Majorization for a Class of Analytic Functions Defined by 𝑞-Differentiation

Abstract

We introduce a new class of multivalent analytic functions defined by using 𝑞-differentiation and fractional 𝑞-calculus operators. Further, we investigate majorization properties for functions belonging to this class. Also, we point out some new and known consequences of our main result.

1. Introduction and Preliminaries

Let 𝒜_𝑝 denote the class of functions 𝑓(𝑧) of the form 𝑓(𝑧)=𝑧^𝑝+^∞_𝑛=𝑝+1𝑎_𝑛𝑧^𝑛,𝑝∈ℕ={1,2,3,…}, (1) which are analytic and 𝑝-valent in the open unit disk 𝒰={𝑧∈ℂ∶|𝑧|<1}. For analytic functions 𝑓(𝑧) and 𝑔(𝑧) in 𝒰, we say that the function 𝑓(𝑧) is majorized by 𝑔(𝑧) in 𝒰 (see [1]) and write 𝑓(𝑧)≪𝑔(𝑧)(𝑧∈𝒰), (2) if there exists a Schwarz function 𝑤(𝑧), analytic in 𝒰, such that |𝑤(𝑧)|≤1,𝑓(𝑧)=𝑤(𝑧)𝑔(𝑧)(𝑧∈𝒰). (3)

For the convenience of the reader, we now give some basic definitions and related details of 𝑞-calculus which are used in the sequel.

For any complex number 𝛼 the 𝑞-shifted factorials are defined as 𝛼;𝑞₀=1,𝛼;𝑞_𝑛=^𝑛−1_𝑘=01−𝛼𝑞^𝑘,𝑛∈ℕ, (4) and in terms of the basic analogue of the gamma function 𝑞^𝛼;𝑞_𝑛=Γ_𝑞(𝛼+𝑛)1−𝑞^𝑛Γ_𝑞(𝛼),(𝑛>0), (5) where the 𝑞-gamma function is defined by Γ_𝑞(𝑥)=𝑞,𝑞_∞1−𝑞^1−𝑥𝑞^𝑥;𝑞_∞,0<𝑞<1. (6) If |𝑞|<1, the definition (4) remains meaningful for 𝑛=∞ as a convergent infinite product 𝛼;𝑞_∞=^∞_𝑗=01−𝛼𝑞^𝑗. (7) In view of the relation lim_𝑞→1−𝑞^𝛼;𝑞_𝑛1−𝑞^𝑛=(𝛼)_𝑛, (8) we observe that the 𝑞-shifted factorial (4) reduces to the familiar Pochhammer symbol (𝛼)_𝑛, where (𝛼)_𝑛=𝛼(𝛼+1)⋯(𝛼+𝑛−1).

It may be noted that the 𝑞-Gauss hypergeometric function ₂Φ₁[−] (see Gasper and Rahman [2, p.3, eqn. (1.2.14)]) is defined by ₂Φ₁𝛼,𝛽;𝛾;𝑞,𝑧=^∞_𝑛=0𝛼;𝑞_𝑛𝛽;𝑞_𝑛𝛾;𝑞_𝑛𝑞;𝑞_𝑛𝑧^𝑛,𝑞<1,|𝑧|<1, (9) and as a special case of the above series for 𝛾=𝛽, we have ₁Φ₀𝛼;−;𝑞,𝑧=^∞_𝑛=0𝛼;𝑞_𝑛𝑞;𝑞_𝑛𝑧^𝑛,𝑞<1,|𝑧|<1. (10) Also, the 𝑞-derivative and 𝑞-integral of a function on a subset of ℂ are, respectively, given by (see [2, pp. 19–22]) 𝐷_𝑞𝑓(𝑧)=𝑓(𝑧)−𝑓𝑧𝑞𝑧,,1−𝑞𝑧≠0,𝑞≠0_𝑧0𝑓(𝑡)𝑑_𝑞𝑡=𝑧1−𝑞^∞_𝑘=0𝑞^𝑘𝑓𝑧𝑞^𝑘. (11) Therefore, the 𝑞-derivative of 𝑓(𝑧)=𝑧^𝑛, where 𝑛 is a positive integer, is given by 𝐷_𝑞𝑧^𝑛=𝑧^𝑛−𝑧𝑞^𝑛𝑧=[𝑛]1−𝑞^𝑞𝑧^𝑛−1, (12) where [𝑛]_𝑞=1−𝑞^𝑛1−𝑞=𝑞^𝑛−1+⋯+1, (13) and is called the 𝑞-analogue of 𝑛. As 𝑞→1, we have [𝑛]_𝑞=𝑞^𝑛−1+⋯+1→1+⋯+1=𝑛. Here we list some relations satisfied by [𝑛]_𝑞: [𝑚+𝑛]_𝑞=[𝑚]_𝑞+𝑞^𝑚[𝑛]_𝑞=𝑞^𝑛[𝑚]_𝑞+[𝑛]_𝑞,[𝑚−𝑛]_𝑞=𝑞^−𝑛[𝑚]_𝑞−𝑞^−𝑛[𝑛]_𝑞,[0]_𝑞=0,[1]_𝑞=1. (14)

Recently, many authors have introduced new classes of analytic functions using 𝑞-calculus operators. For some recent investigations on the classes of analytic functions defined by using 𝑞-calculus operators and related topics, we refer the reader to [3–13] and the references cited therein. In the following, we define the fractional 𝑞-calculus operators of a complex-valued function 𝑓(𝑧), which were recently studied by Purohit and Raina [9].

Definition 1 (fractional 𝑞-integral operator). The fractional 𝑞-integral operator 𝐼^𝛿_𝑞,𝑧 of a function 𝑓(𝑧) of order 𝛿 is defined by 𝐼^𝛿_𝑞,𝑧𝑓(𝑧)≡𝐷^−𝛿_𝑞,𝑧𝑓(𝑧)=1Γ_𝑞(𝛿)^𝑧₀𝑧−𝑡𝑞_𝛿−1𝑓(𝑡)𝑑_𝑞𝑡,(𝛿>0), (15) where 𝑓(𝑧) is analytic in a simply connected region of the 𝑧-plane containing the origin and the 𝑞-binomial function (𝑧−𝑡𝑞)_𝛿−1 is given by 𝑧−𝑡𝑞_𝛿−1=𝑧₁^𝛿−1Φ₀𝑞^−𝛿+1;−;𝑞,𝑡𝑞^𝛿𝑧. (16) The series ₁Φ₀[𝛿;−;𝑞,𝑧] is single valued when |arg(𝑧)|<𝜋 and |𝑧|<1 (see for details [2], pp. 104–106); therefore, the function (𝑧−𝑡𝑞)_𝛿−1 in (15) is single valued when |arg(−𝑡𝑞^𝛿/𝑧)|<𝜋, |𝑡𝑞^𝛿/𝑧|<1, and |arg(𝑧)|<𝜋.

Definition 2 (fractional 𝑞-derivative operator). The fractional 𝑞-derivative operator 𝐷^𝛿_𝑞,𝑧 of a function 𝑓(𝑧) of order 𝛿 is defined by 𝐷^𝛿_𝑞,𝑧𝑓(𝑧)≡𝐷_𝑞,𝑧𝐼^1−𝛿_𝑞,𝑧𝑓(𝑧)=1Γ_𝑞()𝐷1−𝛿^{𝑞,𝑧𝑧}₀𝑧−𝑡𝑞_−𝛿𝑓(𝑡)𝑑_𝑞(),𝑡,0≤𝛿<1 (17) where 𝑓(𝑧) is suitably constrained and the multiplicity of (𝑧−𝑡𝑞)_−𝛿 is removed as in Definition 1.

Definition 3 (extended fractional 𝑞-derivative operator). Under the hypotheses of Definition 2, the fractional 𝑞-derivative for a function 𝑓(𝑧) of order 𝛿 is defined by 𝐷^𝛿_𝑞,𝑧𝑓(𝑧)=𝐷^𝑚_𝑞,𝑧𝐼^𝑚−𝛿_𝑞,𝑧𝑓(𝑧), (18) where 𝑚−1≤𝛿<1,𝑚∈ℕ₀=ℕ∪{0}, and ℕ denotes the set of natural numbers.

Remark 4. It follows from Definition 2 that 𝐷^𝛿_𝑞,𝑧𝑧^𝑛=Γ_𝑞(𝑛+1)Γ_𝑞()𝑧𝑛+1−𝛿^𝑛−𝛿,(𝛿≥0,𝑛>−1). (19)
Using 𝐷^𝛿_𝑞,𝑧, we define a 𝑞-differintegral operator Ω^𝛿_𝑞,𝑝∶𝒜_𝑝→𝒜_𝑝, as follows: Ω^𝛿_𝑞,𝑝𝑓(𝑧)=Γ_𝑞𝑝+1−𝛿Γ_𝑞𝑧𝑝+1^𝛿𝐷^𝛿_𝑞,𝑧𝑓(𝑧),−∞<𝛿<𝑝+1;0<𝑞<1;𝑧∈𝒰, (20) where 𝐷^𝛿_𝑞,𝑧𝑓(𝑧) in (20) represents, respectively, a fractional 𝑞-integral of 𝑓(𝑧) of order 𝛿 when −∞<𝛿<0 and a fractional 𝑞-derivative of 𝑓(𝑧) of order 𝛿 when 0≤𝛿<𝑝+1. It is easy to see from (20) that Ω^𝛿_𝑞,𝑝𝑓(𝑧)=𝑧^𝑝+^∞_𝑛=𝑝+1Γ_𝑞𝑝+1−𝛿Γ_𝑞(𝑛+1)Γ_𝑞𝑝+1Γ_𝑞()𝑎𝑛+1−𝛿^𝑛𝑧^𝑛,𝑞^𝑗𝑧𝐷_𝑞^𝑗+1Ω^𝛿_𝑞,𝑝𝑓(𝑧)=𝑞^𝛿𝑝−𝛿_𝑞𝐷^𝑗_𝑞Ω^𝛿+1_𝑞,𝑝𝑓(𝑧)+𝑞^𝑗𝛿−𝑗_𝑞𝐷^𝑗_𝑞Ω^𝛿_𝑞,𝑝𝑓(𝑧),0≤𝑗≤𝑝;−∞<𝛿<𝑝;𝑧∈𝒰. (21)(22)

Definition 5. A function 𝑓(𝑧)∈𝒜_𝑝 is said to be in the class 𝑆^𝛿,𝑗_𝑞,𝑝(𝑏) of 𝑝-valent functions of complex order 𝑏≠0 in 𝒰 if and only if 1Re1+𝑏𝑧𝐷_𝑞^𝑗+1Ω^𝛿_𝑞,𝑝𝑓(𝑧)𝐷^𝑗_𝑞Ω^𝛿_𝑞,𝑝𝑓(𝑧)−𝑝−𝑗_𝑞>0,𝑧∈𝒰;𝑝∈ℕ;𝑗∈ℕ₀;𝑏∈ℂ−{0};2𝑏𝑞^𝑗−𝛿−𝑝−𝛿_𝑞≤𝑝−𝛿_𝑞. (23)
It can be seen that, by specializing the parameters, the class 𝑆^𝛿,𝑗_𝑞,𝑝(𝑏) reduces to many known subclasses of analytic functions. For instance, if 𝑞→1 then (1)𝒮^0,0_1,1(𝑏)=𝒮(𝑏), the class of starlike functions of complex order 𝑏 (see [14]),(2)𝒮^1,0_1,1(𝑏)=𝒞(𝑏), the class of convex functions of complex order 𝑏 (see [15]),(3)𝒮^0,0_1,1(cos𝛼𝑒^−𝑖𝛼)=𝒮^𝛼,(|𝛼|<𝜋/2), the class of 𝛼-spiral-like functions (see [16]),(4)𝒮^0,0_1,1(1−𝛼)=𝒮^∗(𝛼),(0≤𝛼<1), the class of starlike functions of order 𝛼.

2. Majorization Problem for the Class 𝑆^𝛿,𝑗_𝑞,𝑝(𝑏)

We start by proving the following 𝑞-analogue of the result given by Nehari in [17].

Lemma 6. If 𝑓(𝑧) is analytic and bounded in 𝒰, then 𝐷_𝑞𝑓(𝑧)≤1−𝑓(𝑧)²1−|𝑧|²,(𝑧∈𝒰). (24)

Proof. If 𝑓(𝑧) is bounded in 𝒰, then 𝑔(𝑧)=𝑓(𝑧)−𝑓𝑧𝑞1−𝑓𝑓(𝑧),𝑧𝑞0<𝑞<1 (25) is also bounded in 𝒰. Clearly 𝑔(𝑧) vanishes when 𝑞=1. Therefore, the function ℎ(𝑧)=𝑔(𝑧)𝑧−𝑧𝑞/1−𝑧=𝑓(𝑧)𝑧𝑞−𝑓𝑧𝑞𝑧−𝑧𝑞1−𝑧𝑧𝑞1−𝑓𝑧𝑞𝑓(𝑧) (26) is regular when 𝑞=1 and also at all other points of |𝑧|<1. Furthermore, ℎ(𝑧) is bounded in |𝑧|<1. In fact, lim_|𝑧|→1|𝑔(𝑧)|≤1 and |(𝑧−𝑧𝑞)/(1−𝑧𝑧𝑞)|=1 for |𝑧|=1; hence by maximum principle, |ℎ(𝑧)|≤1 throughout |𝑧|<1. Then from (26) we have 𝐷_𝑞𝑓(𝑧|𝑧|)1−²1−𝑓(𝑧)²≤1whichimplies𝐷_𝑞𝑓(𝑧)≤1−𝑓(𝑧)²1−|𝑧|². (27)

Theorem 7. Let the function 𝑓(𝑧) be in the class 𝒜_𝑝 and suppose that 𝑔(𝑧)∈𝑆^𝛿,𝑗_𝑞,𝑝(𝑏). If 𝐷^𝑗_𝑞(Ω^𝛿_𝑞,𝑝𝑓(𝑧)) is majorized by 𝐷^𝑗_𝑞(Ω^𝛿_𝑞,𝑝𝑔(𝑧)) in 𝒰 for 𝑗∈ℕ₀, then 𝐷^𝑗_𝑞Ω^𝛿+1_𝑞,𝑝𝑓(𝑧)≤𝐷^𝑗_𝑞Ω^𝛿+1_𝑞,𝑝𝑔(𝑧)𝑓𝑜𝑟|𝑧|≤𝑟₁, (28) where 𝑟₁=𝑟_𝑞𝑝,𝑗,𝛿;𝑏∶=𝑘−𝑘²−4𝑝−𝛿_𝑞2𝑏𝑞^𝑗−𝛿−𝑝−𝛿_𝑞22𝑏𝑞^𝑗−𝛿−𝑝−𝛿_𝑞𝑘∶=2𝑞^𝑗−𝛿+𝑝−𝛿_𝑞+2𝑏𝑞^𝑗−𝛿−𝑝−𝛿_𝑞;𝑝∈ℕ;𝑏∈ℂ−{0};0<𝑞<1;𝛿≥0. (29)

Proof. Let ℎ(𝑧)1=1+𝑏𝑧𝐷_𝑞^𝑗+1Ω^𝛿_𝑞,𝑝𝑔(𝑧)𝐷^𝑗_𝑞Ω^𝛿_𝑞,𝑝𝑔(𝑧)−𝑝−𝑗_𝑞𝑧∈𝒰;𝑏∈ℂ−{0};𝑝∈ℕ;𝑗∈ℕ₀and𝑗<𝑝. (30) Since 𝑔(𝑧)∈𝑆^𝛿,𝑗_𝑞,𝑝(𝑏), we have Re(ℎ(𝑧))>0(𝑧∈𝒰) and ℎ(𝑧)=1+𝑤(𝑧)(𝑧),(),1−𝑤𝑤∈𝒫 (31) where 𝒫 denotes the well known class of bounded analytic functions in 𝒰, which satisfy the conditions (cf. [18]) 𝑤(0)=0,|𝑤(𝑧)|≤|𝑧|(𝑧∈𝒰). (32) It follows from (30) and (31) that 𝑧𝐷_𝑞^𝑗+1Ω^𝛿_𝑞,𝑝𝑔(𝑧)𝐷^𝑗_𝑞Ω^𝛿_𝑞,𝑝𝑔(𝑧)=𝑝−𝑗_𝑞+2𝑏−𝑝−𝑗_𝑞𝑤(𝑧)(𝑧).1−𝑤 (33) In view of the identity (22), we have the following inequality from (33) by making some simple calculations: 𝐷^𝑗_𝑞Ω^𝛿_𝑞,𝑝𝑔(𝑧)≤𝑝−𝛿_𝑞(1+|𝑧|)𝑝−𝛿_𝑞−2𝑏𝑞^𝑗−𝛿−𝑝−𝛿_𝑞𝑧|×𝐷_𝑗𝑞|Ω^𝛿+1_𝑞,𝑝𝑔(𝑧). (34) Since 𝐷^𝑗_𝑞(Ω^𝛿_𝑞,𝑝𝑓(𝑧)) is majorized by 𝐷^𝑗_𝑞(Ω^𝛿_𝑞,𝑝𝑔(𝑧)) in 𝒰, there exists an analytic function 𝜑(𝑧) such that 𝐷^𝑗_𝑞Ω^𝛿_𝑞,𝑝𝑓(𝑧)=𝜑(𝑧)𝐷^𝑗_𝑞Ω^𝛿_𝑞,𝑝𝑔(𝑧), (35) and |𝜑(𝑧)|≤1(𝑧∈𝒰). Applying 𝑞-differentiation with respect to 𝑧 and multiplying by 𝑧, we have from (35) 𝑧𝐷_𝑞^𝑗+1Ω^𝛿_𝑞,𝑝𝑓(𝑧)=𝑧𝐷_𝑞𝜑(𝑧)𝐷^𝑗_𝑞Ω^𝛿_𝑞,𝑝𝑔(𝑧)+𝑧𝜑(𝑧)𝐷_𝑞^𝑗+1Ω^𝛿_𝑞,𝑝𝑔(𝑧). (36) Using (22), in the above equation, we get 𝐷^𝑗_𝑞Ω^𝛿+1_𝑞,𝑝𝑓(𝑧)=𝑧𝐷_𝑞𝜑(𝑧)𝑞^𝛿−𝑗𝑝−𝛿_𝑞𝐷^𝑗_𝑞Ω^𝛿_𝑞,𝑝𝑔(𝑧)+𝜑(𝑧)𝐷^𝑗_𝑞Ω^𝛿+1_𝑞,𝑝𝑔(𝑧). (37) Noting that 𝜑(𝑧) is bounded in 𝒰 and using Lemma 6 we obtain 𝐷_𝑞𝜑(𝑧)≤1−𝜑(𝑧)²1−|𝑧|²,(𝑧∈𝒰). (38) Appling (34) and (38) in (37) we get 𝐷^𝑗_𝑞Ω^𝛿+1_𝑞,𝑝𝑓(𝑧)≤𝜑(𝑧)+1−𝜑(𝑧)²|𝑧|⋅𝑞1−_𝑗−𝛿|𝑧|𝑝−𝛿_𝑞−2𝑏𝑞^𝑗−𝛿−𝑝−𝛿_𝑞𝑧|×𝐷_𝑗𝑞|Ω^𝛿+1_𝑞,𝑝𝑔(𝑧)=−𝑞^𝑗−𝛿𝑟𝜌²+(1−𝑟)𝑝−𝛿_𝑞−2𝑏𝑞^𝑗−𝛿−𝑝−𝛿_𝑞𝑟𝜌+𝑞^𝑗−𝛿𝑟(1−𝑟)𝑝−𝛿_𝑞−2𝑏𝑞^𝑗−𝛿−𝑝−𝛿_𝑞𝑟×𝐷^𝑗_𝑞Ω^𝛿+1_𝑞,𝑝𝑔(𝑧)𝑧|𝜑(𝑧=Θ𝜌|=𝑟,)=𝜌,0≤𝜌≤1(1−𝑟)𝑝−𝛿_𝑞−2𝑏𝑞^𝑗−𝛿−𝑝−𝛿_𝑞𝑟×𝐷^𝑗_𝑞Ω^𝛿+1_𝑞,𝑝𝑔(𝑧)(𝑧∈𝒰), (39) where the function Θ(𝜌), defined by Θ𝜌∶=−𝑞^𝑗−𝛿𝑟𝜌²+()×1−𝑟𝑝−𝛿_𝑞−2𝑏𝑞^𝑗−𝛿−𝑝−𝛿_𝑞𝑟𝜌+𝑞^𝑗−𝛿,𝑟,0≤𝜌≤1 (40) takes its maximum value at 𝜌=1 with 𝑟=𝑟_𝑞(𝑝,𝑗,𝛿;𝑏) given by (29). Furthermore, if 0≤𝜎≤𝑟_𝑞(𝑝,𝑗,𝛿;𝑏) where 𝑟_𝑞(𝑝,𝑗,𝛿;𝑏) given by (29), then the function Φ𝜌∶=−𝑞^𝑗−𝛿𝜎𝜌²+()×1−𝜎𝑝−𝛿_𝑞−2𝑏𝑞^𝑗−𝛿−𝑝−𝛿_𝑞𝜎𝜌+𝑞^𝑗−𝛿𝜎 (41) increases in the interval 0≤𝜌≤1, so that Φ(𝜌) does not exceed Φ(1)=(1−𝜎)𝑝−𝛿_𝑞−2𝑏𝑞^𝑗−𝛿−𝑝−𝛿_𝑞𝜎,0≤𝜎≤𝑟_𝑞𝑝,𝑗,𝛿;𝑏. (42) Therefore, from this fact, (39) gives inequality (28).

Letting 𝑞→1,𝑝=1,𝛿=0, and 𝑗=0 in Theorem 7, we have the following.

Corollary 8 (see [19]). Let the function 𝑓(𝑧)∈𝒜 be analytic and univalent in the open unit disk 𝒰 and suppose that 𝑔(𝑧)∈𝒮(𝑏), the class of starlike functions of complex order 𝑏. If 𝑓(𝑧) is majorized by 𝑔(𝑧) in 𝒰, then 𝑓^′(𝑧)≤𝑔^′(𝑧)𝑓𝑜𝑟|𝑧|≤𝑟₂, (43) where 𝑟₂||−∶=3+2𝑏−19+2|2𝑏−1|+|2𝑏−1|²2||.2𝑏−1 (44)

For 𝑏=cos𝛼𝑒^−𝑖𝛼, Corollary 8 reduces to the following result.

Corollary 9. Let the function 𝑓(𝑧)∈𝒜 be analytic and univalent in the open unit disk 𝒰 and suppose that 𝑔(𝑧)∈𝒮^𝛼(|𝛼|<𝜋/2), the class of 𝛼-spiral-like functions. If 𝑓(𝑧) is majorized by 𝑔(𝑧) in 𝒰, then |𝑓^′(𝑧)|≤|𝑔^′(𝑧)| for |𝑧|≤2−√3.

Further setting 𝑏=1, in Corollary 8 we get the following.

Corollary 10 (see [1]). Let the function 𝑓(𝑧)∈𝒜 be analytic and univalent in the open unit disk 𝒰 and suppose that 𝑔(𝑧) is starlike in 𝒰. If 𝑓(𝑧) is majorized by 𝑔(𝑧) in 𝒰, then 𝑓^′(𝑧)≤𝑔^′(𝑧)𝑓𝑜𝑟|𝑧|≤2−√3. (45)

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.