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.