# Certain Subclasses ofβ-Uniformly q-Starlike and β-Uniformly q-Convex Functions

Authors: Eman S. A. AbuJarad; Mohammed H. A. AbuJarad; Thabet Abdeljawad; Fahd Jarad
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, the authors introduced certain subclassesβ-uniformly q-starlike and β-uniformly q-convex functions of order α involving the q-derivative operator defined in the open unit disc. Coefficient bounds were also investigated.


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## Body

## 1. Introduction

Theq-analysis is a generalization of the ordinary analysis. The application of the q-calculus was first introduced by Jackson [1–3]. In geometric function theory, the q-hypergeometric functions were first used by Srivastava [4]. The q-calculus provides valuable tools that have been used to define several subclasses of the normalized analytic function in the open unit disk U. Ismail et al. [5] were the first to study a certain class S∗ of starlike functions by using the q-derivative operator. Recently, new subclasses of analytic functions associated with q-derivative operators are introduced and discussed, see for example [4, 6–18]. Motivated by the importance of q-analysis, in this paper, we introduce the classes of β-uniformly q-starlike and β-uniformly q-convex functions defined by the q-derivative operator in the open unit disc, as a generalization of β-uniformly starlike and β-uniformly convex functions.First, we recall some basic notations and definitions fromq-calculus, which are used in this paper. The q-derivative of the function f is defined as follows [1–3]:
(1)Dqfz=fz−fqz1−qz,z≠0,0<q<1.From equation (1), it is clear that if f and g are the two functions, then
(2)Dqfz+gz=Dqfz+Dqgz,(3)Dqcfz=cDqfz,where c is a constant. We note that Dqfz⟶f′z as q⟶1−, where f′ is the ordinary derivative of the function f.In particular, using equation (1), the q-derivative of the function hz=zn is as follows:
(4)Dqhz=nqzn−1,where nq denotes the q-number and is given as follows:
(5)nq=1−qn1−q,0<q<1.Since we note thatnq⟶n as q⟶1−, therefore, in view of equation (4), Dqhz⟶h′z as q⟶1−, where h′z denotes the ordinary derivative of the function hz with respect to z.In this paper, we consider the classesA and T of the functions, analytic in the open unit disc U=z∈ℂ:z<1, of the following forms, respectively:
(6)fz=z+∑n=2∞anzn,(7)fz=z−∑n=2∞anzn,an≥0.Also, using equations (2), (3), (4), and (6), we get the following q-derivatives of the function f:
(8)Dqfz=1+∑n=2∞nqanzn−1,0<q<1,(9)DqzDqfz=1+∑n=2∞nq2anzn−1,0<q<1,where nq is given by equation (5).The classes of starlike functions of orderα0≤α<1 and convex functions of order α0≤α<1, denoted by S∗α and Kα, respectively, are defined as follows [19]:
(10)S∗α=f∈A:Rzf′zfz>α,(11)Kα=f∈A:R1+zf″zf′z>α.It is clear thatS∗α and Kα are the subclasses of the class A.The classes ofβ-uniformly starlike functions of order α and β-uniformly convex functions of order α, denoted by SDα,β and KDα,β, respectively, are defined as follows [20]:
(12)SDα,β=f∈A:Rzf′zfz−α>βzf′zfz−1,(13)KDα,β=f∈A:Rzf″zf′z−α>βzf″zf′z−1,where z∈U, 0≤α<1, and β≥0.The class ofq-starlike functions of order μ, denoted by Sq∗μ, is defined as follows [13]:
(14)Sq∗μ=f∈A:RzDqfzfz>μ,z∈U;0≤μ<1.Also, the class ofq-convex functions of order μ, denoted by Cqμ, is defined as [13]:
(15)Cqμ=f∈A:RDqzDqfzDqfz>μ,z∈U;0≤μ<1.The analytic functiong is said to be subordinate to the analytic function f in U [21], represented as follows:
(16)gz≺fzorg≺f,if there exists a Schwarz function w, which is analytic in U with
(17)w0=0,wz<1,such that
(18)gz=fwz,z∈U.In the next section, we introduce the classes ofβ-uniformly q-starlike and β-uniformly q-convex functions of order α, denoted by Sqα,β and UCVqα,β, respectively. Also, we obtain the coefficient bounds of the functions belonging to these classes.


## 2. Coefficient Bounds

Since theq-derivative is a generalized form of the ordinary derivative, therefore, in view of definitions of SDα,β and KDα,β, we define the classes of β-uniformly q-starlike and β-uniformly q-convex functions of order α, denoted by Sqα,β and UCVqα,β, respectively, by replacing the ordinary derivative with the q-derivative in equations (12) and (13).We provide the respective definitions of the classesSqα,β and UCVqα,β.Definition 1.
The functionf∈A is said to be β-uniformly q-starlike of order α, if it satisfies the following inequality:
(19)RzDqfzfz−α>βzDqfzfz−1,where 0<q<1, β≥0, 0≤α<1, and z∈U.Definition 2.
The functionf∈A is said to be β-uniformly q-convex of order α, if it satisfies the following inequality:
(20)RDqzDqfzDqfz−α>βDqzDqfzDqfz−1,where 0<q<1, β≥0, 0≤α<1, and z∈U.Further, we define the classesTSqα,β and UCTqα,β containing functions with negative coefficients and satisfying inequalities (19) and (20), respectively, as follows:
(21)TSqα,β=Sqα,β∩T,UCTqα,β=UCVqα,β∩T.Remark 3.
We note that(i)
limq→1−Sqα,β=SDα,β and limq→1−UCVqα,β=KDα,β(ii)
Sqα,β=Sq∗Dα and UCVqα,0=Cqα (see [8]).(iii)
limq→1−Sqα,0=S∗α and limq→1−UCVqα,0=KαNow, the relation between the subclassesSq∗μ and Sqα,βis given by the following result.Theorem 4.
Letf∈Sqα,β, then f∈Sq∗α+β/1+β, where β≥0, 0≤α<1, and 0<q<1.Proof.
Iff∈Sqα,β, then in view of Definition 1 and using the fact that −R<z≤z, we get
(22)RzDqfzfz−α>βzDqfzfz−1≥βRzDqfzfz−1,which implies that
(23)RzDqfzfz−α>βRzDqfzfz+β,then
(24)RzDqfzfz>α+β1+β.
Sinceβ≥0 and 0≤α<1, then 0≤α+β/1+β<1. Hence, in view of equation (14), we obtain f∈Sq∗α+β/1+β.Also, the relation between the subclassesCqα and UCVqα,βis given by the following result.Theorem 5.
Letf∈UCVqα,β, then f∈Cqα+β/1+β, where β≥0, 0≤α<1, and 0<q<1.Proof.
Iff∈UCVqα,β, then in view of Definition 2 and using the fact that −R<z≤z, we get
(25)RDqzDqzDqfz−α>βDqzDqfzDqfz−1≥−βRDqzDqfzDqfz−1,which implies that
(26)RDqzDqzDqfz−α>−βRDqzDqfzDqfz+β,then
(27)RDqzDqfzDqfz>α+β1+β,since β≥0 and 0≤α<1, then 0≤α+β/1+β<1. Hence, in view of equation (15), we obtain f∈Cqα+β/1+β.Next, the coefficient bound of the classSqα,β is given by the following result.Theorem 6.
A functionf∈A belongs to the class Sqα,β if
(28)∑n=2∞nq1+β−α+βan≤1−α,where 0<q<1, β≥0, 0≤α<1, and nq denotes the q-number.Proof.
Now, using the fact that−Rz≤z, we have
(29)βzDqfzfz−1−RzDqfzfz−1≤1+βzDqfzfz−1.
Using equations (6) and (8) in the right hand side of inequality (29), we get
(30)1+βzDqfzfz−1=1+β∑n=2∞nq−1anzn−11+∑n=2∞anzn−1.
Sincez<1, therefore, from the above inequality, we get
(31)1+βzDqfzfz−1<1+β∑n=2∞nq−1an1−∑n=2∞an.
Combining inequalities (29) and (31), we get
(32)βzDqfzfz−1−RzDqfzfz−1<1+β∑n=2∞nq−1an1−∑n=2∞an.
If/1+β∑n=2∞nq−1an/1−∑n=2∞an<1−α, which is equivalent to inequality (28), then from inequality (32) we get
(33)βzDqfzfz−1−RzDqfzfz−1≤1−a,which is equivalent to inequality (19). Thus, in view of Definition 1, the function f∈Sqα,β.Also, we obtain the coefficient bound forf∈TSqα,β in the following result.Theorem 7.
The functionf∈T belongs to the class TSqα,β, if and only if
(34)∑n=2∞nq1+β−α+βan≤1−α,where 0<q<1, β≥0, 0≤α<1, and nq denotes the q-number.Proof.
SinceT is a subclass of class A, therefore in view of Theorem 6, the sufficient condition of our result holds. Now, we need to prove only the necessary condition. Let f∈TSqα,β and taking z real, then from inequality (19), we have
(35)zDqfzfz−α>βzDqfzfz−1.
Now, using equations (7) and (8) in inequality (35), we get
(36)1−∑n=2∞nqanzn−11−∑n=2∞anzn−1−α>∑n=2∞βnqanzn−11−∑n=2∞anzn−1,then, letting z→1 along the real axis, inequality (36), gives the condition (34).The coefficient bound of the classUCVqα,β is given by the following result.Theorem 8.
A functionf∈A belongs to the class UCVqα,β if
(37)∑n=2∞nqnq1+β−α+βan≤1−α,where 0<q<1, β≥0, 0≤α<1, and nq denotes the q-number.Proof.
Now, using the fact that−R<z≤z, we have
(38)βDqzDqfzDqfz−1−RDqzDqfzDqfz−1≤1+βDqzDqfzDqfz−1.
Using equations (8) and (9) in the right hand side of inequality (38), we get
(39)1+βDqzDqfzDqfz−1=1+β1+∑n=2∞nq2anzn−11+∑n=2∞nqanzn−1−1.
Sincez<1, therefore, from the above inequality, we get
(40)1+βDqzDqfzDqfz−1<1+β∑n=2∞nq2−nqan1−∑n=2∞nqan.
Combining inequalities (38) and (40), we get
(41)βDqDqfzDqfz−1−RDqzDqfzDqfz−1<1+β∑n=2∞nq2−nqan1−∑n=2∞nqan.
If1+β∑n=2∞nq2−nqan/1−∑n=2∞nqan<1−α, which is equivalent to inequality (37), then from inequality (41), we get
(42)βDqzDqfzDqfz−1−RDqDqfzDqfz−1≤1−α,which is equivalent to inequality (20). Thus, in view of Definition 2, the function f∈UCVqα,β.The coefficient bound forf∈UCTqα,β is given by the following result.Theorem 9.
The functionf∈T belongs to the class UCTqα,β if and only if
(43)∑n=2∞nqnq1+β−α+βan≤1−α,where 0<q<1, β≥0, 0≤α<1, and nq denotes the q-number.Proof.
SinceT is a subclass of class A, therefore, in view of Theorem 8, the sufficient condition holds. Now, we need to prove only the necessary condition. Let f belong to the class UCTqα,β and taking z real, then from inequality (20), we have
(44)DqzDqfzDqfz−α>βDqzDqfzDqfz−1.
Now, using equations (8) and (9) in inequality (44), we get
(45)1−∑n=2∞nq2anzn−11−∑n=2∞nqanzn−1−α>∑n=2∞βnq2−nqanzn−11−∑n=2∞nqanzn−1,then letting z→1 along real axis, inequality (45) gives condition (43).We note that,q→1− in Theorems 6 and 8, we get the coefficient bounds for the functions belonging to the classes SDα,β and KDα,β in [20], respectively.In the next section, we obtain the extreme points for the functions belonging to the classesTSqα,β and UCTqα,β.


## 3. Extreme Points

The extreme points off∈TSqα,β are given by the following result.Theorem 10.
Letfnzn∈ℕ be sequences of functions such that
(46)f1z=z,fnz=z−1−anq1+β−a+βzn,n≥2,0<q<1,β≥0,0≤α<1,where nq denotes the q-number. Then f belongs to TSqα,β if and only if f can be expressed as the form
(47)fz=∑n=1∞λnfnz,where λn≥0n≥1 and ∑n=1∞λn=1.Proof.
Letf∈TSqα,β, then in view of Theorem 7, inequality (34) holds. Since an≥0n≥1 and 0≤α<1, therefore from inequality (34), we have
(48)nq1+β−α+βan≤1−α,n≥2.
Thus, if we take(49)λnnq1+β−α+β1−αan,n≥2,since λ1≥0, then, λn≥0n≥1.
Substitutinganfrom equation (49) withanfrom equation (7), we get:
(50)fz=z−∑n=2∞1−αnq1+β−α+βλnzn.
Since∑n=1∞λn=1, therefore, we have
(51)fz=λ1z+∑n=2∞λnz−∑n=2∞1−αnq1+β−α+βλnzn=λ1z+∑n=2∞λnz−1−αnq1+β−α+βzn,since f1z=z and fnz is given by equation (46). Therefore, from equation (51), we get the assertion (47). Conversely, let f be expressible in the form (47), which on using equation (46), gives
(52)fz=z−∑n=2∞1−αnq1+β−α+βλnzn,which can be expressed as follows:
(53)fz=z−∑n=2∞ηnzn,where
(54)ηn=1−αnq1+β−α+βλn,n≥2.
Now, to prove that the functionf, given by equation (53), belongs to the class TSqα,β, we need to show that the coefficients ηnn≥2 satisfy the inequality (34).
Sinceλ1≥0 and ∑n=1∞λn=1, therefore from equation (54), we have
(55)∑n=2∞nq1+β−α+β1−αηn=∑n=2∞nq1+β−α+β1−α1−αnq1+β−α+βλn=∑n=2∞λn=1−λ1≤1.
Thus, we get(56)∑n=2∞nq1+β−α+βηn≤1−α.
Therefore, in view of Theorem7 and the above inequality, we proved that the function f, given by equation (53), belongs to the class TSqα,β.Also, the extreme points off∈UCTqα,β are given by the following result.Theorem 11.
Letfnzn∈ℕ be a sequence of functions such that
(57)f1z=z,fn=z−1−αnqnq1+β−α+βzn,where n≥2, 0<q<1, β≥0, and 0≤α<1. Then, f belongs to UCTqα,β if and only if f can be expressed in the form given by equation (47) in terms of functions fnn≥2, given by equation (57), and λn≥0n≥1, ∑n=1∞λn=1.Proof.
Letf∈UCTqα,β, then from inequality (43), we have
(58)nqnq1+β−α+βan≤1−αn≥2.
If we set(59)λn=nqnq1+β−α+β1−αann≥2,since λ1=1, then λn≥0n≥1. Then, substituting an from equation (59) with an equation (7), we get
(60)fz=z−∑n=2∞1−αnqnq1+β−α+βλnzn.
Since∑n=1∞λn=1, therefore, we have
(61)fz=λ1z+∑n=2∞1−αnqnq1+β−α+βzn,since f1z=z and fnz is given by equation (57). Therefore, from equation (61), we get assertion (47).
Conversely, letf be expressible in the form (47), which on using equation (60), gives
(62)fz=z−∑n=2∞1−αnqnq1+β−α+βλnzn,which can be expressed as
(63)fz=z−∑n=2∞ηnzn,where
(64)ηn=1−αnqnq1+β−α+βλn,n≥2.
Now, to prove that functionf is given by equation (63) and belongs to the class UCTqα,β, we need to show that the coefficient ηnn≥2 satisfies inequality (43). Since λ1≥0 and ∑n=1∞λn=1,, therefore from equation (64), we have
(65)∑n=2∞nqnq1+β−α+β1−αηn=∑n=2∞λn=1−λ1≤1.
Thus, we get(66)∑n=2∞nqnq1+β−α+βηn<1−α.
Therefore, in view of Theorem9 and the above inequality, we proved that function f, given by equation (63), belongs to the class ∈UCTqα,β.


## 4. Partial Sums

The sequence of partial sums of the functionfz∈A, is defined as [22].
(67)fkz=z+∑n=2kanznk∈ℕ;z∈U.Now, we find the bounds of the real part of the ratio of the complex valued functionf∈A to its partial sums fkk∈ℕ, for the function to be in the class Sqα,β in the following result.Theorem 12.
Letfz∈A in the form (6) and suppose that
(68)∑n=2∞cnan≤1,where
(69)cnnq1+β+α+β1−αn≥2;0<q<1,β≥0,0≤α<1,then fz∈Sqα,β. Further, the following inequalities hold:
(70)Refzfkz≥1−1ck+1,(71)Refkzfz≥ck+11+ck+1,where
(72)cn≥1,ifn=2,3,⋯,k,ck+1,ifn=k+1,k+2,⋯,Proof.
Sincenqn≥2 is increasing and β≥0, α<1, therefore, in view of equation (69), cnn≥2 is an increasing sequence. Then, cn+1≥cn, ∀n and
(73)cn≥c22q1+β−α+β1+α≥1q1+β−α+β1−α.
Since1q=1, therefore, we have
(74)cn≥1,∀n.
Thus, for the particular valuek of n, condition (72) holds. In view of the first inequality of condition (72), we have
(75)∑n=2kan+ck+1∑n=k+1∞an≤∑n=2∞cnan,which in view of inequality (68), gives
(76)∑n=2kan+ck+1∑n=k+1∞an≤1,or, equivalently
(77)ck+1∑n=k+1∞an≤1−∑n=2kan.
Now, for some fixed positive integerk, we define
(78)h1z≔1+ck+1fz−fkzfkz.
Now, using equations (6) and (67), equation (78) gives
(79)h1z=1+ck+1∑n=k+1∞anzn−11+∑n=2kanzn−1.
From equation (79), we have
(80)h1z−1h1z+1=ck+1∑n=k+1∞anzn−12+2∑n=2kanzn−1+ck+1∑n=k+1∞anzn−1≤ck+1∑n=k+1∞an2−2∑n=2kan−ck+1∑n=k+1∞an.
In view of inequality (77), the above inequality gives h1z−1/h1z+1≤1, which implies
(81)Reh1z≥0.
Since eachcn∈ℝ, therefore, using equation (79) in inequality (81), we get assertion (70).
Again, sincecnn≥2 is an increasing function and cn≥1, ∀n≥2, therefore, we have
(82)∑n=2∞an≤∑n=2∞cnan,which in view of inequality (68), gives
(83)∑n=2∞an≤1.
Now, we define the functionh2z as follows:
(84)h2z=1+ck+1fkzfz−ck+1.
Using equations (6) and (67) in equation (84), we get
(85)h2z=1−ck+1+1∑n=k+1∞anzn−11+∑n=2∞anzn−1.
From equation (85), we have
(86)h2z−1h2z+1=−ck+1+1∑n=k+1∞anzn−12+2∑n=2∞anzn−1−ck+1+1∑n=k+1∞anzn−1=−ck+1+1∑n=k+1∞anzn−12+2∑n=2kanzn−1+1−ck+1∑n=k+1∞anzn−1≤ck+1+1∑n=k+1∞an2−2∑n=2kan−1−ck+1∑n=k+1∞an≤ck+1+1∑n=k+1∞an2−2∑n=2∞an+ck+1+1∑n=k+1∞an,using inequality (83) in inequality (86), we get h2z−1/h2z+1≤1, which implies
(87)Reh2z≥0.
Therefore, using equation (84) in inequality (87), we get assertion (71).Now, we find the bounds of the real part of the ratio of the complex valued functionf∈A to its partial sums fkk∈ℕ, for the function to be in the class UCVqα,β in the following result.Theorem 13.
Letfz∈A be in the form given by equation (6) and
(88)∑n=2∞snan≤1,where
(89)snnqnq1+β−α+β1−αn≥2;β≥0,0≤α<1,0<q<1.Then,fz∈UCVqα,β. Further, the following inequalities hold:
(90)Refzfkz>1−1sk+1,(91)Refkzfz>sk+11+sk+1,where
(92)sn≥1,ifn=2,3,⋯k,sk+1,ifn=k+1,k+2,⋯.Proof.
Using Theorem6 and following the same steps involved in the proof of Theorem 12, we get assertion (90) and (91).In the next section, we discuss the integral means inequality for the functions belonging to the classesTSqα,β and UCTqα,β.


## 5. Integral Means Inequality

Silverman [23] has been using the subordination principle to show that the integral ∫02πfre1θσdθσ>0,0<r<1 attains its maximum value in class T, when f2z=z−z2/2. Then, he applied that principle to solve the integral means inequality ∫02πfreiθσdθ≤∫02πf2reiθσdθ. Also, he found the integral means inequality for the classes S∗α and Kα with negative coefficients.First, we need to mention the following lemma [24].Lemma 14.
Iff and g are two analytic functions in U in the form T and f≺g, then
(93)∫02πfreiθσdθ≤∫02πgreiθσdθ,where σ>0, 0<r<1, and z=reiθ.Now, we establish the integral means inequality for the functions belonging to the classTSqα,β.Theorem 15.
Letf be of the form given by equation (7) that belongs to the class TSqα,β and f2z be defined as follows:
(94)f2z=11−α2q1+β−α+βz,then, for z=reiθ0<r<1, we have
(95)∫02πfzσdθ≤∫02πf2zσdθσ>0.Proof.
We define the functionw1z as follows:
(96)w1z=∑n=2∞2q1+β−α+β1−αanzn−1.
From the above equation, we have(97)w10=0.
Again, from equation (96), we have
(98)w1z=∑n=2∞2q1+β−α+β1−αanzn−1≤∑n=2∞2q1+β−α+β1−αanzn−1,since z=reiθ0<r<1 implies z=r<1, and using inequality (37), therefore, from the above inequality, we have
(99)w1≤∑n=2∞nq1+β−α+β1−αan≤1.
From equation (96), we have
(100)1−∑n=2∞anzn−1=1−1−α2q1+β−α+βw1z.
Sincew1 is analytic in U, therefore in view of equations (18), (96), (97), and (100); inequality (99); and the subordination principle, we have
(101)1−∑n=2∞anzn−1≺1−1−α2q1+β−α+βz.
Since, the function on the both sides of the above relation are analytic inU, therefore, in view of Lemma 14 and equation (94), we get assertion (95).Next, we establish the integral means inequality for the functions belonging to the classUCTqα,β with the positive coefficients.Theorem 16.
Letf belong to the class UCTqα,β and f3z is defined by
(102)f3z=1−1−α2q2q1+β−α+βz,then, for z=reiθ0<r<1, we have
(103)∫02πfzσdθ≤∫02πf3zσdθσ>0.Proof.
We define the functionw2z as follows:
(104)w2z=∑n=2∞2q2q1+β−α+β1−αanzn−1.
From the above equation, we have(105)w20=0.
Again, from equation (104), we have
(106)w2z=∑n=2∞2q2q1+β−α+β1−αanzn−1≤∑n=2∞2q2q1+β−α+β1−αanzn−1,since z=reiθ, then z=r<1 and using inequality (103), therefore, from the above inequality, we have
(107)w2z≤∑n=2∞nqnq1+β−α+β1−αan<1.
From equation (104), we have
(108)1−∑n=2∞anzn−1=1−1−α2q2q1+β−α+βw2z.
Sincew2 is analytic in U, therefore, in view of equations (18), (104), (105), (108); inequality (107); and the subordination principle, we have
(109)1−∑n=2∞anzn−1≺−1−α2q2q1+β−α+βz.
Since, the function on the both sides of the above relation are analytic inU, therefore, in view of Lemma 14 and equation (102), we get assertion (103).



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