This article discusses the theory fundamental theorems of A (z) – analytic functions and establishes analogies for Schwarz inequality as well as Harnack’s theorem for A(z) – analytic functions.

By using the theory of residues of holomorphic functions, one formula is obtained for the product of a finite number of Sines of multiple arcs and one improper integral is computed.

In this paper, we introduce a new class of multivalent harmonic functions defined by liner operator H p λ , we instigate functions in this class have a variety of properties. The bounds for coefficients, distortions, convolution, convex combination, and extreme point are all given.

Some relations in this paper we using  in  new subclass of meromorphically p-valent functions TK( ) defined by integral operator involving  -function  We derived some properties, like, coefficient inequality  , growth and distortion bounds by theorems (2) and (3), Partial sums, convex set, radii of starlikeness and radii  convexity.

In this paper, we will investigate and discuss a new class of meromorphic univalent functions defined by multiplier transformation which is R(c, , y, ), as well as study the coefficient estimates and growth theorems, and then another line in this work, upon to get the close under the convex linear combination