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This paper introduces classes of uniformly geometric functions involving constructed differential operators by means of convolution. Basic properties of those classes are studied, namely, coefficient bounds and inclusion relations.

Throughout this paper, we are dealing with complex functions in the unit disc

The subordination between analytic functions

Let us consider the differential operators

A complex function

On the other hand, a complex function

Notice that the classes

A complex function

The relation between classical starlike and convex functions, obviously, led us to the following relation.

The classes

Also, the classes

The complex functions

On the other hand, the complex functions

Denote by

By few steps of computations,

Involving the operator

The complex functions

On the other hand, we introduce the correspondence class of

The complex functions

It is clear that the complex function

From (

By virtue of (

Conditions (

This section concerns the class

In this subsection, we study the inclusion relations. The following lemmas pave the way for doing so.

Let

Let

Let

Let

Let

Let

Let

Let

Thus,

Let

The result is obtained by using Theorem

Considering the parameters

Consider

Consider

Paving the way to prove next theorem, we provide the forthcoming lemma.

If the complex function

The results follows immediately from (

Let

Let

Let

The results follows by Theorem

Considering the parameters

Consider

where

Consider

where

In this subsection, we obtain the coefficient bounds of those functions belonging to the class

A complex function

It suffices to show that

This section concerns the class

The forthcoming lemma paves the way to provide the inclusion relations in class

Let

In virtue of Lemma

Let

In virtue of Lemma

Let

The result follows by using Theorem

By giving the parameters

Consider

Consider

Let

The results are obtained using Theorem

Let

Let

By giving the parameters

Consider

where

Consider

where

In this subsection, we obtain the coefficient bounds of those functions belonging to the class

A complex function

The result follows from Theorem

This paper introduced two classes of uniformly geometric functions of order

The authors declare that they have no conflicts of interest regarding the publication of this paper.

The work here is supported by MOHE Grant FRGS/1/2016/STG06/UKM/01/1.