Abstract
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.
1. Introduction
Throughout this paper, we are dealing with complex functions in the unit disc . More precisely, we are dealing with analytic functions of the form and we refer to them by .
The subordination between analytic functions and is written as . Conceptually, the complex function is subordinate to if the image under contains the images under . Technically, the complex function is subordinate to if there exists a Schwarz function with and for all ; such that
Let us consider the differential operators and introduced, respectively, in [1, 2]. Then, the convoluted operator of both of them isThe operator can also be written as where
A complex function is said to be in the class of convex functions of order in , if where
On the other hand, a complex function is said to be in the class of starlike functions of order in , if where The classes and are introduced in [3].
Notice that the classes and are the classical classes of starlike and convex functions in , respectively.
A complex function is said to be in the class of uniformly convex function of order and type , denoted by , if where and , and is said to be in a corresponding class denoted by if where and The classes and are introduced in [4].
The relation between classical starlike and convex functions, obviously, led us to the following relation.
The classes and generalised other several classes. For , we obtain the classes and , respectively. The class is known as the uniformly convex functions introduced in [5]. The class is introduced in [6]. The classes and are investigated in [7]. For , the classes and , respectively, are introduced in [8, 9].
Also, the classes and have been studied by Al-Oboudi and Al-Amoudi [10], involving certain differential operators.
2. Geometric Interpretation
The complex functions can be geometrically interpreted as follows. where is the conic domain included in the right half plane such that
On the other hand, the complex functions can be geometrically interpreted as
Denote by the class of functions , such that where denotes the class of positive real part functions in , and . The function provides a conformal mapping between the unit disc and the domain such that and where the boundary of can be parameterised by
By few steps of computations, appear as conic sections that are symmetrical around the real axis. Therefore, domain is an ellipse for , a parabola for , a hyperbola for , and a right half plane for
Involving the operator given by (3), we introduce the following classes.
Definition 1. The complex functions and satisfying is denoted by , where and
On the other hand, we introduce the correspondence class of as follows.
Definition 2. The complex functions and satisfying is denoted by , where and
It is clear that the complex function if and only if and that
From (16) and (17), the complex functions and if and only if and , respectively, laying in the conic domain given in (12). Indeed, the conic domain is lying entirely in the right half plane, which allows us to write conditions (16) and (17) as follows. By virtue of (16) and (17) and the behavior of , we obtain which means that Conditions (19) and (20) led to the following inclusion relations, respectively.
3. Uniformly Starlike Functions
This section concerns the class and its properties, namely, inclusion relation and coefficient bounds.
3.1. Inclusion Relation
In this subsection, we study the inclusion relations. The following lemmas pave the way for doing so.
Lemma 3 (see [11]). Let and be starlike of order Then so is .
Lemma 4 (see [12]). Let and be univalent starlike of order Then, for every function , we have where denotes the closed convex hull.
Lemma 5 (see [12]). Let and , respectively, be in the classes and . Then, for every function , we have
Lemma 6 (see [13]). Let and be complex constants and univalent convex in with and Let be analytic in . Then implies
Lemma 7. Let and Then
Proof. Let . Then and from (22) we see that . We can write in terms of as follows: and, by convolution properties, we obtain Using Lemma 5 we obtain Therefore,
Theorem 8. Let and Then
Proof. Let Then the geometric interpretation (18) can be written in the following subordination relation. By the definition of , we obtainWith the notation of , we have Thus we obtain If , then from (35) and (38) If , we can write by (35) and (38) Thereby, Lemma 6 and condition (20) imply for , since is univalent and convex in .
Thus, Therefore, by Lemma 7.
Corollary 9. Let and Then
Proof. The result is obtained by using Theorem 8.
Remark 10. Considering the parameters , and by certain values, new results are obtained as follows. (1)Consider in Theorem 8; we obtain, for , (2)Consider in Theorem 8; we obtain, for , Paving the way to prove next theorem, we provide the forthcoming lemma.
Lemma 11. If the complex function , then whenever and lie, respectively, in and or and .
Proof. The results follows immediately from (20) where under the restriction of the value of and .
Theorem 12. Let and Then where and or and
Proof. Let . Then by the definition of and the convolution properties, we have By Lemma 11 we have . Using Lemma 4, we obtain Therefore,
Corollary 13. Let . Also let and or and and Then
Proof. The results follows by Theorem 12.
Remark 14. Considering the parameters , and by certain values, new results are obtained as follows.(1)Consider and in Theorem 12; we obtain, for , where and or and (2)Consider in Theorem 12; we obtain, for , where
3.2. Coefficient Bounds
In this subsection, we obtain the coefficient bounds of those functions belonging to the class
Theorem 15. A complex function is in if
Proof. It suffices to show that We have Using condition (53), last expression is bounded above by .
4. Uniformly Convex Functions
This section concerns the class and its properties, namely, inclusion relation and coefficient bounds.
4.1. Inclusion Relation
The forthcoming lemma paves the way to provide the inclusion relations in class .
Lemma 16. Let , andThen
Proof. In virtue of Lemma 7, the following implication is done. Therefore,
Theorem 17. Let and Then
Proof. In virtue of Lemma 3, the following implication is done. Therefore,
Corollary 18. Let and Then
Proof. The result follows by using Theorem 17.
Remark 19. By giving the parameters , and certain values, new results are obtained as follows.(1)Consider in Theorem 17; we obtain, for , (2)Consider in Theorem 17; we obtain, for ,
Theorem 20. Let and Then where and or and
Proof. The results are obtained using Theorem 12 and apply Alexander relation.
Corollary 21. Let and or and . Then
Corollary 22. Let . Then where and or and
Remark 23. By giving the parameters , and certain values, we obtain new results as follows. (1) Consider and in Theorem 20; we obtain for , where and or and (2) Consider in Theorem 20; we obtain for , where
4.2. Coefficient Bounds
In this subsection, we obtain the coefficient bounds of those functions belonging to the class
Theorem 24. A complex function is in if
Proof. The result follows from Theorem 15 and the following relation:
5. Conclusion
This paper introduced two classes of uniformly geometric functions of order type . Literally speaking, convex and starlike uniformly functions of order type were introduced by involving the constructed differential operator . Also, the geometric interpretation of these functions was given. Finally, two properties of each class were investigated, namely, inclusion relations and coefficient bounds.
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.
Acknowledgments
The work here is supported by MOHE Grant FRGS/1/2016/STG06/UKM/01/1.