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We define and investigate a new subclass of Bazilevič type harmonic univalent functions using a linear operator. We investigated the harmonic structures in terms of its coefficient conditions, extreme points, distortion bounds, convolution, and convex combination. So, also, we discussed the subordination properties for the functions in this class.

Let

Here, we recall some definitions and concepts of classes of analytic functions. Let

Also, let

The theory of analytic functions has wide application in many physical problem: problems as in heat conduction, electrostatic potential and fluid flows, and theory of fractals constitute practical examples. The concern of this work is the study of a particular family of analytic functions defined in a given domain by certain geometric conditions which are useful in the above problems.

Let

Sălăgean [

Thus, we obtain the differential operator

Corresponding to the function

For

Now, let

For instance, if we take

In 1992, Abduhalim [

Notable contributors like MacGregor, [

In some general sense, it is possible to further improve work on the function defined by the geometric condition (

The objective of this section is to find the sufficient conditions of functions belonging to the class

For this purpose, the following Lemmas will be necessary.

Let

Let

If

Now, we begin our main results as the following.

Let

Suppose that

For

Let

For

Let

For

Let

For

Let

Let the functions

Let

Letting

Let the function

Let the function

In this section, the authors wish to have a look into the Bazilevič type harmonic univalent functions.

A continuous complex-valued function

In this work, we may express the analytic functions

Furthermore, let the subclass

Assigning specific values to

We first prove a sufficient condition for the function in

Let

If

By (

The harmonic function

Let

Since

Let

In particular, the extreme points of

For functions

Conversely, suppose that

Our next result is on distortion bounds for the functions in the class

Let

We only prove the right-hand inequality. The proof for the left-hand inequality is similar and will be omitted.

Let

The following covering results follow from the left-hand inequality in Theorem

If function

For

Let the functions

We wish to show that the coefficients of

Next, we show that the class

Let functions

Let the functions

According to the definition of

The class

Let the functions

With various special choices of the parameters involved, both the existing classes and new one could be derived.