Certain Subclasses of Harmonic Univalent Functions Associated with Generalized Salagean Operator

In this paper, we investigate necessary and sufficient coefficient conditions, distortion bounds, extreme points and convex combination of a new subclass of harmonic univalent functions defined by a generalization of modified Salagean operator. 2000 Mathematics Subject Classification: 30C45, 30C50. 1.Introduction Let H denote the family of continuous complex valued harmonic functions which are harmonic in the open unit disk U = {z : |z| < 1} and let A be the subclass of H consisting of functions which are analytic in U. A function harmonic in U may be written as f = h+ g, where h and g are members of A. We call h the analytic part and g the co-analytic part of f. In this case, f is sense-preserving if |h′(z)| > |g′(z)| in U. See [4]. Let SH denote the family of functions f = h+ g which are harmonic, univalent, and sense-preserving in U for which f(0) = fz(0)− 1 = 0. To this end, without loss of generality, we may write h(z) = z + ∞ ∑ k=2 akz , g(z) = ∞ ∑ k=1 bkz , |b1| < 1. (1) Note that SH reduces to the class S of normalized analytic univalent functions in U if the co-analytic part of f is identically zero. Let SHj denote the class of all functions f = h+ ḡ ∈ SH such that h and g has the form h(z) = z + ∞ ∑ k=j+1 akz , g(z) = ∞ ∑