)=0 and ( z)  1( z  ) such that g ( z)  f (( z)) (z  ). In particular, if the function f is univalent in ∆, we have the following equivalence([3],[7]) g ( z)  f ( z) (z  )  g(0)  f(0) and g()  f(). Sharma and Raina ([4],[5],[6]) have introduced a multiplier linear operator k ,  for m

m  Z ,   1, k  0 by

  m m  0,  k ,  f ( z )  f ( z ),   1  1   1 1 k z k  2 m 1  z t  k ,  f (t )dt , m  Z -  {1,2,........} (1.3)  mk,  f ( z )  0 k    1 2   11   mk,  f ( z )  k z k d  z k  mk,1 f ( z ) , m  Z   {1,2,...........]     1 dt   m The series representation of k , f ( z) for f(z) of the form (1.1) is given by m

  k (n  1)   a n z n , (1.4) mk,  f ( z )  z   1   1  n2 

Also

(i) mk,0  Dkm , m  0 [1] (ii) 1m,0  D m , m  0 [8] © 2017, IRJET

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