# Majorization for a Class of Analytic Functions Defined byq-Differentiation

Authors: K. A. Selvakumaran; Sunil Dutt Purohit; Aydin Secer
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

We introduce a new class of multivalent analytic functions defined by usingq-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.


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## Body

## 1. Introduction and Preliminaries

LetAp denote the class of functions f(z) of the form
(1)f(z)=zp+∑n=p+1∞‍anzn,(p∈N={1,2,3,…}),
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
(2)f(z)≪g(z)(z∈U),
if there exists a Schwarz function w(z), analytic in U, such that
(3)|w(z)|≤1,f(z)=w(z)g(z)(z∈U).For the convenience of the reader, we now give some basic definitions and related details ofq-calculus which are used in the sequel.For any complex numberα the q-shifted factorials are defined as
(4)(α;q)0=1,(α;q)n=∏k=0n-1‍(1-αqk),n∈N,
and in terms of the basic analogue of the gamma function
(5)(qα;q)n=Γq(α+n)(1-q)nΓq(α),(n>0),
where the q-gamma function is defined by
(6)Γq(x)=(q,q)∞(1-q)1-x(qx;q)∞,(0<q<1).
If |q|<1, the definition (4) remains meaningful for n=∞ as a convergent infinite product
(7)(α;q)∞=∏j=0∞‍(1-αqj).
In view of the relation
(8)lim⁡q→1-(qα;q)n(1-q)n=(α)n,
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 theq-Gauss hypergeometric function 2Φ1[-] (see Gasper and Rahman [2, p.3, eqn. (1.2.14)]) is defined by
(9)2Φ1[α,β;γ;q,z]=∑n=0∞‍(α;q)n(β;q)n(γ;q)n(q;q)nzn,(|q|<1,|z|<1),
and as a special case of the above series for γ=β, we have
(10)1Φ0[α;-;q,z]=∑n=0∞‍(α;q)n(q;q)nzn,(|q|<1,|z|<1).
Also, the q-derivative and q-integral of a function on a subset of C are, respectively, given by (see [2, pp. 19–22])
(11)Dqf(z)=f(z)-f(zq)(1-q)z,(z≠0,q≠0),∫0z‍f(t)dqt=z(1-q)∑k=0∞‍qkf(zqk).
Therefore, the q-derivative of f(z)=zn, where n is a positive integer, is given by
(12)Dqzn=zn-(zq)n(1-q)z=[n]qzn-1,
where
(13)[n]q=1-qn1-q=qn-1+⋯+1,
and is called the q-analogue of n. As q→1, we have [n]q=qn-1+⋯+1→1+⋯+1=n. Here we list some relations satisfied by [n]q:
(14)[m+n]q=[m]q+qm[n]q=qn[m]q+[n]q,[m-n]q=q-n[m]q-q-n[n]q,[0]q=0,[1]q=1.Recently, many authors have introduced new classes of analytic functions usingq-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 (fractionalq-integral operator).
The fractionalq-integral operator Iq,zδ of a function f(z) of order δ is defined by
(15)Iq,zδf(z)≡Dq,z-δf(z)=1Γq(δ)∫0z‍(z-tq)δ-1f(t)dqt,(δ>0),
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
(16)(z-tq)δ-1=zδ-11Φ0[q-δ+1;-;q,tqδz].
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(-tqδ/z)|<π, |tqδ/z|<1, and |arg(z)|<π.Definition 2 (fractionalq-derivative operator).
The fractionalq-derivative operator Dq,zδ of a function f(z) of order δ is defined by
(17)Dq,zδf(z)≡Dq,zIq,z1-δf(z)=1Γq(1-δ)Dq,z∫0z‍(z-tq)-δf(t)dqt,hhhhh(0≤δ<1),
where f(z) is suitably constrained and the multiplicity of (z-tq)-δ is removed as in Definition 1.Definition 3 (extended fractionalq-derivative operator).
Under the hypotheses of Definition2, the fractional q-derivative for a function f(z) of order δ is defined by
(18)Dq,zδf(z)=Dq,zmIq,zm-δf(z),
where m-1≤δ<1,m∈N0=N∪{0}, and N denotes the set of natural numbers.Remark 4.
It follows from Definition2 that
(19)Dq,zδzn=Γq(n+1)Γq(n+1-δ)zn-δ,(δ≥0,n>-1).
UsingDq,zδ, we define a q-differintegral operator Ωq,pδ:Ap→Ap, as follows:
(20)Ωq,pδf(z)=Γq(p+1-δ)Γq(p+1)zδDq,zδf(z),(-∞<δ<p+1;0<q<1;z∈U),
where Dq,zδf(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
(21)Ωq,pδf(z)=zp+∑n=p+1∞‍Γq(p+1-δ)Γq(n+1)Γq(p+1)Γq(n+1-δ)anzn,(22)qjzDqj+1(Ωq,pδf(z))=qδ[p-δ]qDqj(Ωq,pδ+1f(z))+qj[δ-j]qDqj(Ωq,pδf(z)),h(0≤j≤p;-∞<δ<p;z∈U).Definition 5.
A functionf(z)∈Ap is said to be in the class Sq,pδ,j(b) of p-valent functions of complex order b≠0 in U if and only if
(23)Re{1+1b(zDqj+1(Ωq,pδf(z))Dqj(Ωq,pδf(z))-[p-j]q)}>0,([p-δ]qz∈U;p∈N;j∈N0;b∈C-{0};|2bqj-δ-[p-δ]q|≤[p-δ]q).
It can be seen that, by specializing the parameters, the classSq,pδ,j(b) reduces to many known subclasses of analytic functions. For instance, if q→1 then (1)
S1,10,0(b)=S(b), the class of starlike functions of complex order b (see [14]),(2)
S1,11,0(b)=C(b), the class of convex functions of complex order b (see [15]),(3)
S1,10,0(cos⁡αe-iα)=Sα,(|α|<π/2), the class of α-spiral-like functions (see [16]),(4)
S1,10,0(1-α)=S*(α),(0≤α<1), the class of starlike functions of order α.


## 2. Majorization Problem for the ClassSq,pδ,j(b)

We start by proving the followingq-analogue of the result given by Nehari in [17].Lemma 6.
Iff(z) is analytic and bounded in U, then
(24)|Dq(f(z))|≤1-|f(z)|21-|z|2,(z∈U).Proof.
Iff(z) is bounded in U, then
(25)g(z)=f(z)-f(zq)1-f(zq)¯f(z),(0<q<1)
is also bounded in U. Clearly g(z) vanishes when q=1. Therefore, the function
(26)h(z)=g(z)((z-zq)/(1-zz-q))=(f(z)-f(zq)z-zq)(1-zz-q1-f(zq)¯f(z))
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-zz-q)|=1 for |z|=1; hence by maximum principle, |h(z)|≤1 throughout |z|<1. Then from (26) we have
(27)|Dq(f(z))|(1-|z|21-|f(z)|2)≤1whichimplies|Dq(f(z))|≤1-|f(z)|21-|z|2.Theorem 7.
Let the functionf(z) be in the class Ap and suppose that g(z)∈Sq,pδ,j(b). If Dqj(Ωq,pδf(z)) is majorized by Dqj(Ωq,pδg(z)) in U for j∈N0, then
(28)|Dqj(Ωq,pδ+1f(z))|≤|Dqj(Ωq,pδ+1g(z))|for|z|≤r1,
where
(29)r1=rq(p,j,δ;b)≔k-k2-4[p-δ]q|2bqj-δ-[p-δ]q|2|2bqj-δ-[p-δ]q|(k≔2qj-δ+[p-δ]q+|2bqj-δ-[p-δ]q|;p∈N;b∈C-{0};0<q<1;δ≥0[p-δ]q).Proof.
Let(30)h(z)=1+1b(zDqj+1(Ωq,pδg(z))Dqj(Ωq,pδg(z))-[p-j]q)(z∈U;b∈C-{0};p∈N;j∈N0andj<p).
Since g(z)∈Sq,pδ,j(b), we have Re(h(z))>0(z∈U) and
(31)h(z)=1+w(z)1-w(z),(w∈P),
where P denotes the well known class of bounded analytic functions in U, which satisfy the conditions (cf. [18])
(32)w(0)=0,|w(z)|≤|z|(z∈U).
It follows from (30) and (31) that
(33)zDqj+1(Ωq,pδg(z))Dqj(Ωq,pδg(z))=[p-j]q+(2b-[p-j]q)w(z)1-w(z).
In view of the identity (22), we have the following inequality from (33) by making some simple calculations:
(34)|Dqj(Ωq,pδg(z))|≤[p-δ]q(1+|z|)[p-δ]q-|2bqj-δ-[p-δ]q||z|×|Dqj(Ωq,pδ+1g(z))|.
Since Dqj(Ωq,pδf(z)) is majorized by Dqj(Ωq,pδg(z)) in U, there exists an analytic function φ(z) such that
(35)Dqj(Ωq,pδf(z))=φ(z)Dqj(Ωq,pδg(z)),
and |φ(z)|≤1(z∈U). Applying q-differentiation with respect to z and multiplying by z, we have from (35)
(36)zDqj+1(Ωq,pδf(z))=zDq(φ(z))Dqj(Ωq,pδg(z))+zφ(z)Dqj+1(Ωq,pδg(z)).
Using (22), in the above equation, we get
(37)Dqj(Ωq,pδ+1f(z))=zDq(φ(z))qδ-j[p-δ]qDqj(Ωq,pδg(z))+φ(z)Dqj(Ωq,pδ+1g(z)).
Noting that φ(z) is bounded in U and using Lemma 6 we obtain
(38)|Dq(φ(z))|≤1-|φ(z)|21-|z|2,(z∈U).
Appling (34) and (38) in (37) we get
(39)|Dqj(Ωq,pδ+1f(z))|≤{qj-δ|z|[p-δ]q-|2bqj-δ-[p-δ]q||z|φ(z)+1-|φ(z)|21-|z|·qj-δ|z|[p-δ]q-|2bqj-δ-[p-δ]q||z|}×|Dqj(Ωq,pδ+1g(z))|=-qj-δrρ2+(1-r)([p-δ]q-|2bqj-δ-[p-δ]q|r)ρ+qj-δr(1-r)([p-δ]q-|2bqj-δ-[p-δ]q|r)×|Dqj(Ωq,pδ+1g(z))|(|z|=r,|φ(z)|=ρ,0≤ρ≤1)=Θ(ρ)(1-r)([p-δ]q-|2bqj-δ-[p-δ]q|r)×|Dqj(Ωq,pδ+1g(z))|(z∈U),
where the function Θ(ρ), defined by
(40)Θ(ρ)≔-qj-δrρ2+(1-r)×([p-δ]q-|2bqj-δ-[p-δ]q|r)ρ+qj-δr,hhhhh(0≤ρ≤1),
takes its maximum value at ρ=1 with r=rq(p,j,δ;b) given by (29). Furthermore, if 0≤σ≤rq(p,j,δ;b) where rq(p,j,δ;b) given by (29), then the function
(41)Φ(ρ)≔-qj-δσρ2+(1-σ)×([p-δ]q-|2bqj-δ-[p-δ]q|σ)ρ+qj-δσ
increases in the interval 0≤ρ≤1, so that Φ(ρ) does not exceed
(42)Φ(1)=(1-σ)([p-δ]q-|2bqj-δ-[p-δ]q|σ),hhh(0≤σ≤rq(p,j,δ;b)).
Therefore, from this fact, (39) gives inequality (28).Lettingq→1,p=1,δ=0, and j=0 in Theorem 7, we have the following.Corollary 8 (see [19]).
Let the functionf(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
(43)|f′(z)|≤|g′(z)|for|z|≤r2,
where
(44)r2≔3+|2b-1|-9+2|2b-1|+|2b-1|22|2b-1|.Forb=cos⁡αe-iα, Corollary 8 reduces to the following result.Corollary 9.
Let the functionf(z)∈A be analytic and univalent in the open unit disk U and suppose that g(z)∈Sα(|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 settingb=1, in Corollary 8 we get the following.Corollary 10 (see [1]).
Let the functionf(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
(45)|f′(z)|≤|g′(z)|for|z|≤2-3.



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