JCA Journal of Complex Analysis 2314-4971 2314-4963 Hindawi 10.1155/2017/5916805 5916805 Research Article Uniformly Geometric Functions Involving Constructed Operators Al-Kaseasbeh Mohammad 1 http://orcid.org/0000-0001-9138-916X Darus Maslina 1 Grinshpan Arcadii Z. School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia 43600 Bangi Selangor Malaysia ukm.my 2017 1642017 2017 22 02 2017 28 03 2017 1642017 2017 Copyright © 2017 Mohammad Al-Kaseasbeh and Maslina Darus. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

MOHE FRGS/1/2016/STG06/UKM/01/1
1. Introduction

Throughout this paper, we are dealing with complex functions in the unit disc U={zC:z<1}. More precisely, we are dealing with analytic functions of the form (1)fz=z+k=2akzk,and we refer to them by A.

The subordination between analytic functions f(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 zU; such that (2)fz=gwz,zU.

Let us consider the differential operators Rα,λn and Dλn introduced, respectively, in [1, 2]. Then, the convoluted operator of both of them is(3)D~α,λnfz=DλnfzRα,λnfz=z+k=21+λk-1nakzkz+k=21+λk-1nCα,kakzk=z+k=21+λk-12nCα,kak2zk.The operator D~α,λn can also be written as (4)D~α,λnfz=φzφz2n-timesfzz1-zα+1fz=φzφz2n-timesfzRαfz,where (5)φz=z1-z+λz1-z2-λz1-z.

A complex function fA is said to be in the class C(η) of convex functions of order η in U, if (6)R1+zfzfz>η,zU,where η[0,1).

On the other hand, a complex function fA is said to be in the class S(η) of starlike functions of order η in U, if (7)Rzfzfz>η,zU,where η[0,1). The classes S(η) and C(η) are introduced in .

Notice that the classes SS(0) and CC(0) are the classical classes of starlike and convex functions in U, respectively.

A complex function fA 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+zfzfz>ζzfzfz+η,zU,where ζ0,η[0,1) and ζ+η0, and is said to be in a corresponding class denoted by SP(ζ,η) if (9)Rzfzfz>ζzfzfz-1+η,zU,where ζ0,η[0,1) and ζ+η0. The classes UCV(ζ,η) and SP(ζ,η) are introduced in .

The relation between classical starlike and convex functions, obviously, led us to the following relation. (10)fUCVζ,ηzfSPζ,η.

The classes SP(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 . The class SP(1,0)SP is introduced in . The classes UCV(1,η)UCV(η) and SP(1,η)SP(η) are investigated in . For η=0, the classes UCV(ζ,0)ζ-UCV and SP(ζ,0)ζ-SP, respectively, are introduced in [8, 9].

Also, the classes SP(ζ,η) and UCV(ζ,η) have been studied by Al-Oboudi and Al-Amoudi , involving certain differential operators.

2. Geometric Interpretation

The complex functions fSP(ζ,η) can be geometrically interpreted as follows. (11)fUCVζ,η1+zfzfz  lies  in  Rζ,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 functions fUCV(ζ,η) can be geometrically interpreted as (13)fSPζ,η(14)zfzfz  lies  in  Rζ,η.

Denote by P(Pζ,η)  (ζ0,-1η<1) the class of functions p, such that pPζ,η where P denotes the class of positive real part functions in U, and pP. The function Pζ,η provides a conformal mapping between the unit disc and the domain Rζ,η such that 1Rζ,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 operator D~α,λn given by (3), we introduce the following classes.

Definition 1.

The complex functions fA and satisfying (16)R1+zD~α,λnfzD~α,λnfz>ζzD~α,λnfzD~α,λnfz+η,zU,is denoted by UCVα,λn(ζ,η), where ζ0,η[0,1) and ζ+η0.

On the other hand, we introduce the correspondence class of SPα,λn(ζ,η) as follows.

Definition 2.

The complex functions fA and satisfying (17)RzD~α,λnfzD~α,λnfz>ζzD~α,λnfzD~α,λnfz-1+η,zU,is denoted by SPα,λn(ζ,η), where ζ0,η[0,1) and ζ+η0.

It is clear that the complex function fUCVα,λn(ζ,η) if and only if zfSPα,λn(ζ,η) and that UCVα,λn(ζ,η)SPα,λn(ζ,η).

From (16) and (17), the complex functions fUCVα,λn(ζ,η) and fSPα,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)pPζ,η.

By virtue of (16) and (17) and the behavior of Rζ,η, we obtain (19)R1+zD~α,λnfzD~α,λnfz>ζ+η1+ζ,zU,(20)RzD~α,λnfzD~α,λnfz>ζ+η1+ζ,zU,which means that (21)fUCVα,λnζ,ηD~α,λnfCζ+η1+ζC,(22)fSPα,λnζ,ηD~α,λnfSζ+η1+ζS.

Conditions (19) and (20) led to the following inclusion relations, respectively. (23)UCVα,λnζ,ηCα,λnζ+η1+ζ,SPα,λnζ,ηSα,λnζ+η1+ζ.

3. Uniformly Starlike Functions

This section concerns the class SPα,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 [<xref ref-type="bibr" rid="B14">11</xref>]).

Let f and g be starlike of order 1/2. Then so is fg.

Lemma 4 (see [<xref ref-type="bibr" rid="B13">12</xref>]).

Let f and g be univalent starlike of order 1/2. Then, for every function FA, we have (24)fzgzFzfzgzco¯FU,where co¯ denotes the closed convex hull.

Lemma 5 (see [<xref ref-type="bibr" rid="B13">12</xref>]).

Let f and g, respectively, be in the classes C and S. Then, for every function FA, we have (25)fzgzFzfzgzco¯FU.

Lemma 6 (see [<xref ref-type="bibr" rid="B9">13</xref>]).

Let a 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=1bkzk be analytic in U. Then (27)gz+zgzagz+bhz.implies g(z)h(z).

Lemma 7.

Let Rαf(z)SPα,λn(ζ,η) and (28)k=2k2Cα,kak<1.Then fSPα,λn(ζ,η).

Proof.

Let Rαf(z)SPα,λn(ζ,η). Then (29)zD~α,λnRαfD~α,λnRαfURζ,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-1fzD~α,λnRαfz,and, by convolution properties, we obtain (31)zD~α,λnfz=Rα-1fzzD~α,λnRαfz.Using Lemma 5 we obtain (32)zD~α,λnfzD~α,λnfz=Rα-1fzzD~α,λnRαfzRα-1fzD~α,λnRαfz=Rα-1fzzD~α,λnRαfz/D~α,λnRαfzD~α,λnRαfzRα-1fzD~α,λnRαfzco¯zD~α,λnRαfD~α,λnRαfURζ,η.Therefore, fSPα,λn(ζ,η).

Theorem 8.

Let 0λ1+ζ/1-η and (33)k=2k2Cα,kak<1.Then (34)SPα,λn+1ζ,ηSPα,λnζ,η.

Proof.

Let f(z)SPα,λn+1(ζ,η). Then the geometric interpretation (18) can be written in the following subordination relation. (35)zD~α,λn+1fzD~α,λn+1fzPζ,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)zpzpz=1-pz+zD~α,λnRαfzD~α,λnRαfz.Thus we obtain (38)zD~α,λn+1fzD~α,λn+1fz=pz+λzpz1-λ+λpz.If λ=0, then from (35) and (38) (39)Rα,λnfzSPα,λnζ,η.If λ0, we can write by (35) and (38) (40)pz+11-λ/λ+pz·zpzPζ,η.Thereby, Lemma 6 and condition (20) imply pPζ,η for 0λ1+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.

Let 0λ1+ζ/1-η and (41)k=2k2Cα,kak<1.Then (42)SPα,λnζ,ηSPα,λζ,η.

Proof.

The result is obtained by using Theorem 8.

Remark 10.

Considering the parameters n,α, and ζ by certain values, new results are obtained as follows.

Consider α=0 in Theorem 8; we obtain, for 0λ1+ζ/1-η, (43)SPλn+1ζ,ηSPλnζ,η.

Consider ζ=0 in Theorem 8; we obtain, for 0λ1+ζ/1-η, (44)Sα,λn+10,ηSα,λn0,η.

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

Lemma 11.

If the complex function fSPα,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+ζ1/2 under the restriction of the value of ζ and η.

Theorem 12.

Let 0μα<1 and (45)k=2k2Cμ,kCα,kak<1.Then (46)SPα,λnζ,ηSPμ,λnζ,η,where [(0ζ<1 and 1/2η) or (ζ1 and 0η<1)].

Proof.

Let fSPα,λn(ζ,η). Then by the definition of D~α,λn and the convolution properties, we have (47)D~μ,λnfz=z1-zμ+1Rα-1fzfzφφ2n-timesz1-zα+1fz=z1-zμ+1Rα-1fzD~α,λnfz,zD~μ,λnfz=z1-zμ+1Rα-1fzzD~α,λnfz.By Lemma 11 we have D~α,λnf(z)S(1/2). Using Lemma 4, we obtain (48)zD~μ,λnfzD~μ,λnfz=z/1-zμ+1Rα-1fzfzzD~α,λnfzz/1-zμ+1Rα-1fzfzD~α,λnfz=z/1-zμ+1Rα-1fzfzzD~α,λnfz/D~α,λnfzD~α,λnfzz/1-zμ+1Rα-1fzfzD~α,λnfzco¯zD~α,λnfD~α,λnfURζ,η.Therefore, fSPμ,λn(ζ,η).

Corollary 13.

Let μ=0. Also let [(0ζ<1 and 1/2η) or (ζ1 and 0η<1)] and (49)k=2k2Cα,kak<1.Then (50)SPα,λnζ,ηSPλnζ,η.

Proof.

The results follows by Theorem 12.

Remark 14.

Considering the parameters n,α,λ, and ζ by certain values, new results are obtained as follows.

Consider n=1 and λ=0 in Theorem 12; we obtain, for 0μα<1, (51)SPα,01ζ,ηSPμ,01ζ,η,

where [(0ζ<1 and 1/2η) or (ζ1 and 0η<1)].

Consider ζ=0 in Theorem 12; we obtain, for 0μα<1, (52)Sα,λn0,ηSμ,λn0,η,

where 0η<1/2.

3.2. Coefficient Bounds

In this subsection, we obtain the coefficient bounds of those functions belonging to the class SPα,λn(ζ,η).

Theorem 15.

A complex function fA is in SPα,λn(ζ,η) if (53)k=2k1+ζ-ζ+η1+λk-12nCα,kak21-η.

Proof.

It suffices to show that (54)ζzD~α,λnfzD~α,λnfz-1-RzD~α,λnfzD~α,λnfz-1<1-η.We have (55)ζzD~α,λnfzD~α,λnfz-1-RzD~α,λnfzD~α,λnfz-11+ζzD~α,λnfzD~α,λnfz-11+ζk=2k-11+λk-12nCα,kak2zk-11-k=21+λk-12nCα,kak2zk-1<1+ζk=2k-11+λk-12nCα,kak21-k=21+λk-12nCα,kak2.Using condition (53), last expression is bounded above by (1-η).

4. Uniformly Convex Functions

This section concerns the class UCVα,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 class UCVα,λn(ζ,η).

Lemma 16.

Let Rα,λnf(z)UCVα,λn(ζ,η), and(56)k=2k2Cα,kak<1.Then fUCVα,λn(ζ,η).

Proof.

In virtue of Lemma 7, the following implication is done. (57)RαfzUCVα,λnζ,ηzRαfzSPα,λnζ,ηzRαfzSPα,λnζ,ηzfzSPα,λnζ,ηfzUCVα,λnζ,η.Therefore, f(z)UCVα,λn(ζ,η).

Theorem 17.

Let 0λ1+ζ/1-η and (58)k=2k2Cα,kak<1.Then (59)UCVα,λn+1ζ,ηUCVα,λnζ,η.

Proof.

In virtue of Lemma 3, the following implication is done. (60)fzUCVα,λn+1ζ,ηzfzSPα,λn+1ζ,ηzfzSPα,λnζ,ηfzUCVα,λnζ,η.Therefore, f(z)UCVα,λn(ζ,η).

Corollary 18.

Let 0λ1+ζ/1-η and (61)k=2k2Cα,kak<1.Then (62)UCVα,λnζ,ηUCVα,λζ,η.

Proof.

The result follows by using Theorem 17.

Remark 19.

By giving the parameters n,α, and ζ certain values, new results are obtained as follows.

Consider α=0 in Theorem 17; we obtain, for 0λ1+ζ/1-η, (63)UCVλn+1ζ,ηUCVλnζ,η.

Consider ζ=0 in Theorem 17; we obtain, for 0λ1+ζ/1-η, (64)Cα,λn+1ηCα,λnη.

Theorem 20.

Let 0μα<1 and (65)k=2k2Cα,kak<1.Then (66)UCVα,λnζ,ηUCVμ,λnζ,η,where [(0ζ<1 and 1/2η) or (ζ1 and 0η<1)].

Proof.

The results are obtained using Theorem 12 and apply Alexander relation.

Corollary 21.

Let [(0ζ<1 and 1/2η) or (ζ1 and 0η<1)]. Then (67)UCVα,λnζ,ηUCVλζ,η.

Corollary 22.

Let 0μα<1. Then (68)UCVα,λnζ,ηUCVμ,λζ,η,where [(0ζ<1 and 1/2η) or (ζ1 and 0η<1)].

Remark 23.

By giving the parameters n,α,λ, and ζ certain values, we obtain new results as follows.

Consider n=1 and λ=0 in Theorem 20; we obtain for 0μα<1, (69)UCVα,01ζ,ηUCVμ,01ζ,η,

where [(0ζ<1 and 1/2η) or (ζ1 and 0η<1)].

Consider ζ=0 in Theorem 20; we obtain for 0μα<1, (70)Cα,λn0,ηCμ,λn0,η,

where 0η<1/2.

4.2. Coefficient Bounds

In this subsection, we obtain the coefficient bounds of those functions belonging to the class UCVα,λn(ζ,η).

Theorem 24.

A complex function fA is in UCVα,λn(ζ,η) if (71)k=2kk1+ζ-ζ+η1+λk-12nCα,kak21-η.

Proof.

The result follows from Theorem 15 and the following relation: (72)fUCVα,λnζ,ηzfSPα,λnζ,η.

5. Conclusion

This paper introduced two classes of uniformly geometric functions of order η 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.

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.

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