# Faber Polynomial Coefficient Bounds form-Fold Symmetric Analytic and Bi-univalent Functions Involving q-Calculus

Authors: Zeya Jia; Shahid Khan; Nazar Khan; Bilal Khan; Muhammad Asif
Journal: Journal of Function Spaces
 (2021)

---

## Abstract

In our present investigation, by applyingq-calculus operator theory, we define some new subclasses of m-fold symmetric analytic and bi-univalent functions in the open unit disk U=z∈ℂ:z<1 and use the Faber polynomial expansion to find upper bounds of amk+1 and initial coefficient bounds for am+1 and a2m+1 as well as Fekete-Szego inequalities for the functions belonging to newly defined subclasses. Also, we highlight some new and known corollaries of our main results.


---

## Body

## 1. Introduction, Definitions, and Motivation

LetA denote the class of all analytic functions fz in the open unit disk U=z:z<1 and have the series expansion of the form
(1)fz=z+∑n=2∞anzn.ByS, we mean the subclass of A consisting of univalent functions. The inverse f−1 of univalent function f can be defined as
(2)f−1fz=z,z∈U,ff−1w=w,w<r0f,r0f≥14,where
(3)g1w=f−1w=w−a2w2+2a22−a3w3−5a23−5a2a3+a4w4+.⋯According to the Koebe one-quarter theorem [1], an analytic function f is called bi-univalent in U if both f and f−1 are univalent in U. Let Σ denote the class all bi-univalent functions in U. For f∈Σ, Lewin [2] showed that a2<1.51 and Brannan and Cluni [3] proved that a2≤2. Netanyahu [4] showed that maxa2=4/3. Brannan and Taha [5] introduced a certain subclass of bi-univalent functions for class Σ. In recent years, Srivastava et al. [6], Frasin and Aouf [7], Altinkaya and Yalcin [8, 9], and Hayami and Owa [10] studied the various subclasses of analytic and bi-univalent function. For a brief history, see [11].In [12], Faber introduced Faber polynomials, and after that, Gong [13] studied Faber polynomials in geometric function theory. In their published works, some contributions have been made to finding the general coefficient bounds ∣an∣ by applying Faber polynomial expansions. By using Faber polynomial expansions, very little work has been done for the coefficient bounds an for n≥4 of Maclaurin’s series. For more studies, see [14–17].A domainU is said to be m-fold symmetric if
(4)fei2π/mz=ei2π/mfz,z∈U,f∈A,m∈ℕ.The univalent functionhz maps the unit disk U into a region with m-fold symmetry and can be defined as
(5)hz=fzmm,f∈S.A functionf is said to be m-fold symmetric [18] if it has the series expansion of the form
(6)fz=z+∑k=1∞amk+1zmk+1.The class of allm-fold symmetric univalent functions is denoted by Sm, and for m=1, then Sm=S.In [19], Srivastava et al. proved the inverse fm−1 series expansion for f∈Σm, which is given as follows:
(7)gw=fm−1w=w−am+1wm+1+m+1am+12−a2m+1w2m+1−12m+13m+2am+13−3m+2am+1a2m+1+a3m+1w3m+1+.⋯Here, we will denotem-fold symmetric bi-univalent functions by Σm. For m=1, equation (7) coincides with equation (3) of the class Σ. The coefficient problem for f∈Σm is one of the favorite subjects of geometric function theory in these days (see [20–23]).The quantum (orq-) calculus has great importance because of its applications in several fields of mathematics, physics, and some related areas. The importance of q-derivative operator Dq is pretty recognizable by its applications in the study of numerous subclasses of analytic functions. Initially, in 1908, Jackson [24] introduced a q-derivative operator and studied its applications. Further, in [25], Ismail et al. defined a class of q-starlike functions; after that, Srivastava [26] studied q-calculus in the context of univalent function theory; also, numerous mathematicians studied q-calculus in the context of univalent function theory. Further, the q-analogue of the Ruscheweyh differential operator was defined by Kanas and Raducanu [27] and Arif et al. [28] discussed some of its applications for multivalent functions while Zhang et al. in [29] studied q-starlike functions related with the generalized conic domain. Srivastava et al. published the articles (see [30, 31]) in which they studied the class of q-starlike functions. For some more recent investigations about q-calculus, we may refer to [32–34].For a better understanding of the article, we recall some concept details and definitions of theq-difference calculus. Throughout the article, we presume that
(8)0<q<1.Definition 1.
Theq-factorial nq! is defined as
(9)nq!=∏k=1nkqn∈ℕ,and the q-generalized Pochhammer symbol tn,q, t∈ℂ, is defined as
(10)tn,q=tqt+1qt+2q⋯t+n−1qn∈ℕ.Remark 2.
Forn=0, then nq!=1, and tn,q=1.Definition 3.
Theq-number tq for q∈0,1 is defined as
(11)tq=1−qt1−qt∈ℂ,∑k=0n−1qkt=n∈ℕ.Definition 4 (see [24]).
Theq-derivative (or q-difference) operator Dq of a function fis defined, in a given subset of ℂ, by
(12)Dqfz=fz−fqz1−qz,z≠0,f′0,z=0,provided that f′0 exists.From Definition4, we can observe that
(13)limq⟶1−Dqfz=limq⟶1−fz−fqz1−qz=f′z,for a differentiable function f in a given subset of ℂ. It is also known from (1) and (12) that
(14)Dqfz=1+∑n=2∞nqanzn−1.Here, in this paper, we use theq-difference operator to define new subclasses of m-fold symmetric analytic and bi-univalent functions and then apply the Faber polynomial expansion technique to determine the general coefficient bounds amk+1 and initial coefficient bounds am+1 and a2m+1 as well as Fekete-Szego inequalities.Definition 5.
A functionf∈Σm is said to be in the class Rbφ,m,q if and only if
(15)1+1bDqfz−1≺φz,1+1bDqgw−1≺φw,where φ∈P, b∈ℂ\0, andz,w∈U, andgw=fm−1w is defined by (7).Remark 6.
Forq⟶1− and m=1, then the class Rbφ,m,q reduces into the class Rbφ introduced by Hamidi and Jahangiri in [35].Definition 7.
A functionf∈Σm is said to be in the class SΣm∗φ,q if and only if
(16)zDqfzfz≺φz,wDqgwgw≺φw,where φ∈P, b∈ℂ\0, andz,w∈U, andgw=fm−1w is defined by (7).Remark 8.
Forq⟶1−, m=1, and φz=1+Az/1+Bz, then the class SΣm∗φ,q reduces into the class SA,B, introduced by Hamidi and Jahangiri in [36].


## 2. Main Results

Using the Faber polynomial expansion of functionsf∈A of the form (1), the coefficients of its inverse map g=f−1 may be expressed as [15] given by
(17)gw=f−1w=w+∑n=2∞1nKn−1−na2,a3,⋯wn,for an expansion of Kn−1−n (see [37]). In particular, the first three terms of Kn−1−n are
(18)12K1−2=−a2,13K2−3=2a22−a3,14K3−4=−5a23−5a2a3+a4.In general, for anyp∈ℕ and n≥2, an expansion of Kn−1p is as (see [15])
(19)Kn−1p=pan+pp−12En−12+p!p−3!3!En−13+⋯+p!p−n+1!n−1!En−1n−1,where En−1p=En−1pa2,a3,⋯, and by [37],
(20)En−1ma2,⋯,an=∑n=2∞m!a2μ1⋯anμn−1μ1!,⋯,μn−1!,form≤n,while a1=1, and the sum is taken over all nonnegative integers μ1,⋯,μn satisfying
(21)μ1+μ2+⋯+μn=m,μ1+2μ2+⋯+n−1μn−1=n−1.Evidently,En−1n−1a2,⋯,an=a2n−1 (see [14]), or equivalently,
(22)Enma1,a2,⋯,an=∑n=1∞m!a1μ1⋯anμnμ1!,⋯,μn!,form≤n,while a1=1, and the sum is taken over all nonnegative integers μ1,⋯,μn satisfying
(23)μ1+μ2+⋯+μn=m,μ1+2μ2+⋯+nμn=n.It is clear thatEnna1,⋯,an=E1n, and the first and last polynomials are Enn=a1n and En1=an.Similarly, using the Faber polynomial expansion of functionsf∈A of the form (6), that is,
(24)fz=z+∑k=1∞Kk1/ma2,a3,⋯,ak+1zmk+1.The coefficients of its inverse mapg=fm−1 may be expressed as
(25)gz=fm−1z=w+∑k=1∞1mk+1Kk−mk+1am+1,a2m+1,⋯,amk+1wmk+1.Theorem 9.
Forb∈ℂ\0, let f∈Rbφ,m,qbe given by (6), and ifamj+1=0, 1≤j≤k−1, then
(26)amk+1≤2b1+mk,fork≥2.Proof.
By definition, for the functionf∈Rbφ,m,q of the form (6), we have
(27)1+1bDqfz−1=1+∑k=1∞1+mkqbamk+1zmk,and for its inverse map g=fm−1, we have
(28)1+1bDqgw−1=1+∑k=1∞1+mkqbAmk+1wmk,where
(29)Amk+1=1mk+1Kk−mk+1am+1,a2m+1,⋯,amk+1,k≥1.
On the other hand, sincef∈Rbφ,m,q and g=fm−1∈Rbφ,m,q by definition, we have
(30)pz=c1zm+c2z2m+⋯=∑k=1∞ckzmk,qw=d1wm+d2w2m+⋯=∑k=1∞dkwmk,where
(31)φpz=1+∑k=1∞∑l=1kφlKklc1,c2,⋯,ckzmk,(32)φqw=1+∑k=1∞∑l=1kφlKkld1,d2,⋯,dkwmk.
Comparing the coefficients of (27) and (31), we have
(33)1b1+mkqamk+1=∑l=1k−1φlKklc1,c2,⋯,ck.
Similarly, comparing coefficients of (28) and (32), we have
(34)1b1+mkqAmk+1=∑l=1k−1φlKkld1,d2,⋯,dk.
Note that foramj+1=0, 1≤j≤k−1, we have
(35)Amk+1=−amk+1,and so
(36)1b1+mkqamk+1=φ1ck,(37)−1b1+mkqamk+1=φ1dk.
Now taking the absolute of (36) and (37) and using the fact that φ1≤2, ck≤1, and dk≤1, we have
(38)amk+1≤b1+mkqφ1ck=b1+mkqφ1dk,amk+1≤2b1+mkq,which completes the proof of Theorem 9.Form=1 and k=n−1, in Theorem 9, we obtain the following corollary.Corollary 10.
Forb∈ℂ\0, let f∈Rbφ,q, and ifaj+1=0, 1≤j≤n, then
(39)an≤2bnq,forn≥3.Forq⟶1−, m=1, and k=n−1, in Theorem 9, we obtain the following known corollary.Corollary 11 (see [35]).
Forb∈ℂ\0, let f∈Rbφ, and ifaj+1=0, 1≤j≤n, then
(40)an≤2bn,forn≥3.Theorem 12.
Forb∈ℂ\0, let f∈Bbφ,m,q be given by (6), and then
(41)am+1≤2bm+1q,ifb<8m+12m+1q,8bm+12m+1q,ifb≥8m+12m+1q,a2m+1≤2b2m+1q+2m+1b2m+1q2,ifb<22m+1q,4b2m+1q,ifb≥22m+1q,a2m+1−m+1am+12≤4b2m+1q,a2m+1−m+12am+12≤2b2m+1q.Proof.
Replacingk by 1 and 2 in (33) and (34), respectively, we have
(42)1bm+1qam+1=φ1c1,(43)1b2m+1qa2m+1=φ1c2+φ2c12,(44)−1bm+1qam+1=φ1d1,(45)1b2m+1qm+1am+12−a2m+1=φ1d2+φ2d12.
From (42) and (44), we have
(46)am+1≤bm+1qφ1c1=bm+1qφ1d1≤2bm+1q.
Adding (43) and (45), we have
(47)am+12=bφ1c2+d2+φ2c12+d12m+12m+1q.
Taking the absolute value (47), we have
(48)am+1≤8bm+12m+1q.
Now, the bounds given foram+1 can be justified since
(49)b<8bm+12m+1q,forb<8m+12m+1q.
From (43), we have
(50)a2m+1=bφ1c2+φ2c122m+1q≤4b2m+1q.
Next, we subtract (45) from (43), and we have
(51)22m+1qba2m+1−m+12am+12=φ1c2−d2+φ2c12−d12=φ1c2−d2,or
(52)a2m+1=m+12am+12+φ1bc2−d222m+1q.
After some simple calculation and by taking the absolute, we have(53)a2m+1≤φ1bc2−d222m+1+m+12am+12.
Using the assertion (46), we have
(54)a2m+1≤2b2m+1q+2m+1b2m+1q2.
From (50) and (54), we note that
(55)2b2m+1q+2m+1b2m+1q2≤4b2m+1q,ifb<22m+1q.
Now, we rewrite (45) as
(56)1b2m+1qm+1am+12−a2m+1=φ1d2+φ2d12.
Taking the absolute value, we have(57)a2m+1−m+1am+12≤4b2m+1q.
Finally, from (51), we have
(58)22m+1qba2m+1−m+12am+12=φ1c2−d2.
Taking the absolute value, we have(59)a2m+1−m+12am+12≤2b2m+1q.Form=1 and k=n−1, in Theorem 12, we obtain the following corollary.Corollary 13.
Forb∈ℂ\0, let f∈Bbφ,q be given by (1), and then
(60)a2≤2b2q,ifb<43q,4b3q,ifb≥43q,a3≤2b3q+4b22q2,ifb<23q,4b3q,ifb≥23q,a3−2am+12≤4b3q,a2m+1−am+12≤2b3q.Forq⟶1−, m=1, and k=n−1, in Theorem 12, we obtain the following corollary.Corollary 14 (see [35]).
Forb∈ℂ\0, let f∈Bbφ be given by (1), and then
(61)a2≤b,ifb<43,4b3,ifb≥43,a3≤2b3+b2,ifb<23,4b3,ifb≥23,a3−2a22≤4b3,a3−a22≤2b3.Theorem 15.
Letf∈SΣm∗φ,qbe given by (6), and ifamj+1=0, 1≤j≤k−1, then
(62)amk+1≤2mkq,fork≥2.Proof.
By definition, for the functionf∈SΣm∗φ,q of the form (6), we have
(63)zDqfzfz=1−∑k=1∞Fkam+1,a2m+1,⋯,amk+1zmk,where the first few coefficients of Fkam+1,a2m+1,⋯,amk+1 are
(64)F1=−am+1,F2=am+12−m+1a2m+1,F3=−am+13+2m+1am+1a2m+1−2m+1a3m+1.
In general,(65)Fkam+1,a2m+1,⋯,amk+1=∑i1+2i2+⋯kink=kAi1,i2,i2,⋯,ikam+1i1a2m+1i2⋯amk+1ik,where
(66)Ai1,i2,i2,⋯,ik=−1k+2i1+⋯k+1iki1+i2+i2⋯+ik−1!ki1!i2!⋯ik!.
For the inverse mapg=fm−1∈SΣm∗φ,q, we obtain
(67)zDqgwgw=1−∑k=1∞Fkbm+1,b2m+1,⋯,bmk+1wmk,
where(68)Amk+1=1mk+1Kk−mk+1am+1,a2m+1,⋯,amk+1,k≥1.
On the other hand, sincef∈SΣm∗φ,q and g=fm−1∈SΣm∗φ,q by definition, we have
(69)pz=c1zm+c2z2m+⋯=∑k=1∞ckzmk,qw=d1wm+d2w2m+⋯=∑k=1∞dkwmk,where
(70)φpz=1+∑k=1∞∑l=1kφlKklc1,c2,⋯,ckzmk,(71)φqw=1+∑k=1∞∑l=1kφlKkld1,d2,⋯,dkwmk.
Comparing the coefficients of (63) and (70), we have
(72)−mkqamk+1=∑l=1k−1φlKklc1,c2,⋯,ck.
Similarly, comparing the coefficients of (67) and (71), we have
(73)−mkqbmk+1=∑l=1k−1φlKkld1,d2,⋯,dk.
Note that foramj+1=0, 1≤j≤k−1, we have
(74)Amk+1=−amk+1,and so
(75)−mkqamk+1=φ1ck,(76)mkqamk+1=φ1dk.
Taking the absolute values of (75) and (76) and using the fact that φ1≤2, ck≤1, and dk≤1, we have
(77)amk+1≤1mkqφ1ck=1mkqφ1dk,amk+1≤2mkq.
Hence, Theorem15 is complete.Forq⟶1−, m=1, and k=n−1, in Theorem 15, we obtain the following corollary.Corollary 16.
f∈S∗φ, and ifaj+1=0, 1≤j≤n, then
(78)an≤2n−1,forn≥3.Theorem 17.
Letf∈SΣm∗φ,qbe given by (6), and then
(79)am+1≤2mq,a2m+1≤4m+1m2mq+22mq,a2m+1−mq2m+12mqam+12≤42mq,a2m+1−mqm+12mqam+12≤22mq.Proof.
Replacingk by 1 and 2 in (72) and (73), respectively, we have
(80)mqam+1=φ1c1,(81)2mqa2m+1−mqam+12=φ1c2+φ2c12,(82)−mqam+1=φ1d1,(83)mq2m+1am+12−2mqa2m+1=φ1d2+φ2d12.
From (80) and (82), we have
(84)am+1≤1mqφ1c1=1mqφ1d1≤2mq.
Adding (81) and (83), we have
(85)am+12=φ1c2+d2+φ2c12+d122mmq.
Taking the absolute value (85), we have
(86)am+1≤2mmq.
Next, we subtract (83) from (81), and we have
(87)22mqa2m+1−2mqm+1am+12=φ1c2−d2+φ2c12−d12,or
(88)a2m+1=mqm+12mqam+12+φ1c2−d222mq.
After some simple calculation of (88) and by taking the absolute, we have
(89)a2m+1≤φ1c2−d222mq+mqm+12mqam+12.
Using the assertion (86), we have
(90)a2m+1≤4m+1m2mq+22mq.
For the third part, we rewrite (83) as
(91)mq2m+1am+12−2mqa2m+1=φ1d2+φ2d12.
Taking the absolute value, we have(92)a2m+1−mq2m+12mqam+12≤42mq.
Finally, from (87), we have
(93)22mqa2m+1−mqm+12mqam+12=φ1c2−d2.
Taking the absolute value, we have(94)a2m+1−mqm+12mqam+12≤22mq.Forq⟶1−, m=1, and k=n−1, in Theorem 17, we get the following corollary.Corollary 18.
Letf∈S∗φ be given by (1), and then
(95)a2≤2,a3≤5,a3−32a22≤2,a3−a22≤1.


## 3. Conclusion

In this paper, we have appliedq-calculus operator theory to define some new subclasses of m-fold symmetric analytic and bi-univalent functions in open unit disk U and used the Faber polynomial expansion to find upper bounds amk+1 and initial coefficient bounds am+1 and a2m+1 as well as Fekete-Szego inequalities for the functions belonging to newly defined subclasses of m-fold symmetric analytic and bi-univalent function. Also, we highlighted some new and known consequences of our main results.



---

*Source: 5232247-2021-10-26.xml*