# Fekete–Szegö Problems for Certain Classes of Meromorphic Functions Involvingq-Al-Oboudi Differential Operator

Authors: M.K. Aouf; Fatma Z. El-Emam
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we introduce a new derivative operator involvingq-Al-Oboudi differential operator for meromorphic functions. By using this new operator, we define a new subclass of meromorphic functions and obtain the Fekete–Szegő inequalities.


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## Body

## 1. Introduction

For two analytic functionsf and g in U, we say that fz is subordinate to gz, written f≺g , if there is a Schwarz function wz with w0=0, wz<1z∈U, such that fz=gwz,z∈U. If g is univalent, then(1)f≺gif and only iff0=g0and fU⊂gU,(see [8, 24]). A function f is meromorphic if it is analytic throughout a domain D, except possibly for poles in D (see [40]).Let∑ denote the class of meromorphic functions of the form(2)fz=1z+∑k=0∞akzk,which are analytic in the open punctured unit disc U∗=z∈ℂ:0<z<1=U/0. A function f∈∑ is said to be a meromorphic starlike function of order ζ, denoted by ∑∗ζ, if(3)−ℜzf′zfz>ζ,0≤ζ<1.The class∑∗ζ was studied by Pommerenke [29], Miller [23], and many others (see [9, 25]).Letφz be an analytic function with a positive real part on U satisfying φ0=1 and φ′0>0 which maps U onto a region starlike with respect to 1 and symmetric with respect to the real axis.Let∑b∗φ be the class of functions f∈∑ for which(4)1−1bzf′zfz+1≺φz,b∈ℂ∗=ℂ/0.The class∑b∗φ was introduced and studied by Mohammed and Darus [26] (see also Reddy and Sharma [30], with γ=1).We note that for suitable choices ofb and φz, we obtain the following subclasses:(1)
∑1∗φ=∑∗φ (see [4], with α=1] and [33]);(2)
∑b∗1+z/1−z=F∗b (see [6]);(3)
∑1∗1+1−2ζz/1−z=∑∗ζ,0≤ζ<1 (see [29]);(4)
∑1∗1+Az/1−Az=K1A,B0≤B≤1,−B≤A<B (see [17]);(5)
∑1−ρe−iθcosθ∗1+z/1−z=∑θ∗ρ,0≤ρ<1,θ<π/2 (see [16, 31]).LetMb∗φ be the class of functions f∈∑ for which(5)1−1bzf″zf′z+2≺φz,b∈ℂ∗.We note that(1)
.Mb∗1+z/1−z=G∗b (see Aouf [6]);(2)
M1−ρe−iθcosθ∗φ=Mρ,θ∗φ,0≤ρ<1,θ<π/2=f∈∑:−eiθ1+zf″z/f′z−ρcosθ−isinθ/1−ρcosθ≺φz,z∈U∗.In geometric function theory, operators play an important role. Many authors present differential and integral operators, for example ([1, 20, 32, 37]). For a function f∈∑ given by (2), the δ-derivative of a function fz is defined by [3, 11] (see also [14, 15])(6)Dδ∗fz=fz−fδz1−δz=−1δz2+∑k=0∞kδakzk−1,0<δ<1,where(7)kδ=1−δk1−δ=1+δ+⋯+δk,0<δ<1.Asδ⟶1−, we have kδ⟶k and limδ⟶1−Dδ∗fz=f′z.Due to its use in numerous fields of mathematics and physics, theδ-derivative operator Dδ∗ has fascinated and inspired many researchers. Jackson [14] was among the key contributors of all the scientists who introduced and developed the δ-calculus theory. In 1991, Ismail [13] was the first to demonstrate a crucial link between geometric function theory and the δ-derivative operator, but a solid and comprehensive foundation was provided in 1989 in a book chapter by Srivastava [34]. Several recent works on this operator can be found in ([7, 18, 19, 35, 36]).Forfz∈∑, n∈ℕ0=ℕ∪0,ℕ=1,2,…,λ≥0 and 0<δ<1, we define the following operator Dλ,δ∗n as follows:(8)Dλ,δ∗0fz=fz,Dλ,δ∗1fz=1−λfz+λzDδ∗z2fz⋮Dλ,δ∗nfz=1−λDλ,δ∗n−1fz+λzDδ∗z2Dλ,δ∗n−1fz,n∈ℕ.From (2) and (8), we obtain(9)Dλ,δ∗nfz=1z+∑k=0∞1+λk+2δ−1nakzk,n∈ℕ0=1z+∑k=0∞1+λδδkδ+1nakzk,n∈ℕ0.From (9), it is easy to see that, for fz∈∑,(10)λδ2zDδ∗Dλ,δ∗nfz=Dλ,δ∗n+1fz−λδ+1Dλ,δ∗nfz,λ>0.We note that(i)
limδ⟶1−Dλ,δ∗nfz =Dλ∗nfz=1/z+∑k=0∞1+λk+1nakzk (see [5], with p=1).(ii)
limδ⟶1−D1,δ∗nfz =  D∗nfz=1/z+∑k=0∞k+2nakzk (see [21, 38], with p=1).Making use ofDλ,δ∗n, we define the following class ∑λ,δ,α∗nb,φ as follows:Definition 1.
Forn∈ℕ0,λ≥0,0<δ<1,b∈ℂ∗, and0≤α<δ/δ+1, we say that a functionf∈∑is in the class∑λ,δ,α∗nb,φif and only if(11)1+1b−1−α/δδzDδ∗Dλ,δ∗nfz+αδzDδ∗zDδ∗Dλ,δ∗nfz1−α/δDλ,δ∗nfz−αzDδ∗Dλ,δ∗nfz−1≺φz.Noting that(1)
∑λ,δ,0∗nb,φ=∑λ,δ∗nb,φ=f∈∑:1−1/b(δzDδ∗(Dλ,δ∗nfz)/Dλ,δ∗nfz)+1≺φz,z∈U∗;(2)
limδ⟶1−∑λ,δ,0∗nb,φ=∑λ∗nb,φ=f∈∑:1−1/bzDλ∗nfz′/Dλ∗nfz+1≺φz,z∈U∗;(3)
∑λ,δ,0∗0b,φ=∑δ∗b,φ=f∈∑:1−1/b(δzDδ∗(fz)/fz)+1≺φz,z∈U∗;(4)
∑λ,δ,0∗01,φ=∑δ∗φ (see [12]);(5)
∑λ,δ,0∗n1−ρe−iθcosθ,φ=∑λ,δ∗nρ,θ,φ,0≤ρ<1,θ<π/2=f∈∑:−eiθδzDδ∗Dλ,δ∗nfz/Dλ,δ∗nfz−ρcosθ−isinθ/1−ρcosθ≺φz,z∈U∗;(6)
limδ⟶1−∑λ,δ,0∗01−ρe−iθcosθ,φ=∑∗ρ,θ,φ=f∈∑:−eiθzf′z/fz−ρcosθ−isinθ/1−ρcosθ≺φz,z∈U∗;(7)
∑λ,δ,α∗0b,φ=∑δ,α∗b,φ=f∈∑:1+1/b−1−α/δδzDδ∗fz+αδzDδ∗zDδ∗fz/1−α/δfz−αzDδ∗fz−1≺φz,z∈U∗;(8)
∑λ,δ,α∗01,φ=∑δ,α∗φ (see [12]);(9)
∑λ,δ,α∗n1,φ=∑λ,δ,α∗nφ=f∈∑:−1−α/δδzDδ∗Dλ,δ∗nfz+αδzDδ∗zDδ∗Dλ,δ∗nfz/1−α/δDλ,δ∗nfz−αzDδ∗Dλ,δ∗nfz≺φz,z∈U∗;(10)
limδ⟶1−∑λ,δ,0∗nb,φ=∑λ∗nb,φ=f∈∑:1−1/bzDλ∗nfz′/Dλ∗nfz+1≺φz,z∈U∗;(11)
∑λ,δ,0∗n1,φ=∑λ,δ∗nφ=f∈∑:−δzDδ∗Dλ,δ∗nfz/Dλ,δ∗nfz≺φz,z∈U∗.A Fekete–Szegő inequalitya3−ζa22 is one of the inequalities for coefficients of univalent analytic functions found by Fekete and Szegő (1933) (see [10]), which is related to the Bieberbach conjecture. Providing similar estimates for other classes of functions is called the Fekete–Szegő problem. This problem has been considered by many authors for typical classes of univalent functions (see [2, 39]). In this paper, several properties, such as coefficient inequalities and Fekete–Szegő functionals, for the currently established families are derived.


## 2. Fekete–Szeg Problems

We need the following lemmas, which will be used in our investigation.Lemma 1 (see [22]).
Letpz=1+c1z+c2z2+⋯, be analytic inUand satisfy the following condition(12)ℜpz>0,then for a complex number μ, we have(13)c2−μc12≤2max1;2μ−1.The result is sharp for the functions given by(14)pz=1+z21−z2,pz=1+z1−z.Lemma 2 (see [22]).
Ifhz=1+c1z+c2z2+⋯is a function with positive real part inU, then(15)c2−νc12≤−4ν+2,ifν≤0,2,if 0≤ν≤1,4ν−2,ifν≥1.
Whenν<0 or ν>1, the equality holds if and only if hz is 1+z/1−z or one of its rotations. If 0<ν<1, then the equality holds if and only if hz is 1+z2/1−z2 or one of its rotations. If ν=0, the equality holds if and only if(16)hz=12+β21+z1−z+12−β21−z1+z,0≤β≤1,or one of its rotations. If ν=1, equality holds if and only if(17)1hz=12+β21+z1−z+12−β21−z1+z,0≤β≤1,or one of its rotations. Also, the above upper bound is sharp and it can be improved as follows when 0<ν<1:(18)c2−νc12+νc12≤2,0<ν≤12,c2−νc12+1−νc12≤2,12<ν≤1.Unless otherwise mentioned, we assume throughout this paper thatb∈ℂ∗, 0≤α<δ/δ+1,λ≥0, n∈ℕ0, and 0<δ<1.Theorem 1.
Letfzbe defined by (2) andφz=1+d1z+d2z2+⋯d1≥0. Iffz∈∑λ,δ,α∗nb,φandμ∈ℂ, then(19)a1−μa02≤d1δδ+11+λδδ+1nbδ−αδ+1×max1,d2d1−1−μδδ+1δ−αδ+11+λδδ+1nδ−α21+λδ2nd1b,d1≠0,(20)a1≤δδ+11+λδδ+1nd2bδ−αδ+1,d1=0.
The result is sharp.Proof.
Iffz∈∑λ,δ,α∗nb,φ, then there is a Schwarz function wz in U with w0=0 and wz<1 in U, such that(21)1+1b−1−α/δδzDδ∗Dλ,δ∗nfz+αδzDδ∗zDδ∗Dλ,δ∗nfz1−α/δDλ,δ∗nfz−αzDδ∗Dλ,δ∗nfz−1=φwz.
If we set(22)hz=1+wz1−wz=1+c1z+c2z2+⋯.
Sincewz is a Schwarz function, then ℜhz>0 and h0=1. Let us define(23)pz=1+1b−1−α/δδzDδ∗Dλ,δ∗nfz+αδzDδ∗zDδ∗Dλ,δ∗nfz1−α/δDλ,δ∗nfz−αzDδ∗Dλ,δ∗nfz−1=1+b1z+b2z2+⋯.
Using (21)–(23), we get(24)pz=φhz−1hz+1.
Since(25)hz−1hz+1=12c1z+c2−c122z2+c3+c134−c1c2z3+⋯.
Then,(26)φhz−1hz+1=1+12d1c1z+12d1c2−c122+14d2c12z2+⋯.
From (24) and (26), we have(27)b1=12d1c1,b2=12d1c2−c122+14d2c12.
Then, from (2) and (23), we see that b1=−1/b1−α/δ1+λδna0 and b2=1/b1−α/δ21+λδ2na02−δ+11−α−α/δ1+λδδ+1na1, or, equivalently, we obtain(28)a0=−δd1c1b2δ−α1+λδn,a1=−δd1b2δ+1δ−αδ+11+λδδ+1nc2−c1221−d2d1+d1b.
Therefore(29)a1−μa02=−δd1b2δ+1δ−αδ+11+λδδ+1nc2−νc12,whereν=121−d2d1+d1b1−μδδ+1δ−αδ+11+λδδ+1nδ−α1+λδ2n.
Applying Lemma1, we obtain the result (19). Also, if d1=0, then(30)a0=0,a1=−δd2c12b4δ+1δ−αδ+11+λδδ+1n.
Sinceℜpz>0, then c1≤2 (see [28]), hence(31)a1≤δδ+11+λδδ+1nd2bδ−αδ+1,this proves (20). The result is sharp for the functions(32)1+1b−1−α/δδzDδ∗Dλ,δ∗nfz+αδzDδ∗zDδ∗Dλ,δ∗nfz1−α/δDλ,δ∗nfz−αzDδ∗Dλ,δ∗nfz−1=φz2,1+1b−1−α/δδzDδ∗Dλ,δ∗nfz+αδzDδ∗zDδ∗Dλ,δ∗nfz1−α/δDλ,δ∗nfz−αzDδ∗Dλ,δ∗nfz−1=φz,which completes the proof of Theorem 1.Takingα=0 in Theorem 1, we getCorollary 1.
Letfz be defined by (2) and φz=1+d1z+d2z2+⋯d1≥0. If fz∈∑λ,δ∗nb,φ and μ∈ℂ, then(33)a1−μa02≤d1bδ+11+λδδ+1n⋅max1,d2d1−1−μδ+11+λδδ+1n1+λδ2nd1b,d1≠0,a1≤d2bδ+11+λδδ+1n,d1=0.
The result is sharp.Takingα=n=0 in Theorem 1, we getCorollary 2.
Letfz be defined by (2) and φz=1+d1z+d2z2+⋯d1≥0. If fz∈∑δ∗b,φ and μ∈ℂ, then(34)a1−μa02≤d1bδ+1⋅max1,d2d1−1−μδ+1d1b,d1≠0,a1≤d2bδ+1,d1=0.
The result is sharp.Remark 1.
(1)
Forb=1 in Corollary 2, we get the result obtained by [12], [Theorem 2.3].(2)
Forδ⟶1− in Corollary 2, we get the results obtained by [26, 30].(3)
Forδ⟶1− and b=1 in Corollary 2, we get the results obtained by [33] and [4], [Theorem 5.2].Takingn=0 in Theorem 1, we getCorollary 3.
Letfz be defined by (2) and φz=1+d1z+d2z2+⋯d1≥0. If fz∈∑δ,α∗b,φ and μ∈ℂ, then(35)a1−μa02≤d1δδ+1bδ−αδ+1×max1,d2d1−1−μδδ+1δ−αδ+1δ−α2d1b,d1≠0,a1≤δδ+1d2bδ−αδ+1,d1=0.The result is sharp.Remark 2.
(1)
Takingb=1 in Corollary 3, we get the result obtained by [12], [Theorem 2.8].(2)
Lettingδ⟶1− in Corollary 3, we get the result obtained by [30].Takingb=1 in Theorem 1, we getCorollary 4.
Letfz be defined by (2) and φz=1+d1z+d2z2+⋯d1≥0. If fz∈∑λ,δ,α∗nφ and μ∈ℂ, then(36)a1−μa02≤d1δδ+11+λδδ+1n1δ−αδ+1×max1,d2d1−1−μδδ+1δ−αδ+11+λδδ+1nδ−α21+λδ2nd1,d1≠0,a1≤δδ+11+λδδ+1nd2δ−αδ+1,d1=0.
The result is sharp.Takingα=0 and b=1−ρe−iθcosθ0≤ρ<1,θ<π/2 in Theorem 1, we getCorollary 5.
Letfz be defined by (2) and φz=1+d1z+d2z2+⋯d1≥0. If fz∈∑λ,δ,0∗n1−ρe−iθcosθ,φ and μ∈ℂ, then(37)a1−μa02≤d11−ρcosθδ+11+λδδ+1n×max1,d2d1eiθ−1−μδ+11+λδδ+1n1+λδ2nd11−ρcosθ,d1≠0,a1≤1−ρcosθδ+11+λδδ+1nd2,d1=0.
The result is sharp.Remark 3.
Lettingδ⟶1− and takingn=0 and φz=1+z/1−z in Corollary 5, we get the result obtained by [27], [Example 1.1].By using Lemma2, we can obtain the following theorem.Theorem 2.
For realμ, let φz=1+d1z+d2z2+⋯di>0,i∈1,2. If fz given by (2) belongs to the class ∑λ,δ,α∗n1,φ=∑λ,δ,α∗nφ, then(38)a1−μa02≤δδ+1δ−αδ+11+λδδ+1n×−d2+1−μδδ+1δ−αδ+11+λδδ+1nδ−α21+λδ2nd12,ifμ≤σ1δd1δ+1δ−αδ+11+λδδ+1n,ifσ1≤μ≤σ2δδ+1δ−αδ+11+λδδ+1n×d2−1−μδδ+1δ−αδ+11+λδδ+1nδ−α21+λδ2nd12,ifμ≥σ2,whereσ1=δ−α21+λδ2n−d1−d2+d12δδ+1δ−αδ+11+λδδ+1nd12,σ2=δ−α21+λδ2nd1−d2+d12δδ+1δ−αδ+11+λδδ+1nd12.
The result is sharp. Further, letσ3=δ−α21+λδ2n−d2+d12/δδ+1δ−αδ+11+λδδ+1nd12.(i)
Ifσ1≤μ<σ3, then(39)a1−μa02+δ−α21+λδ2nδδ+1δ−αδ+11+λδδ+1nd12×d1+d2−1−μδδ+1δ−αδ+11+λδδ+1nδ−α21+λδ2nd12a02≤δd1δ+1δ−αδ+11+λδδ+1n.(ii)
Ifσ3≤μ<σ2, then(40)a1−μa02+δ−α21+λδ2nδδ+1δ−αδ+11+λδδ+1nd12×d1−d2+1−μδδ+1δ−αδ+11+λδδ+1nδ−α21+λδ2nd12a02≤δd1δ+1δ−αδ+11+λδδ+1n.Proof.
First, letμ≤σ1. Then(41)a1−μa02≤δd1δ+1δ−αδ+11+λδδ+1n−d2d1+1−μδδ+1δ−αδ+11+λδδ+1nδ−α21+λδ2nd1=δδ+1δ−αδ+11+λδδ+1n−d2+1−μδδ+1δ−αδ+11+λδδ+1nδ−α21+λδ2nd12.
Let,σ1≤μ≤σ2. Then, we obtain(42)a1−μa02≤δd1δ+1δ−αδ+11+λδδ+1n.
Finally, ifμ≥σ2. Then(43)a1−μa02≤δd1δ+1δ−αδ+11+λδδ+1nd2d1−1−μδδ+1δ−αδ+11+λδδ+1nδ−α21+λδ2nd1=δδ+1δ−αδ+11+λδδ+1nd2−1−μδδ+1δ−αδ+11+λδδ+1nδ−α21+λδ2nd12.
The sharpness is an immediate consequence of Lemma2. This completes the proof of Theorem 2.Remark 4.
Takingn=0 in Theorem 2, we get the result obtained by [12], [Theorem 2.10].Takingα=0 in Theorem 2, we getCorollary 6.
For realμ, let φz=1+d1z+d2z2+⋯di>0,i=1,2. If fz given by (2) belongs to the class ∑λ,δ,0∗n1,φ, then(44)a1−μa02≤1δ+11+λδδ+1n×−d2+1−μδ+11+λδδ+1n1+λδ2nd12,ifμ≤σ4d1δ+11+λδδ+1n,ifσ4≤μ≤σ51δ+11+λδδ+1n×d2−1−μδ+11+λδδ+1n1+λδ2nd12,ifμ≥σ5,whereσ4=1+λδ2n−d1−d2+d12δ+11+λδδ+1nd12,σ5=1+λδ2nd1−d2+d12δ+11+λδδ+1nd12.The result is sharp. Further, letσ6=1+λδ2n−d2+d12/δ+11+λδδ+1nd12.(i)
Ifσ4≤μ<σ6, then(45)a1−μa02+1+λδ2nδ+11+λδδ+1nd12×d1+d2−1−μδ+11+λδδ+1n1+λδ2nd12a02≤d1δ+11+λδδ+1n.(ii)
Ifσ6≤μ<σ5, then(46)a1−μa02+1+λδ2nδ+11+λδδ+1nd12×d1−d2+1−μδ+11+λδδ+1n1+λδ2nd12a02≤d1δ+11+λδδ+1n.Remark 5.
(1)
Takingn=0 in Corollary 6, we get the result obtained by [12].(2)
Lettingδ⟶1− and takingn=0 in Corollary 6, we get the result obtained by [4].


## 3. Conclusion

In the fields of combinatorics and quantum calculus, theδ-derivative introduced by Frank Hilton Jackson [14] plays an important role in the theory of functions of a complex variable and other fields of mathematics. In this paper, we define a new differential operator for meromorphic functions. By using this new operator, we define and study a new family of meromorphic functions. Several properties of the abovementioned family of functions are investigated, including coefficient inequalities and Fekete–Szegő functionals.



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