# Uniformly Geometric Functions Involving Constructed Operators

Authors: Mohammad Al-Kaseasbeh; Maslina Darus
Journal: Journal of Complex Analysis
 (2017)

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## 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.


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## Body

## 1. Introduction

Throughout this paper, we are dealing with complex functions in the unit discU={z∈C:z<1}. More precisely, we are dealing with analytic functions of the form (1)fz=z+∑k=2∞akzk,and we refer to them by A.The subordination between analytic functionsf(z) and g(z) is written as f(z)≺g(z). Conceptually, the complex function f(z) is subordinate to g(z) if the image under g(z) contains the images under f(z). Technically, the complex function f(z) is subordinate to g(z) if there exists a Schwarz function w with w(0)=0 and wz<1 for all z∈U; such that (2)fz=gwz,z∈U.Let us consider the differential operatorsRα,λn and Dλn introduced, respectively, in [1, 2]. Then, the convoluted operator of both of them is(3)D~α,λnfz=Dλnfz∗Rα,λnfz=z+∑k=2∞1+λk-1nakzk∗z+∑k=2∞1+λk-1nCα,kakzk=z+∑k=2∞1+λk-12nCα,kak2zk.The operator D~α,λn can also be written as (4)D~α,λnfz=φz∗⋯∗φz︸2n-times∗fz∗z1-zα+1∗fz=φz∗⋯∗φz︸2n-times∗fz∗Rαfz,where (5)φz=z1-z+λz1-z2-λz1-z.A complex functionf∈A is said to be in the class C(η) of convex functions of order η in U, if (6)R1+zf′′zf′z>η,z∈U,where η∈[0,1).On the other hand, a complex functionf∈A is said to be in the class S∗(η) of starlike functions of order η in U, if (7)Rzf′zfz>η,z∈U,where η∈[0,1). The classes S∗(η) and C(η) are introduced in [3].Notice that the classesS∗≡S∗(0) and C≡C(0) are the classical classes of starlike and convex functions in U, respectively.A complex functionf∈A is said to be in the class of uniformly convex function of order η and type ζ, denoted by UCV(ζ,η), if (8)R1+zf′′zf′z>ζzf′′zf′z+η,z∈U,where ζ≥0,η∈[0,1) and ζ+η≥0, and is said to be in a corresponding class denoted by SP(ζ,η) if (9)Rzf′zfz>ζzf′zfz-1+η,z∈U,where ζ≥0,η∈[0,1) and ζ+η≥0. The classes UCV(ζ,η) and SP(ζ,η) are introduced in [4].The relation between classical starlike and convex functions, obviously, led us to the following relation.(10)f∈UCVζ,η⟺zf′∈SPζ,η.The classesSP(ζ,η) and UCV(ζ,η) generalised other several classes. For ζ=0, we obtain the classes S∗(η) and C(η), respectively. The class UCV(1,0)≡UCV is known as the uniformly convex functions introduced in [5]. The class SP(1,0)≡SP is introduced in [6]. The classes UCV(1,η)≡UCV(η) and SP(1,η)≡SP(η) are investigated in [7]. For η=0, the classes UCV(ζ,0)≡ζ-UCV and SP(ζ,0)≡ζ-SP, respectively, are introduced in [8, 9].Also, the classesSP(ζ,η) and UCV(ζ,η) have been studied by Al-Oboudi and Al-Amoudi [10], involving certain differential operators.


## 2. Geometric Interpretation

The complex functionsf∈SP(ζ,η) can be geometrically interpreted as follows. (11)f∈UCVζ,η⟺1+zf′′zf′zlies  inRζ,η,where Rζ,η is the conic domain included in the right half plane such that (12)Rζ,η=u+iv:u>ζu-12+v2+η.On the other hand, the complex functionsf∈UCV(ζ,η) can be geometrically interpreted as (13)f∈SPζ,η⟺(14)zf′zfzlies  inRζ,η.Denote byP(Pζ,η)(ζ≥0,-1≤η<1) the class of functions p, such that p≺Pζ,η where P denotes the class of positive real part functions in U, and p∈P. The function Pζ,η provides a conformal mapping between the unit disc and the domain Rζ,η such that 1∈Rζ,η and where the boundary of Rζ,η can be parameterised by (15)∂Rζ,η=u+iv:u2=ζu-12+v2+η2.By few steps of computations,∂Rζ,η appear as conic sections that are symmetrical around the real axis. Therefore, domain Rζ,η is an ellipse for ζ>1, a parabola for ζ=1, a hyperbola for 0<ζ<1, and a right half plane u>η for ζ=0.Involving the operatorD~α,λn given by (3), we introduce the following classes.Definition 1.
The complex functionsf∈A and satisfying (16)R1+zD~α,λnf′′zD~α,λnf′z>ζzD~α,λnf′′zD~α,λnf′z+η,z∈U,is denoted by UCVα,λn(ζ,η), where ζ≥0,η∈[0,1) and ζ+η≥0.
On the other hand, we introduce the correspondence class ofSPα,λn(ζ,η) as follows.Definition 2.
The complex functionsf∈A and satisfying (17)RzD~α,λnf′zD~α,λnfz>ζzD~α,λnf′zD~α,λnfz-1+η,z∈U,is denoted by SPα,λn(ζ,η), where ζ≥0,η∈[0,1) and ζ+η≥0.
It is clear that the complex functionf∈UCVα,λn(ζ,η) if and only if zf′∈SPα,λn(ζ,η) and that UCVα,λn(ζ,η)⊆SPα,λn(ζ,η).
From (16) and (17), the complex functions f∈UCVα,λn(ζ,η) and f∈SPα,λn(ζ,η) if and only if 1+zD~α,λnf′′(z)/D~α,λnf′(z) and zD~α,λnf′(z)/D~α,λnf(z), respectively, laying in the conic domain Rζ,η given in (12). Indeed, the conic domain Rζ,η is lying entirely in the right half plane, which allows us to write conditions (16) and (17) as follows. (18)p≺Pζ,η.
By virtue of (16) and (17) and the behavior of Rζ,η, we obtain (19)R1+zD~α,λnf′′zD~α,λnf′z>ζ+η1+ζ,z∈U,(20)RzD~α,λnf′zD~α,λnfz>ζ+η1+ζ,z∈U,which means that (21)f∈UCVα,λnζ,η⟹D~α,λnf∈Cζ+η1+ζ⊆C,(22)f∈SPα,λnζ,η⟹D~α,λnf∈S∗ζ+η1+ζ⊆S∗.
Conditions (19) and (20) led to the following inclusion relations, respectively. (23)UCVα,λnζ,η⊆Cα,λnζ+η1+ζ,SPα,λnζ,η⊆Sα,λ∗nζ+η1+ζ.


## 3. Uniformly Starlike Functions

This section concerns the classSPα,λn(ζ,η) 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]).
Letf and g be starlike of order 1/2. Then so is f∗g.Lemma 4 (see [12]).
Letf and g be univalent starlike of order 1/2. Then, for every function F∈A, we have (24)fz∗gzFzfz∗gz∈co¯FU,where co¯ denotes the closed convex hull.Lemma 5 (see [12]).
Letf and g, respectively, be in the classes C and S∗. Then, for every function F∈A, we have (25)fz∗gzFzfz∗gz∈co¯FU.Lemma 6 (see [13]).
Leta and b be complex constants and h univalent convex in U with h(0)=c and (26)Rahz+b>0.Let g(z)=c+∑k=1∞bkzk be analytic in U. Then (27)gz+zg′zagz+b≺hz.implies g(z)≺h(z).Lemma 7.
LetRαf(z)∈SPα,λn(ζ,η) and (28)∑k=2∞k2Cα,kak<1.Then f∈SPα,λn(ζ,η).Proof.
LetRαf(z)∈SPα,λn(ζ,η). Then (29)zD~α,λnRαf′D~α,λnRαfU⊆Rζ,ηand from (22) we see that D~α,λnRαf(z)∈S∗. We can write D~α,λnf(z) in terms of D~α,λnRα as follows: (30)D~α,λnfz=Rα-1fz∗D~α,λnRαfz,and, by convolution properties, we obtain (31)zD~α,λnfz′=Rα-1fz∗zD~α,λnRαfz′.Using Lemma 5 we obtain (32)zD~α,λnfz′D~α,λnfz=Rα-1fz∗zD~α,λnRαfz′Rα-1fz∗D~α,λnRαfz=Rα-1fz∗zD~α,λnRαfz′/D~α,λnRαfzD~α,λnRαfzRα-1fz∗D~α,λnRαfz∈co¯zD~α,λnRαf′D~α,λnRαfU⊆Rζ,η.Therefore, f∈SPα,λn(ζ,η).Theorem 8.
Let0≤λ≤1+ζ/1-η and (33)∑k=2∞k2Cα,kak<1.Then (34)SPα,λn+1ζ,η⊆SPα,λnζ,η.Proof.
Letf(z)∈SPα,λn+1(ζ,η). Then the geometric interpretation (18) can be written in the following subordination relation. (35)zD~α,λn+1fz′D~α,λn+1fz≺Pζ,η.By the definition of D~α,λnf(z), we obtain(36)D~α,λn+1fz=1-λD~α,λnRαfz+λzD~α,λnRαfz′=D~α,λnRαfz-λD~α,λnRαfz+λzD~α,λnRαfz′,D~α,λn+1fz′=D~α,λnRαfz′-λD~α,λnRαfz′+λzD~α,λnRαfz′′+λD~α,λnRαfz′=D~α,λnRαfz′+λzD~α,λnRαfz′′.With the notation of p(z)=zD~α,λnRαfz′/D~α,λnRαf(z), we have (37)zp′zpz=1-pz+zD~α,λnRαfz′′D~α,λnRαfz′.Thus we obtain (38)zD~α,λn+1fz′D~α,λn+1fz=pz+λzp′z1-λ+λpz.If λ=0, then from (35) and (38) (39)Rα,λnfz∈SPα,λnζ,η.If λ≠0, we can write by (35) and (38) (40)pz+11-λ/λ+pz·zp′z≺Pζ,η.Thereby, Lemma 6 and condition (20) imply p≺Pζ,η for 0≤λ≤1+ζ/1-η, since Pζ,η is univalent and convex in U.
Thus,Rα,λnf(z)∈SPα,λn(ζ,η). Therefore, f(z)∈SPα,λn(ζ,η) by Lemma 7.Corollary 9.
Let0≤λ≤1+ζ/1-η and (41)∑k=2∞k2Cα,kak<1.Then (42)SPα,λnζ,η⊆SPα,λζ,η.Proof.
The result is obtained by using Theorem8.Remark 10.
Considering the parametersn,α, and ζ by certain values, new results are obtained as follows. (1)
Considerα=0 in Theorem 8; we obtain, for 0≤λ≤1+ζ/1-η, (43)SPλn+1ζ,η⊆SPλnζ,η.(2)
Considerζ=0 in Theorem 8; we obtain, for 0≤λ≤1+ζ/1-η, (44)Sα,λ∗n+10,η⊆Sα,λ∗n0,η.
Paving the way to prove next theorem, we provide the forthcoming lemma.Lemma 11.
If the complex functionf∈SPα,λn(ζ,η), then D~α,λnf(z)∈S∗ whenever ζ and η lie, respectively, in [0,1) and [1/2,1) or [0,∞) and [0,1).Proof.
The results follows immediately from (20) where ζ+η/1+ζ≥1/2 under the restriction of the value of ζ and η.Theorem 12.
Let0≤μ≤α<1 and (45)∑k=2∞k2Cμ,kCα,kak<1.Then (46)SPα,λnζ,η⊆SPμ,λnζ,η,where [(0≤ζ<1 and 1/2≤η) or (ζ≥1 and 0≤η<1)].Proof.
Letf∈SPα,λn(ζ,η). Then by the definition of D~α,λn and the convolution properties, we have (47)D~μ,λnfz=z1-zμ+1∗Rα-1fz∗fz∗φ∗⋯∗φ︸2n-times∗z1-zα+1∗fz=z1-zμ+1∗Rα-1∗fz∗D~α,λnfz,zD~μ,λnfz′=z1-zμ+1∗Rα-1∗fz∗zD~α,λnfz′.By Lemma 11 we have D~α,λnf(z)∈S∗(1/2). Using Lemma 4, we obtain (48)zD~μ,λnfz′D~μ,λnfz=z/1-zμ+1∗Rα-1fz∗fz∗zD~α,λnfz′z/1-zμ+1∗Rα-1fz∗fz∗D~α,λnfz=z/1-zμ+1∗Rα-1fz∗fz∗zD~α,λnfz′/D~α,λnfzD~α,λnfzz/1-zμ+1∗Rα-1fz∗fz∗D~α,λnfz∈co¯zD~α,λnf′D~α,λnfU⊆Rζ,η.Therefore, f∈SPμ,λn(ζ,η).Corollary 13.
Letμ=0. Also let [(0≤ζ<1 and 1/2≤η) or (ζ≥1 and 0≤η<1)] and (49)∑k=2∞k2Cα,kak<1.Then (50)SPα,λnζ,η⊆SPλnζ,η.Proof.
The results follows by Theorem12.Remark 14.
Considering the parametersn,α,λ, and ζ by certain values, new results are obtained as follows.(1)
Considern=1 and λ=0 in Theorem 12; we obtain, for 0≤μ≤α<1, (51)SPα,01ζ,η⊆SPμ,01ζ,η,where[(0≤ζ<1 and 1/2≤η) or (ζ≥1 and 0≤η<1)].(2)
Considerζ=0 in Theorem 12; we obtain, for 0≤μ≤α<1, (52)Sα,λ∗n0,η⊆Sμ,λ∗n0,η,where0≤η<1/2.


### 3.2. Coefficient Bounds

In this subsection, we obtain the coefficient bounds of those functions belonging to the classSPα,λn(ζ,η).Theorem 15.
A complex functionf∈A is in SPα,λn(ζ,η) if (53)∑k=2∞k1+ζ-ζ+η1+λk-12nCα,kak2≤1-η.Proof.
It suffices to show that(54)ζzD~α,λnfz′D~α,λnfz-1-RzD~α,λnfz′D~α,λnfz-1<1-η.We have (55)ζzD~α,λnfz′D~α,λnfz-1-RzD~α,λnfz′D~α,λnfz-1≤1+ζzD~α,λnfz′D~α,λnfz-1≤1+ζ∑k=2∞k-11+λk-12nCα,kak2zk-11-∑k=2∞1+λk-12nCα,kak2zk-1<1+ζ∑k=2∞k-11+λk-12nCα,kak21-∑k=2∞1+λk-12nCα,kak2.Using condition (53), last expression is bounded above by (1-η).


## 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]).
Letf and g be starlike of order 1/2. Then so is f∗g.Lemma 4 (see [12]).
Letf and g be univalent starlike of order 1/2. Then, for every function F∈A, we have (24)fz∗gzFzfz∗gz∈co¯FU,where co¯ denotes the closed convex hull.Lemma 5 (see [12]).
Letf and g, respectively, be in the classes C and S∗. Then, for every function F∈A, we have (25)fz∗gzFzfz∗gz∈co¯FU.Lemma 6 (see [13]).
Leta and b be complex constants and h univalent convex in U with h(0)=c and (26)Rahz+b>0.Let g(z)=c+∑k=1∞bkzk be analytic in U. Then (27)gz+zg′zagz+b≺hz.implies g(z)≺h(z).Lemma 7.
LetRαf(z)∈SPα,λn(ζ,η) and (28)∑k=2∞k2Cα,kak<1.Then f∈SPα,λn(ζ,η).Proof.
LetRαf(z)∈SPα,λn(ζ,η). Then (29)zD~α,λnRαf′D~α,λnRαfU⊆Rζ,ηand from (22) we see that D~α,λnRαf(z)∈S∗. We can write D~α,λnf(z) in terms of D~α,λnRα as follows: (30)D~α,λnfz=Rα-1fz∗D~α,λnRαfz,and, by convolution properties, we obtain (31)zD~α,λnfz′=Rα-1fz∗zD~α,λnRαfz′.Using Lemma 5 we obtain (32)zD~α,λnfz′D~α,λnfz=Rα-1fz∗zD~α,λnRαfz′Rα-1fz∗D~α,λnRαfz=Rα-1fz∗zD~α,λnRαfz′/D~α,λnRαfzD~α,λnRαfzRα-1fz∗D~α,λnRαfz∈co¯zD~α,λnRαf′D~α,λnRαfU⊆Rζ,η.Therefore, f∈SPα,λn(ζ,η).Theorem 8.
Let0≤λ≤1+ζ/1-η and (33)∑k=2∞k2Cα,kak<1.Then (34)SPα,λn+1ζ,η⊆SPα,λnζ,η.Proof.
Letf(z)∈SPα,λn+1(ζ,η). Then the geometric interpretation (18) can be written in the following subordination relation. (35)zD~α,λn+1fz′D~α,λn+1fz≺Pζ,η.By the definition of D~α,λnf(z), we obtain(36)D~α,λn+1fz=1-λD~α,λnRαfz+λzD~α,λnRαfz′=D~α,λnRαfz-λD~α,λnRαfz+λzD~α,λnRαfz′,D~α,λn+1fz′=D~α,λnRαfz′-λD~α,λnRαfz′+λzD~α,λnRαfz′′+λD~α,λnRαfz′=D~α,λnRαfz′+λzD~α,λnRαfz′′.With the notation of p(z)=zD~α,λnRαfz′/D~α,λnRαf(z), we have (37)zp′zpz=1-pz+zD~α,λnRαfz′′D~α,λnRαfz′.Thus we obtain (38)zD~α,λn+1fz′D~α,λn+1fz=pz+λzp′z1-λ+λpz.If λ=0, then from (35) and (38) (39)Rα,λnfz∈SPα,λnζ,η.If λ≠0, we can write by (35) and (38) (40)pz+11-λ/λ+pz·zp′z≺Pζ,η.Thereby, Lemma 6 and condition (20) imply p≺Pζ,η for 0≤λ≤1+ζ/1-η, since Pζ,η is univalent and convex in U.
Thus,Rα,λnf(z)∈SPα,λn(ζ,η). Therefore, f(z)∈SPα,λn(ζ,η) by Lemma 7.Corollary 9.
Let0≤λ≤1+ζ/1-η and (41)∑k=2∞k2Cα,kak<1.Then (42)SPα,λnζ,η⊆SPα,λζ,η.Proof.
The result is obtained by using Theorem8.Remark 10.
Considering the parametersn,α, and ζ by certain values, new results are obtained as follows. (1)
Considerα=0 in Theorem 8; we obtain, for 0≤λ≤1+ζ/1-η, (43)SPλn+1ζ,η⊆SPλnζ,η.(2)
Considerζ=0 in Theorem 8; we obtain, for 0≤λ≤1+ζ/1-η, (44)Sα,λ∗n+10,η⊆Sα,λ∗n0,η.
Paving the way to prove next theorem, we provide the forthcoming lemma.Lemma 11.
If the complex functionf∈SPα,λn(ζ,η), then D~α,λnf(z)∈S∗ whenever ζ and η lie, respectively, in [0,1) and [1/2,1) or [0,∞) and [0,1).Proof.
The results follows immediately from (20) where ζ+η/1+ζ≥1/2 under the restriction of the value of ζ and η.Theorem 12.
Let0≤μ≤α<1 and (45)∑k=2∞k2Cμ,kCα,kak<1.Then (46)SPα,λnζ,η⊆SPμ,λnζ,η,where [(0≤ζ<1 and 1/2≤η) or (ζ≥1 and 0≤η<1)].Proof.
Letf∈SPα,λn(ζ,η). Then by the definition of D~α,λn and the convolution properties, we have (47)D~μ,λnfz=z1-zμ+1∗Rα-1fz∗fz∗φ∗⋯∗φ︸2n-times∗z1-zα+1∗fz=z1-zμ+1∗Rα-1∗fz∗D~α,λnfz,zD~μ,λnfz′=z1-zμ+1∗Rα-1∗fz∗zD~α,λnfz′.By Lemma 11 we have D~α,λnf(z)∈S∗(1/2). Using Lemma 4, we obtain (48)zD~μ,λnfz′D~μ,λnfz=z/1-zμ+1∗Rα-1fz∗fz∗zD~α,λnfz′z/1-zμ+1∗Rα-1fz∗fz∗D~α,λnfz=z/1-zμ+1∗Rα-1fz∗fz∗zD~α,λnfz′/D~α,λnfzD~α,λnfzz/1-zμ+1∗Rα-1fz∗fz∗D~α,λnfz∈co¯zD~α,λnf′D~α,λnfU⊆Rζ,η.Therefore, f∈SPμ,λn(ζ,η).Corollary 13.
Letμ=0. Also let [(0≤ζ<1 and 1/2≤η) or (ζ≥1 and 0≤η<1)] and (49)∑k=2∞k2Cα,kak<1.Then (50)SPα,λnζ,η⊆SPλnζ,η.Proof.
The results follows by Theorem12.Remark 14.
Considering the parametersn,α,λ, and ζ by certain values, new results are obtained as follows.(1)
Considern=1 and λ=0 in Theorem 12; we obtain, for 0≤μ≤α<1, (51)SPα,01ζ,η⊆SPμ,01ζ,η,where[(0≤ζ<1 and 1/2≤η) or (ζ≥1 and 0≤η<1)].(2)
Considerζ=0 in Theorem 12; we obtain, for 0≤μ≤α<1, (52)Sα,λ∗n0,η⊆Sμ,λ∗n0,η,where0≤η<1/2.


## 3.2. Coefficient Bounds

In this subsection, we obtain the coefficient bounds of those functions belonging to the classSPα,λn(ζ,η).Theorem 15.
A complex functionf∈A is in SPα,λn(ζ,η) if (53)∑k=2∞k1+ζ-ζ+η1+λk-12nCα,kak2≤1-η.Proof.
It suffices to show that(54)ζzD~α,λnfz′D~α,λnfz-1-RzD~α,λnfz′D~α,λnfz-1<1-η.We have (55)ζzD~α,λnfz′D~α,λnfz-1-RzD~α,λnfz′D~α,λnfz-1≤1+ζzD~α,λnfz′D~α,λnfz-1≤1+ζ∑k=2∞k-11+λk-12nCα,kak2zk-11-∑k=2∞1+λk-12nCα,kak2zk-1<1+ζ∑k=2∞k-11+λk-12nCα,kak21-∑k=2∞1+λk-12nCα,kak2.Using condition (53), last expression is bounded above by (1-η).


## 4. Uniformly Convex Functions

This section concerns the classUCVα,λn(ζ,η) 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 classUCVα,λn(ζ,η).Lemma 16.
LetRα,λnf(z)∈UCVα,λn(ζ,η), and(56)∑k=2∞k2Cα,kak<1.Then f∈UCVα,λn(ζ,η).Proof.
In virtue of Lemma7, the following implication is done. (57)Rαfz∈UCVα,λnζ,η⟺zRαfz′∈SPα,λnζ,η⟺zRαf′z∈SPα,λnζ,η⟹zf′z∈SPα,λnζ,η⟺fz∈UCVα,λnζ,η.Therefore, f(z)∈UCVα,λn(ζ,η).Theorem 17.
Let0≤λ≤1+ζ/1-η and (58)∑k=2∞k2Cα,kak<1.Then (59)UCVα,λn+1ζ,η⊆UCVα,λnζ,η.Proof.
In virtue of Lemma3, the following implication is done. (60)fz∈UCVα,λn+1ζ,η⟺zf′z∈SPα,λn+1ζ,η⟹zf′z∈SPα,λnζ,η⟺fz∈UCVα,λnζ,η.Therefore, f(z)∈UCVα,λn(ζ,η).Corollary 18.
Let0≤λ≤1+ζ/1-η and (61)∑k=2∞k2Cα,kak<1.Then (62)UCVα,λnζ,η⊆UCVα,λζ,η.Proof.
The result follows by using Theorem17.Remark 19.
By giving the parametersn,α, and ζ certain values, new results are obtained as follows.(1)
Considerα=0 in Theorem 17; we obtain, for 0≤λ≤1+ζ/1-η, (63)UCVλn+1ζ,η⊆UCVλnζ,η.(2)
Considerζ=0 in Theorem 17; we obtain, for 0≤λ≤1+ζ/1-η, (64)Cα,λn+1η⊆Cα,λnη.Theorem 20.
Let0≤μ≤α<1 and (65)∑k=2∞k2Cα,kak<1.Then (66)UCVα,λnζ,η⊆UCVμ,λnζ,η,where [(0≤ζ<1 and 1/2≤η) or (ζ≥1 and 0≤η<1)].Proof.
The results are obtained using Theorem12 and apply Alexander relation.Corollary 21.
Let[(0≤ζ<1 and 1/2≤η) or (ζ≥1 and 0≤η<1)]. Then (67)UCVα,λnζ,η⊆UCVλζ,η.Corollary 22.
Let0≤μ≤α<1. Then (68)UCVα,λnζ,η⊆UCVμ,λζ,η,where [(0≤ζ<1 and 1/2≤η) or (ζ≥1 and 0≤η<1)].Remark 23.
By giving the parametersn,α,λ, and ζ certain values, we obtain new results as follows. (1)
Considern=1 and λ=0 in Theorem 20; we obtain for 0≤μ≤α<1, (69)UCVα,01ζ,η⊆UCVμ,01ζ,η,where[(0≤ζ<1 and 1/2≤η) or (ζ≥1 and 0≤η<1)].(2)
Considerζ=0 in Theorem 20; we obtain for 0≤μ≤α<1, (70)Cα,λn0,η⊆Cμ,λn0,η,where0≤η<1/2.


### 4.2. Coefficient Bounds

In this subsection, we obtain the coefficient bounds of those functions belonging to the classUCVα,λn(ζ,η).Theorem 24.
A complex functionf∈A is in UCVα,λn(ζ,η) if (71)∑k=2∞kk1+ζ-ζ+η1+λk-12nCα,kak2≤1-η.Proof.
The result follows from Theorem15 and the following relation: (72)f∈UCVα,λnζ,η⟺zf′∈SPα,λnζ,η.


## 4.1. Inclusion Relation

The forthcoming lemma paves the way to provide the inclusion relations in classUCVα,λn(ζ,η).Lemma 16.
LetRα,λnf(z)∈UCVα,λn(ζ,η), and(56)∑k=2∞k2Cα,kak<1.Then f∈UCVα,λn(ζ,η).Proof.
In virtue of Lemma7, the following implication is done. (57)Rαfz∈UCVα,λnζ,η⟺zRαfz′∈SPα,λnζ,η⟺zRαf′z∈SPα,λnζ,η⟹zf′z∈SPα,λnζ,η⟺fz∈UCVα,λnζ,η.Therefore, f(z)∈UCVα,λn(ζ,η).Theorem 17.
Let0≤λ≤1+ζ/1-η and (58)∑k=2∞k2Cα,kak<1.Then (59)UCVα,λn+1ζ,η⊆UCVα,λnζ,η.Proof.
In virtue of Lemma3, the following implication is done. (60)fz∈UCVα,λn+1ζ,η⟺zf′z∈SPα,λn+1ζ,η⟹zf′z∈SPα,λnζ,η⟺fz∈UCVα,λnζ,η.Therefore, f(z)∈UCVα,λn(ζ,η).Corollary 18.
Let0≤λ≤1+ζ/1-η and (61)∑k=2∞k2Cα,kak<1.Then (62)UCVα,λnζ,η⊆UCVα,λζ,η.Proof.
The result follows by using Theorem17.Remark 19.
By giving the parametersn,α, and ζ certain values, new results are obtained as follows.(1)
Considerα=0 in Theorem 17; we obtain, for 0≤λ≤1+ζ/1-η, (63)UCVλn+1ζ,η⊆UCVλnζ,η.(2)
Considerζ=0 in Theorem 17; we obtain, for 0≤λ≤1+ζ/1-η, (64)Cα,λn+1η⊆Cα,λnη.Theorem 20.
Let0≤μ≤α<1 and (65)∑k=2∞k2Cα,kak<1.Then (66)UCVα,λnζ,η⊆UCVμ,λnζ,η,where [(0≤ζ<1 and 1/2≤η) or (ζ≥1 and 0≤η<1)].Proof.
The results are obtained using Theorem12 and apply Alexander relation.Corollary 21.
Let[(0≤ζ<1 and 1/2≤η) or (ζ≥1 and 0≤η<1)]. Then (67)UCVα,λnζ,η⊆UCVλζ,η.Corollary 22.
Let0≤μ≤α<1. Then (68)UCVα,λnζ,η⊆UCVμ,λζ,η,where [(0≤ζ<1 and 1/2≤η) or (ζ≥1 and 0≤η<1)].Remark 23.
By giving the parametersn,α,λ, and ζ certain values, we obtain new results as follows. (1)
Considern=1 and λ=0 in Theorem 20; we obtain for 0≤μ≤α<1, (69)UCVα,01ζ,η⊆UCVμ,01ζ,η,where[(0≤ζ<1 and 1/2≤η) or (ζ≥1 and 0≤η<1)].(2)
Considerζ=0 in Theorem 20; we obtain for 0≤μ≤α<1, (70)Cα,λn0,η⊆Cμ,λn0,η,where0≤η<1/2.


## 4.2. Coefficient Bounds

In this subsection, we obtain the coefficient bounds of those functions belonging to the classUCVα,λn(ζ,η).Theorem 24.
A complex functionf∈A is in UCVα,λn(ζ,η) if (71)∑k=2∞kk1+ζ-ζ+η1+λk-12nCα,kak2≤1-η.Proof.
The result follows from Theorem15 and the following relation: (72)f∈UCVα,λnζ,η⟺zf′∈SPα,λnζ,η.


## 5. Conclusion

This paper introduced two classes of uniformly geometric functions of orderη type ζ. Literally speaking, convex and starlike uniformly functions of order η type ζ were introduced by involving the constructed differential operator D~α,λn. Also, the geometric interpretation of these functions was given. Finally, two properties of each class were investigated, namely, inclusion relations and coefficient bounds.



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*Source: 5916805-2017-04-16.xml*