# Applications ofq-Symmetric Derivative Operator to the Subclass of Analytic and Bi-Univalent Functions Involving the Faber Polynomial Coefficients

Authors: Mohammad Faisal Khan; Shahid Khan; Nazar Khan; Jihad Younis; Bilal Khan
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this paper, using the basic concepts of symmetricq-calculus operator theory, we define a symmetric q-difference operator for m-fold symmetric functions. By considering this operator, we define a new subclass ℛbφ,m,q of m-fold symmetric bi-univalent functions in open unit disk U. As in applications of Faber polynomial expansions for fm∈ℛbφ,m,q, we find general coefficient amk+1 for n≥4, Fekete–Szegő problems, and initial coefficients am+1 and a2m+1. Also, we construct q-Bernardi integral operator for m-fold symmetric functions, and with the help of this newly defined operator, we discuss some applications of our main results. For validity of our result, we have chosen to give some known special cases of our main results in the form of corollaries and remarks.


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## Body

## 1. Introduction and Definitions

LetA denote the set of all analytic functions f1z in the open unit disk U=z:z<1, and thus each analytic function can be written in terms of power series:(1)f1z=z+∑n=2∞anzn.A functionf1∈A in open unit disk U is considered to be normalized function if it fulfills the condition of normalization, that is,(2)f10=0,f′0=1.A functionf1z is said to be univalent in the open unit disk U at opints z1,z2∈D if (3)z1≠z2⇒f1z1≠f1z2.and here S represent the class of univalent function. We know that every f1∈S has an inverse f1−1, which is given as(4)f1−1f1z=z,z∈U,f1f1−1w=w,w<r0f,r0f≥14,where(5)g1w=f1−1w=w−a2w2+2a22−a3w3−5a23−5a2a3+a4w4+….An analytic functionf1 and its inverse are univalent in U; then, f1 is called bi-univalent. An analytic function f1 is called bi-Bazilevic in U if f1 and f1−1 are Bazilevic in U (see [1]). The behavior of these types of functions is unpredictable and not much is known about their coefficients. Let Σ denote the class of analytic and bi-univalent functions in U. For f1∈Σ, in [2], Levin showed that a2<1.51, and after that, Branan and Clunie [3] investigated a2≤2. Furthermore, in paper [4], Netanyahu showed that maximum of f1∈Σ is maxa2=4/3.In 1986, Branan and Taha [5] defined the subclass of the bi-univalent functions of class Σ, and also, Srivastava et al. in [6] investigated various subclasses of class Σ. After that, Frasin and Aouf [7], Xu et al. [8, 9], and Hayami and Owa [10] followed their work by introducing new subclasses of class Σ. For more studies, we refer the readers to [11, 12]. The performance of the coefficients of the functions f1 and f1−1 is unpredictable due to the bi-univalency requirements; therefore, we cannot much investigate about general coefficient an for n≥4.In [13], Faber introduced Faber polynomials and used it to determine the general coefficient bounds an for n≥4. Furthermore, Gong [14] explained importance of Faber polynomials in mathematical sciences, especially in geometric function theory. Recently, Airault [15] gave some remarks on Faber polynomials, and in [16], he discussed more about Faber polynomials on differential calculus.By using the Faber polynomial expansion technique, Hamidi and Jahangiri [17, 18] investigated some new coefficient estimates for analytic bi-close-to-convex functions. In the literature, there are only a few works determining the general coefficient bounds an for f1∈Σ given by (1) by using Faber polynomial expansions. By using Faber polynomial expansion, we have known very little about the bounds of Maclaurin’s series coefficient an for n≥4. Recently, many authors have conducted some studies about Faber polynomial expansion and determined general coefficient bounds an for n≥4 (for details, see [19–22]).A domainU is recognized m-fold symmetric if(6)fmei2π/mz=ei2π/mfmz,z∈U,f∈A,where m is a positive integer. The function fm∈S, of the form(7)hz=fzmm,is univalent and maps the unit disk U into a region with m-fold symmetry.A functionfm is called m-fold symmetric if it has the following normalized form:(8)fmz=z+∑k=1∞amk+1zmk+1.LetSm denote the set of all m-fold symmetric univalent functions and note that S1=S (for details, see [14, 23]).The univalent functionfm of form (8) is said to be m-fold symmetric bi-univalent function in U if its inverse fm−1 is univalent and the series expansion for fm−1 is given as follows:(9)gmw=fm−1w=w−am+1wm+1+ϒ1a,mw2m+1−ϒ2a,mw3m+1+…,where(10)ϒ1a,m=m+1am+12−a2m+1,ϒ2a,m=12m+13m+2am+13−3m+2am+1a2m+1+a3m+1,and series for fm−1w was proved by Srivastava et al. in [24] and denoted by Σm. For m=1, the series in (9) corresponds with (5) of the class Σ. Srivastava et al. [24] defined a subclass of class Σm and investigated initial coefficient bounds while Hamidi and Jahangiri [25] also defined m-fold symmetric bi-starlike functions.Many authors investigated several subclasses ofA by using the basic concepts of q-calculus and fractional q-calculus. Ismail et al. [26] were the first ones to introduce a q-difference operator Dq for the class S of normalized starlike functions in U. A number of researchers have got inspired by the q-calculus because of its divers applications in mathematics and physics [24]. Historically, Srivastava [27] was the first who made used of q-calculus in the context of geometric function theory. In 1909, Jackson [28, 29] defined the q-analogue of derivative and integral operator and discussed some of its applications. Aral and Gupta [30, 31] introduced the q-Baskakov–Durrmeyer operator while the author in [32] studied the q-Picard and q-Gauss–Weierstrass singular integral operators. Recently, Kanas and Raducanu [33] introduced the q-analogue of Ruscheweyh differential operator while Aldweby and Darus [34] and Mahmood and Sokoł [35] discussed some applications of this differential operator. In [36], using the symmetric q-derivative operator, a subclass of analytic and bi-univalent functions has been introduced and discussed. Some properties of q-close-to-convex functions have been obtained in [37], while Hu et al. [38] considered some subclasses of starlike functions and have obtained some sufficiency criteria for their defined functions class. For some recent investigations, we refer the readers to [39, 40].The theory ofq-symmetric calculus has been used in different fields of mathematics and physics, for example, Lavagno studied quantum mechanics in [41] while Da Cruz et al. [42] discussed q-symmetric variation calculus. After that, several authors used basic concepts of q-symmetric calculus in geometric function theory from different aspects and defined some new subclasses of analytic functions and investigated some new results. Kanas et al. [43] implemented some basics concepts of q-symmetric calculus and defined q-symmetric derivative. They discussed some applications of this operator on new subclasses of analytic functions. Furthermore, Khan et al. [44] defined symmetric conic domain by using the concepts of q-symmetric calculus and used this domain to investigate some new subclasses of analytic functions. But in geometric function theory, using q-symmetric calculus very little work has been done. Especially, very few articles have been published so far on this topic.Here we recall few basic concepts and definitions of theq-symmetric difference calculus. Throughout this paper, we suppose that 0<q<1 and(11)ℕ=1,2,3,….Theq-symmetric number frequently occurs in the study of q-deformed quantum mechanical simple harmonic oscillator (see [45]) and can be defined as follows.Definition 1.
Forn∈ℕ, the q-symmetric number is defined by(12)n,q˜=q−n−qnq−1−q,0,q˜=0.Remark 1.
We note that theq-symmetric number does not reduce to the q-number.Definition 2.
For anyn∈ℤ+∪0, the q-symmetric number shift factorial is defined by(13)n,q˜!=n,q˜n−1,q˜n−2,q˜…2,q˜1,q˜,n≥1,0,q˜!=1.
Note that(14)limq⟶1−n,q˜!=n!.Definition 3.
(see [46]). The q-symmetric derivative (q-difference) operator for f1∈A is defined by(15)D˜qf1z=1zf1qz−f1q−1zq−q−1,z∈U=1+∑n=1∞n,q˜anzn−1,z≠0,q≠1,D˜qzn=n,q˜zn−1,D˜q∑n=1∞anzn=∑n=1∞n,q˜anzn−1.
We can observe that(16)limq⟶1−D˜qf1z=f1′z.
Here we defineq-symmetric derivative (q-difference) operator for m-fold symmetric analytic functions.Definition 4.
Letfm∈Σm of form (8); then, q-symmetric derivative (q-difference) operator for fm can be defined as(17)D˜qfmz=fmqz−fmq−1zq−q−1z=1+∑k=1∞mk+1,q˜amk+1zmk.
Note that forn∈ℕ=1,2,3,… and z∈U,(18)D˜qzmk+1=mk+1,q˜zmk,D˜q∑k=1∞amk+1zmk+1=∑k=1∞mk+1,q˜amk+1zmk.Recently, Bulut [47] used a Faber polynomial technique on fm∈Σm and investigated some useful results. Here, in this article, we define a new subclass ℛ˜bφ,m,q of m-fold symmetric analytic bi-univalent functions associated with q-symmetric derivative (q-difference) operator. We shall implement a Faber polynomial expansions technique to determine the estimates for the general coefficient bounds amk+1, as well as initial coefficients am+1, a2m+1 and Fekete–Szegő problem for ℛ˜bφ,m,q.Definition 5.
A functionfm∈Σm is said to be in the class ℛ˜bφ,m,q, as fm∈ℛ˜bφ,m,q if and only if(19)1+1bD˜qfmz−1≺φz,1+1bD˜qgmw−1≺φw,where φ∈, b∈ℂ0, z, w∈U, and gmw=fm−1w is defined by (9).Remark 2.
Forq⟶1−, m=1, ℛ˜bφ,m,q=ℛbφ introduced by Hamidi and Jahangiri in [18].Here in this paper, the symmetricq-difference operator for m-fold symmetric functions is defined. Then, by using this newly defined operator, we defined a new subclass ℛbφ,m,q of m-fold symmetric bi-univalent functions in open unit disk U. As in applications of Faber polynomial expansions for fm∈ℛbφ,m,q, we find general coefficient amk+1 for n≥4, Fekete–Szegő problems, initial and coefficients am+1 and a2m+1. Also, q-Bernardi integral operator for m-fold symmetric functions is constructed, and with the help of this operator, we discuss some applications of our main results.


## 2. Main Results

Using the Faber polynomial expansion of functionsf1∈A of form (1), the coefficients of its inverse map g=f1−1 may be expressed as [16](20)g1w=f1−1w=w+∑n=2∞1nKn−1−na2,a3,…wn,where(21)Kn−1−n=−n!−2n+1!n−5!a2n−1+−n!2−n+1!n−3!a2n−3a3+−n!−2n+3!n−4!a2n−4a4+−n!2−n+2!n−5!a2n−5a5+−n+2a32+−n!−2n+5!n−6!a2n−6a6+−2n+5a3a4+∑j≥7a2n−jVj,such that Vj with 7≤j≤n is a homogeneous polynomial in the variables a2,a3,…an [48]. In particular, the first three terms of Kn−1−n are(22)12K1−2=−a2,13K2−3=2a22−a3,14K3−4=−5a23−5a2a3+a4.For more details, see [15, 16, 48].Similarly, Bulut [47] used the Faber polynomial expansion on (8) and obtained the series of the form(23)fmz=z+∑k=1∞amk+1zmk+1=z+∑k=1∞Kk1/ma2,a3,…ak+1zmk+1.The coefficients of its inverse mapgm=fm−1 can be expressed as(24)gmz=fm−1z=w+∑k=1∞1mk+1Kk−mk+1am+1,a2m+1,…amk+1wmk+1.Theorem 1.
Forb∈ℂ∖0, let fm∈ℛ˜bφ,m,q be given by (8); if amj+1=0, 1≤j≤k−1, then(25)amk+1≤2bmk+1,q˜,for k≥2.Proof.
Letfm∈ℛ˜bφ,m,q of form (8); then,(26)1+1bD˜qfmz−1=1+∑k=1∞mk+1,q˜bamk+1zmk,and for its inverse map gm=fm−1, we have(27)1+1bD˜qgmw−1=1+∑k=1∞mk+1,q˜bAmk+1wmk,where Amk+1=1/mk+1Kk−mk+1am+1,a2m+1,…amk+1, k≥1.
On the other hand, sincefm∈ℛ˜bφ,m and gm=fm−1∈ℛ˜bφ,m,q by definitions pz and rw, we have(28)pz=c1zm+c2z2m+…=∑k=1∞ckzmk,rw=d1wm+d2w2m+⋯=∑k=1∞dkwmk,where(29)φpz=1+∑k=1∞∑l=1kφlKklc1,c2,…,ckzmk,(30)φrw=1+∑k=1∞∑l=1kφlKkld1,d2,…,dkwmk.
Equating the coefficients of (26) and (29), we have(31)1bmk+1,q˜amk+1=∑l=1k−1φlKklc1,c2,…,ck.
Similarly, from (27) and (30), we have(32)1bmk+1,q˜Amk+1=∑l=1k−1φlKkld1,d2,…,dk.
Sinceamj+1=0, 1≤j≤k−1, we have(33)Amk+1=−amk+1,(34)1bmk+1,q˜amk+1=φ1ck,(35)1bmk+1,q˜Amk+1=φ1dk.
Taking the absolute values of (34) and (35), we have(36)1bmk+1,q˜amk+1=φ1ck,−1bmk+1,q˜amk+1=φ1dk.
Now using the fact thatφ1≤2, ck≤1, and dk≤1, we have(37)amk+1≤bmk+1,q˜φ1ck=bmk+1,q˜φ1dk,amk+1≤2bmk+1,q˜.
Hence, Theorem1 is complete.Forq⟶1−, m=1, and k=n−1, in Theorem 1, we obtain the following known corollary.Corollary 1.
(see [18]). For b∈ℂ\0, let f∈ℛ˜bφ; if aj+1=0, 1≤j≤n, then(38)an≤2bn,forn≥3.Theorem 2.
Forb∈ℂ\0, let fm∈ℛ˜bφ,m,q be given by (8); then,(39)am+1≤2bm+1,q˜,ifb<8m+1,q˜2m+1,q˜,8bm+1,q˜2m+1,q˜,ifb≥8m+1,q˜2m+1,q˜,a2m+1≤2b2m+1,q˜+2b2m+1,q˜,ifb<22m+1,q˜,4b2m+1,q˜,ifb≥22m+1,q˜,(40)a2m+1−m+1,q˜am+12≤4b2m+1,q˜,(41)a2m+1−m+1,q˜2am+12≤2b2m+1,q˜.Proof.
Takingk=1 in (31) and k=2 in (32), then we have(42)1bm+1,q˜am+1=φ1c1,(43)1b2m+1,q˜a2m+1=φ1c2+φ2c12,(44)−1bm+1,q˜am+1=φ1d1,(45)1b2m+1,q˜m+1,q˜am+12−a2m+1=φ1d2+φ2d12.
From (42) and (44) and using the fact that φ1≤2, ck≤1, and dk≤1, we have(46)am+1≤bm+1,q˜φ1c1=bm+1,q˜φ1d1≤2bm+1,q˜.
Adding (43) and (45), we have(47)am+12=bφ1c2+d2+φ2c12+d12m+1,q˜2m+1,q˜.
Taking modulus on (47), we have(48)am+1≤8bm+1,q˜2m+1,q˜.
Now the boundsam+1 can be justified since(49)b<8bm+1,q˜2m+1,q˜forb<8m+1,q˜2m+1,q˜.
From (43), we have(50)a2m+1=bφ1c2+φ2c122m+1,q˜≤4b2m+1,q˜.
Next we subtract (45) from (43), and we have(51)22m+1,q˜ba2m+1−m+1,q˜2am+12=φ1c2−d2+φ2c12−d12=φ1c2−d2,or(52)a2m+1=m+1,q˜2am+12+φ1bc2−d222m+1,q˜.
After some simple calculation for (52) and taking the absolute values, we get(53)a2m+1≤φ1bc2−d222m+1,q˜+m+1,q˜2am+12.
Using assertion (46) on (53), we have(54)a2m+1≤2b2m+1,q˜+2b2m+1,q˜.
It follows from (50) and (54) that(55)2b2m+1,q˜+2b2m+1,q˜≤2b2m+1,q˜ifb<22m+1,q˜.
Again, we rewrite (45) for the result of (40) as follows:(56)1b2m+1,q˜m+1,q˜am+12−a2m+1=φ1d2+φ2d12.
Taking the absolute value and using the fact thatφ1≤2, ck≤1, and dk≤1, we have(57)a2m+1−m+1,qam+12≤4b2m+1,q˜.
Finally, from (51), we have(58)22m+1,q˜ba2m+1−m+1,q˜2am+12=φ1c2−d2.
Taking the absolute value and using the fact thatφ1≤2, ck≤1, and dk≤1, we have(59)a2m+1−m+1,q˜2am+12≤2b2m+1,q˜.Puttingq⟶1−, m=1, and k=n−1 in Theorem (29), we obtain the following known corollary.Corollary 2.
(see [18]). For b∈ℂ\0, let f∈ℛ˜bφ be given by (1); then,(60)a2≤b,ifb<43,4b3,ifb≥43,a3≤2b3+b2,ifb<23,4b3,ifb≥23,a3−2a22≤4b3,a3−m+12a22≤2b3.


## 3. Applications of the Main Results

In this section, firstly we define theq-Bernardi integral operator ℒfm=ℬ˜β,mq for m-fold symmetric analytic functions and then use it to discuss some applications of our main results.Letfm∈Am of form (8); then, ℒ:Am⟶Am is called the q-Bernardi integral operator for functions defined by ℒfm=ℬ˜β,mq with β>−1, and ℬ˜β,mq is given by(61)ℬ˜β,mqfmz=1+β,q˜zβ∫0ztβ−1fmtdqt,(62)=z+∑k=1∞β+1,q˜mk+1+β,q˜amk+1zmk+1,z∈U=z+∑k=1∞ℬ˜mk+1amk+1zmk+1,where(63)ℬ˜mk+1=β+1,q˜mk+1+β,q˜.Remark 3.
If we takeq⟶1− and m=1, in (61), then we obtain Bernardi integral operator introduced by Bernardi in [49].Theorem 3.
Forb∈ℂ\0, let fm∈ℛ˜bφ,m,q be given by (8); if amj+1=0, 1≤j≤k−1, and(64)ℬ˜β,mqfmz=z+∑k=1∞ℬ˜mk+1amk+1zmk+1,where ℬβ,mqfm is the integral operator given by (61), then(65)amk+1≤2bℬ˜mk+1mk+1,q˜,fork≥2.Proof.
The proof of Theorem3 follows by using (62) and Theorem 1.Theorem 4.
Forb∈ℂ\0, let fm∈ℛ˜bφ,m,q be given by (8); in addition, ℬ˜β,mq is defined by (61) and of form (62); then,(66)am+1≤2bℬ˜m+1m+1,q˜,ifb<8m+1,q˜2m+1,q˜ℬ˜2m+1,8bℬ˜2m+1m+1,q˜2m+1,q˜,ifb≥8m+1,q˜2m+1,q˜ℬ˜2m+1,a2m+1≤2b2m+1,q˜ℬ˜2m+1+2b2ℬ˜m+1m+1,q˜,ifb<22m+1,q˜ℬ˜2m+1,4bℬ˜2m+12m+1,q˜,ifb≥2ℬ˜2m+12m+1,q˜,a2m+1−ℬ˜m+1m+1,q˜am+12≤4bℬ˜2m+12m+1,q˜,a2m+1−ℬ˜m+1m+1,q˜2am+12≤2bℬ˜2m+12m+1,q˜.Proof.
The proof of Theorem4 follows by using (62) and Theorem 2.


## 4. Conclusion

The applications of operators in geometric function theory are quite significant. Many new subclasses of analytic functions have been defined with the help of operators. In our present investigations, we were motivated by the recent research on operator theory and have defined a newq-symmetric derivative (difference) operator for m-fold symmetric functions. With the help of this newly defined operator, we have systematically defined a new subclass ℛ˜bφ,m,q of m-fold symmetric bi-univalent functions. We have then successfully used the Faber polynomial expansion technique to find general coefficient amk+1 for n≥4, Fekete–Szegő problems, and initial coefficients am+1 and a2m+1 for the function f∈ℛ˜bφ,m,q in the open unit disk U. Also, we have defined a q-symmetric Bernardi integral operator for m-fold symmetric functions and have used it to discuss some applications of our main results. In future, researchers can define certain new subclasses related to m-fold symmetric functions associated with (17).



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