In this article, a new class of harmonic univalent functions, defined by the differential operator, is introduced. Some geometric properties, like, coefficient estimates, extreme points, convex combination and convolution (Hadamard product) are obtained.

In this paper, we introduce and discuss a certain subclass A(α,β) of univalent functions in the open unit disc, we obtain some properties like coefficient estimates and results of integral means by using differential subordination.

In this paper, we introduce and study a new class A^* (λ,μ,ν,η,p,ζ,τ) of meromorphic multivalent functions defined by fractional calculus operator of the punctured unit disc U^*. On this class we obtain several results like, coefficient inequality, modified Hadamard product, (N,δ)-neighborhood, partial sums, convex linear combination and integral operator.

In this work, using the differential operator and the concept of meromorephic analytic functions we introduce and investigate the class Σ β , λ α , m ( A , B , Y , β ) , of functions of the form f ( z ) = z − 1 + ∑ n = 1 ∞ a n z n , a n ≥ 0 , which are analytic and meromorphic univalent in the punctured unit disk U * = { z ∈ C : 0 < | z | < 1 } , satisfying | A [ γ z 2 ( D β , λ α , m f ( z ) ) ′ ′ ′ + z ( 3 Υ + β ) ( D β , λ α , m f ( z ) ) ′ ′ + ( Υ + β ) ( D β , λ α , m f ( z ) ) ′ ] B [ Υ z ( D β , λ α , m f ( z ) ) ′ ′ + ( Υ + β ) ( D β , λ α , m f ( z ) ) ′ ] | < 1 . Further, coefficient bounds, Hadamard product, radius of close to-convexity, inclusive properties and neighbourhoods of functions in our class are obtain.