A Different Approach to Normalized Analytic Functions Through Meromorphic Functions Defined by Extended Multiplier Transformations Operator

Let A(n) denote the class of analytic functions f, in the open unit disk U = {z : |z| < 1} normalized by f(0) = f'(0)-1 = 0, and let Sigma(p,k) denote the class of Meromorphic functions in U. = U\{0} . In this paper, we introduce and study the classes of functions z(p+1) f(z). A, such that f(z). Sigma(p,k) . We also introduce and study the class V (lambda,k) (alpha,beta)( n, mu, delta,xi,epsilon) of functions f(z)is an element of Sigma(p,k) satisfying |D-lambda(n+1) (alpha,beta,mu)z(p+1) f(z)/z (z/D-lambda(n+1) (alpha,beta,mu)z(p+1) f(z))(delta) -1/xi| <1-epsilon Where delta > 0, lambda >0, xi is an element of(0,1], epsilon is an element of[0,1) and D lambda n(alpha,beta,mu)f(z): A -> A, is the Extended Multiplier Transformations Operator, newly defined as follows D-lambda(n)(alpha,beta,mu) f (z) -z +Sigma(infinity)(k=2) (alpha+(mu+lambda) (k-1) + beta/alpha+beta)n alpha(k)z(k) Several inclusion, subordination, and superordination properties have been discussed for the above said classes of normalized analytic and meromorphic functions.