Coefficients Estimates of New Subclasses for Univalent Functions Related to Complex Order

We introduce the class of analytic and univalent functions of the form 𝑓(𝑧

Our aim is to introduce new subclasses of univalent and analytic functions and study their coefficients estimates and other geometric properties such as, distortion theorems and maximization theorem.Now, let the new subclass  μ (τ, λ, L, M) of functions of the form where L and M are fixed numbers such that, −1 ≤ M < L ≤ 1 and μ ≠ 0 is arbitrary complex number or, equivalently (1.3) can be rewritten as where  λ τ is the generalized Jung-Kim-Srivastava integral operator [6] defined by for τ ≥ 0, λ > −1, we observe that for τ = 0, we have  λ 0 () = ().By giving specific values to μ, τ, L and  in (1.4), we obtain some subclasses that are previously studied by several authors as follows; (i) for τ = 0, we obtain the subclass of functions () satisfying the condition studied by Dixit and Pal in [3].
(v) For μ = 1 and τ = 0, we obtain the subclass of functions () satisfying the condition studied by Goel and Mehrok in [4].We refer the interested readers to [5,9,11] concerning study of certain subclasses of analytic functions.
We state the following lemma that needed in our results.

Coefficient Estimates Theorem 2.1
Let the function () given with (1.2) be in the class   (, , , ), then the estimates are sharp.
We found that the sharpness is obtained for the function

Conclusion
We determined the coefficients estimates for the new subclass  μ (τ, λ, L, M).Also we obtained the distortion theorem and maximization theorem for this new class.This study will