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We define and investigate a new subclass of Salagean-type harmonic univalent functions. We obtain coefficient conditions, extreme points, distortion bounds, convolution, and convex combination for the above subclass of harmonic functions.which are analytic in the open unit disk Í {z ∈ : |z| < 1}.We denote the subclass of A consisting of analytic and univalent functions f z in the unit disk Í by S.The following classes of functions and many others are well known and have been studied repeatedly by many authors, namely, Sȃlȃgean 1 , Abdul Halim 2 , and Darus 3 to mention but a few.2 Abstract and Applied Analysis iii δ α {f z ∈ A : Re{f z } > α, 0 ≤ α < 1, z ∈ Í}.

Let S H denote the class of functions f h-g that are harmonic univalent and sense-preserving in the unit disk U {z : |z| < 1}, where h z z ∞ k 2 a k z k , g z ∞ k 1 b k z k |b 1 | < 1. In this paper, we introduce the class M H n, λ, α of functions f h-g which are harmonic in U. A sufficient coefficient of this class is determined. It is shown that this coefficient bound is also necessary for the class M-H n, λ, α if f n z h-g n ∈ M H n, λ, α , where h z z − ∞ k 2 |a k |z k , g n z −1 n ∞ k 1 |b k |z k and n ∈ N 0. Coefficient conditions, such as distortion bounds, convolution conditions, convex combination, extreme points, and neighborhood for the class M-H n, λ, α , are obtained.

By using the polylogarithm function, a new integral operator is introduced. Strong differential subordination and superordination properties are determined for some families of univalent functions in the open unit disk which are associated with new integral operator by investigating appropriate classes of admissible functions. New strong differential sandwich-type results are also obtained.

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