Citation: Akhter Rasheed, Saqib Hussain, Syed Ghoos Ali Shah, Maslina Darus, Saeed Lodhi. Majorization problem for two subclasses of meromorphic functions associated with a convolution operator[J]. AIMS Mathematics, 2020, 5(5): 5157-5170. doi: 10.3934/math.2020331
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Let ∑ denote the class of meromorphic function of the form:
λ(ω)=1ω+∞∑t=0atωt, | (1.1) |
which are analytic in the punctured open unit disc U∗={ω:ω∈C and 0<|ω|<1}=U−{0}, where U=U∗∪{0}. Let δ(ω)∈∑, be given by
δ(ω)=1ω+∞∑t=0btωt, | (1.2) |
then the Convolution (Hadamard product) of λ(ω) and δ(ω) is denoted and defined as:
(λ∗δ)(ω)=1ω+∞∑t=0atbtωt. |
In 1967, MacGregor [17] introduced the concept of majorization as follows.
Definition 1. Let λ and δ be analytic in U∗. We say that λ is majorized by δ in U∗ and written as λ(ω)≪δ(ω)ω∈U∗, if there exists a function φ(ω), analytic in U∗, satisfying
|φ(ω)|≤1, and λ(ω)=φ(ω)δ(ω), ω∈U∗. | (1.3) |
In 1970, Robertson [19] gave the idea of quasi-subordination as:
Definition 2. A function λ(ω) is subordinate to δ(ω) in U and written as: λ(ω)≺δ(ω), if there exists a Schwarz function k(ω), which is holomorphic in U∗ with |k(ω)|<1, such that λ(ω)=δ(k(ω)). Furthermore, if the function δ(ω) is univalent in U∗, then we have the following equivalence (see [16]):
λ(ω)≺δ(ω)andλ(U)⊂δ(U). | (1.4) |
Further, λ(ω) is quasi-subordinate to δ(ω) in U∗ and written is
λ(ω)≺qδ(ω) ( ω∈U∗), |
if there exist two analytic functions φ(ω) and k(ω) in U∗ such that λ(ω)φ(ω) is analytic in U∗ and
|φ(ω)|≤1 and k(ω)≤|ω|<1 ω∈U∗, |
satisfying
λ(ω)=φ(ω)δ(k(ω)) ω∈U∗. | (1.5) |
(ⅰ) For φ(ω)=1 in (1.5), we have
λ(ω)=δ(k(ω)) ω∈U∗, |
and we say that the λ function is subordinate to δ in U∗, denoted by (see [20])
λ(ω)≺δ(ω) ( ω∈U∗). |
(ⅱ) If k(ω)=ω, the quasi-subordination (1.5) becomes the majorization given in (1.3). For related work on majorization see [1,4,9,21].
Let us consider the second order linear homogenous differential equation (see, Baricz [6]):
ω2k′′(ω)+αωk′(ω)+[βω2−ν2+(1−α)]k(ω)=0. | (1.6) |
The function kν,α,β(ω), is known as generalized Bessel's function of first kind and is the solution of differential equation given in (1.6)
kν,α,β(ω)=∞∑t=0(−β)tΓ(t+1)Γ(t+ν+1+α+12)(ω2)2t+ν. | (1.7) |
Let us denote
Lν,α,βλ(ω)=2νΓ(ν+α+12)ων2+1kν,α,β(ω12), =1ω+∞∑t=0(−β)t+1Γ(ν+α+12)4t+1Γ(t+2)Γ(t+ν+1+α+12)(ω)t, |
where ν,α and β are positive real numbers. The operator Lν,α,β is a variation of the operator introduced by Deniz [7] (see also Baricz et al. [5]) for analytic functions. By using the convolution, we define the operator Lν,α,β as follows:
( Lν,α,βλ)(ω)=Lν,α,β(ω)∗λ(ω),=1ω+∞∑t=0(−β)t+1Γ(ν+α+12)4t+1Γ(t+2)Γ(t+ν+1+α+12)at(ω)t. | (1.8) |
The operator Lν,α,β was introduced and studied by Mostafa et al. [15] (see also [2]). From (1.8), we have
ω(Lν,α,βλ(ω))j+1=(ν−1+α+12)(Lν−1,α,βλ(ω))j−(ν+α+12)(Lν,α,βλ(ω))j. | (1.9) |
By taking α=β=1, the above operator reduces to ( Lνλ)(ω) studied by Aouf et al. [2].
Definition 3. Let −1≤B<A≤1,η∈C−{0},j∈W and ν,α,β>0. A function λ∈∑ is said to be in the class Mν,jα,β(η,ϰ;A,B) of meromorphic functions of complex order η≠0 in U∗ if and only if
1−1η(ω(Lν,α,βλ(ω))j+1(Lν,α,βλ(ω))j+ν+j)−ϰ|−1η(ω(Lν,α,βλ(ω))j+1(Lν,α,βλ(ω))j+ν+j)|≺1+Aω1+Bω. | (1.10) |
Remark 1.
(i). For A=1,B=−1 and ϰ=0, we denote the class
Mν,jα,β(η,0;1,−1)=Mν,jα,β(η). |
So, λ∈Mν,jα,β(η,ϰ;A,B) if and only if
ℜ[1−1η(ω(Lν,α,βλ(ω))j+1(Lν,α,βλ(ω))j+ν+j)]>0. |
(ii). For α=1,β=1, Mν,j1,1(η,0;1,−1) reduces to the class Mν,j(η).
ℜ[1−1η(ω(Lνλ(ω))j+1(Lνλ(ω))j+ν+j)]>0. |
Definition 4. A function λ∈∑ is said to be in the class Nν,jα,β(θ,b;A,B) of meromorphic spirllike functions of complex order b≠0 in U∗, if and only if
1−eiθbcosθ(ω(Lν,α,βλ(ω))j+1(Lν,α,βλ(ω))j+j+1)≺1+Aω1+Bω, | (1.11) |
where,
(−π2<θ<π2, −1≤β<A≤1,η∈C−{0}, j∈W, ν,α,β>0andω∈U∗ ). |
(i). For A=1 and B=−1, we set
Nν,jα,β(θ,b;1,−1)=Nν,jα,β(θ,b), |
where Nν,jα,β(θ,b) denote the class of functions λ∈∑ satisfying the following inequality:
ℜ[eiθbcosθ(ω(Lν,α,βλ(ω))j+1(Lν,α,βλ(ω))j+j+1)]<1. |
(ii). For θ=0 and α=β=1 we write
Nν,j1,1(0,b;1,−1)=Nν,j(b), |
where Nν,j(b) denote the class of functions λ∈∑ satisfying the following inequality:
ℜ[1b(ω(Lνλ(ω))j+1(Lνλ(ω))j+j+1)]<1. |
A majorization problem for the normalized class of starlike functions has been examined by MacGregor [17] and Altintas et al. [3,4]. Recently, Eljamal et al. [8], Goyal et al. [12,13], Goswami et al. [10,11], Li et al. [14], Tang et al. [21,22] and Prajapat and Aouf [18] generalized these results for different classes of analytic functions.
The objective of this paper is to examined the problems of majorization for the classes Mν,jα,β(η,ϰ;A,B) and Nν,jα,β(θ,b;A,B).
In Theorem 1, we prove majorization property for the class Mν,jα,β(η,ϰ;A,B).
Theorem 1. Let the function λ∈∑ and suppose that δ∈Mν,jα,β(η,ϰ;A,B). If (Lν,α,βλ(ω))j is majorized by (Lν,α,βδ(ω))j in U∗, then
|(Lν,α,βλ(ω))j+1|≤|(Lν,α,βδ(ω))j+1|,(|ω|<r0), | (2.1) |
where r0=r0(η,ϰ,ν,α,β,A,B) is the smallest positive roots of the equation
−ρ(ν−1+α+12)[(A−B)|η|1−ϰ−(α+12)|B|]r3−(ν−1+α+12)[ρ(α+12)+ρ2|B|−|B|]r2−(ν−1+α+12)[(A−B)|η|1−ϰ−(α+12)|B|+ρ2|B|−1]r+ρ(ν−1+α+12)(α+12)=0. | (2.2) |
Proof. Since δ∈Mν,jα,β(η,ϰ;A,B), we have
1−1η(ω(Lν,α,βδ(ω))j+1(Lν,α,βδ(ω))j+ν+j)−ϰ|−1η(ω(Lν,α,βδ(ω))j+1(Lν,α,βδ(ω))j+ν+j)|=1+Ak(ω)1+Bk(ω), | (2.3) |
where k(ω)=c1ω+c2ω2+..., is analytic and bounded functions in U∗ with
|k(ω)|≤|ω| (ω∈U∗). | (2.4) |
Taking
§(ω)=1−1η(ω(Lν,α,βδ(ω))j+1(Lν,α,βδ(ω))j+ν+j), | (2.5) |
In (2.3), we have
§(ω)−ϰ|§(ω)−1|=1+Ak(ω)1+Bk(ω), |
which implies
§(ω)=1+(A−Bϰe−iθ1−ϰe−iθ)k(ω)1+Bk(ω). | (2.6) |
Using (2.6) in (2.5), we get
ω(Lν,α,βδ(ω))j+1(Lν,α,βδ(ω))j=−ν+j+[(A−B)η1−ϰe−iθ+(ν+j)B]k(ω)1+Bk(ω). | (2.7) |
Application of Leibnitz's Theorem on (1.9) gives
ω(Lν,α,βδ(ω))j+1=(ν−1+α+12)(Lν−1,α,βδ(ω))j−(ν+j+α+12)(Lν,α,βδ(ω))j. | (2.8) |
By using (2.8) in (2.7) and making simple calculations, we have
(Lν−1,α,βδ(ω))j(Lν,α,βδ(ω))j=α+12−[(A−B)η1−ϰe−iθ−(α+12)B]k(ω)(1+Bk(ω))(ν−1+α+12). | (2.9) |
Or, equivalently
(Lν,α,βδ(ω))j=(1+Bk(ω))(ν−1+α+12)α+12−[(A−B)η1−ϰe−iθ−(α+12)B]k(ω)(Lν−1,α,βδ(ω))j. | (2.10) |
Since |k(ω)|≤|ω|, (2.10) gives us
|(Lν,α,βδ(ω))j|≤[1+|B||ω|](ν−1+α+12)α+12−|(A−B)η1−ϰe−iθ−(α+12)B||ω||(Lν−1,α,βδ(ω))j|≤[1+|B||ω|](ν−1+α+12)α+12−[(A−B)|η|1−ϰ−(α+12)|B|]|ω||(Lν−1,α,βδ(ω))j| | (2.11) |
Since (Lν,α,βλ(ω))j is majorized by (Lν,α,βδ(ω))j in U∗. So from (1.3), we have
(Lν,α,βλ(ω))j=φ(ω)(Lν,α,βδ(ω))j. | (2.12) |
Differentiating (2.12) with respect to ω then multiplying with ω, we get
(Lν,α,βλ(ω))j=ωφ′(ω)(Lν,α,βδ(ω))j+ωφ(ω)(Lν,α,βδ(ω))j+1. | (2.13) |
By using (2.8), (2.12) and (2.13), we have
(Lν,α,βλ(ω))j+1=1(ν−1+α+12)ωφ′(ω)(Lν,α,βδ(ω))j+φ(ω)(Lν−1,α,βδ(ω))j+1. | (2.14) |
On the other hand, noticing that the Schwarz function φ satisfies the inequality
|φ′(ω)|≤1−|φ(ω)|21−|ω|2 (ω∈U∗). | (2.15) |
Using (2.8) and (2.15) in (2.14), we get
|(Lν,α,βλ(ω))j|≤[|φ(ω)|+ω(1−|φ(ω)|2)[1+|B||ω|](ν−1+α+12)(ν−1+α+12)(1−|ω|2)(α+12−[(A−B)|η|1−ϰ−(α+12)B]|ω|)]×|(Lν−1,α,βδ(ω))j|, |
By taking
|ω|=r, |φ(ω)|=ρ (0≤ρ≤1), |
reduces to the inequality
|(Lν,α,βλ(ω))j|≤Φ1(ρ)(ν−1+α+12)(1−r2)(α+12−[(A−B)|η|1−ϰ−(α+12)B]r)|(Lν−1,α,βδ(ω))j|, |
where
Φ1(ρ)=[ρ(ν−1+α+12)(1−r2)(α+12−[(A−B)|η|1−ϰ−(α+12)B]r)+r(1−ρ2)[1+|B|r](ν−1+α+12)]=−r[1+|B|r](ν−1+α+12)ρ2+ρ(ν−1+α+12)(1−r2)(α+12−[(A−B)|η|1−ϰ−(α+12)B]r)+r[1+|B|r](ν−1+α+12), | (2.16) |
takes in maximum value at ρ=1 with r0=r0(η,ϰ,ν,α,β,A,B) where r0 is the least positive root of the (2.2). Furthermore, if 0≤ξ0≤r0(η,ϰ,ν,α,β,A,B), then the function ψ1(ρ) defined by
ψ1(ρ)=−ξ0[1+|B|ξ0](ν−1+α+12)ρ2+ρ(ν−1+α+12)(1−ξ20)(α+12−[(A−B)|η|1−ϰ−(α+12)B]ξ0)+ξ0[1+|B|ξ0](ν−1+α+12), | (2.17) |
is an increasing function on the interval (0≤ρ≤1), so that
ψ1(ρ)≤ψ1(1)=(ν−1+α+12)(1−ξ20)[α+12−((A−B)|η|1−ϰ−(α+12)B)ξ0](0≤ρ≤1, 0≤ξ0≤r0(η,ϰ,A,B)). |
Hence, upon setting ρ=1 in (2.17), we achieve (2.1).
Special Cases: Let A=1 and B=−1 in Theorem 1, we obtain the following corollary.
Corollary 1. Let the function λ∈∑ and suppose that δ∈Mν,jα,β(η,ϰ;A,B). If (Lν,α,βλ(ω))j is majorized by (Lν,α,βδ(ω))j in U∗, then
|(Lν,α,βλ(ω))j+1|≤|(Lν,α,βδ(ω))j+1|,(|ω|<r1), |
where r1=r1(η,ϰ,ν,α,β) is the least positive roots of the equation
ρ(ν−1+α+12)[2|η|1−ϰ−(α+12)]r3−(ν−1+α+12)[ρ(α+12)+ρ2−1]r2−(ν−1+α+12)[ρ{2|η|1−ϰ−(α+12)}+ρ2−1]r+ρ(ν−1+α+12)(α+12)=0. | (2.18) |
Here, r=−1 is one of the roots (2.18) and the other roots are given by
r1=k0−√k20−4ρ2(ν−1+α+12)[2|η|1−ϰ−(α+12)](ν−1+α+12)(α+12)2ρ(ν−1+α+12)[2|η|1−ϰ−(α+12)], |
where
k0=(ν−1+α+12)[ρ{2|η|1−ϰ−2(α+12)}+ρ2−1]. |
Taking ϰ=0 in corollary 1, we state the following:
Corollary 2. Let the function λ∈∑ and suppose that δ∈Mν,jα,β(η,ϰ;A,B). If (Lν,α,βλ(ω))j is majorized by (Lν,α,βδ(ω))j in U∗, then
|(Lv,α,βλ(ω))j+1|≤|(Lv,α,βδ(ω))j+1|,(|ω|<r2), |
where r2=r2(η,ν,α,β) is the lowest positive roots of the equation
ρ(ν−1+α+12)[2|η|−(α+12)]r3−(ν−1+α+12)[ρ(α+12)+ρ2−1]r2−(ν−1+α+12)[ρ{2|η|−(α+12)}+ρ2−1]r+ρ(ν−1+α+12)(α+12)=0, | (2.19) |
given by
r2=k1−√k21−4ρ2(ν−1+α+12)[2|η|−(α+12)](ν−1+α+12)(α+12)2ρ(ν−1+α+12)[2|η|−(α+12)], |
where
k1=(ν−1+α+12)[ρ{2|η|−2(α+12)}+ρ2−1]. |
Taking α=β=1 in corollary 2, we get the following:
Corollary 3. Let the function λ∈∑ and suppose that δ∈Mν,jα,β(η,ϰ;A,B). If (Lν,α,βλ(ω))j is majorized by (Lν,α,βδ(ω))j in U∗, then
|(Lν,α,βλ(ω))j+1|≤|(Lν,α,βδ(ω))j+1|,(|ω|<r3), |
where r3=r3(η,ν) is the lowest positive roots of the equation
ρν[2|η|−1]r3−ν[ρ+ρ2−1]r2−ν[ρ(2|η|−1)+ρ2−1]r+ρν=0, | (2.20) |
given by
r3=k2−√k22−4ρ2ν[2|η|−1]ν2ρν[2|η|−1], |
where
k2=ν[ρ{2|η|−2}+ρ2−1]. |
Secondly, we exam majorization property for the class Nν,jα,β(θ,b;A,B).
Theorem 2. Let the function λ∈∑ and suppose that δ∈Nν,jα,β(θ,b;A,B). If
(Lν,α,βλ(ω))j≪(Lν,α,βδ(ω))j,(j∈0,1,2...), |
then
|(Lν,α,βλ(ω))j+1|≤|(Lν,α,βδ(ω))j+1|,(|ω|<r4), | (3.1) |
where r4=r4(θ,b,ν,α,β,A,B) is the smallest positive roots of the equation
−ρ[|(B−A)bcosθ+(ν+α+12−1)|B||]r3−[ρ{ν+α+12−1}−|B|(1−ρ2)(ν−1+α+12)]r2+[ρ{|(B−A)bcosθ+(ν+α+12−1)|B||}+(1−ρ2)(ν−1+α+12)]r+ρ[ν+α+12−1]=0,(−π2<θ<π2,−1≤β<A≤1,η∈C−{0},ν,α,β>0,andω∈U∗). | (3.2) |
Proof. Since δ∈Nν,jα,β(θ,b;A,B), so
1−eiθbcosθ(ω(Lν,α,βλ(ω))j+1(Lν,α,βλ(ω))j+j+1)=1+Aω1+Bω, | (3.3) |
where, k(ω) is defined as (2.4).
From (3.3), we have
ω(Lν,α,βδ(ω))j+1(Lν,α,βδ(ω))j=[(B−A)bcosθ−(j+1)Beiθ]k(ω)−(j+1)eiθeiθ(1+Bk(ω)). | (3.4) |
Now, using (2.8) in (3.4) and making simple calculations, we obtain
(Lν−1,α,βδ(ω))j(Lν,α,βδ(ω))j=[(B−A)bcosθ+(ν+α+12−1)Beiθ]k(ω)+[(ν+j+α+12)−1]eiθeiθ(1+Bk(ω))(ν−1+α+12), | (3.5) |
which, in view of |k(ω)|≤|ω| (ω∈U∗), immediately yields the following inequality
|(Lν,α,βδ(ω))j|≤|eiθ|(1+|B||k(ω)|)(ν−1+α+12)[|(B−A)bcosθ+(ν+α+12−1)Beiθ|]|k(ω)|+[(ν+α+12)−1]|eiθ|×|(Lν−1,α,βδ(ω))j|. | (3.6) |
Now, using (2.15) and (3.6) in (2.14) and working on the similar lines as in Theorem 1, we have
|(Lν−1,α,βλ(ω))j|≤[|φ(ω)|+|ω|(1−|φ(ω)|2)(1+|B||ω|)(ν−1+α+12)(1−|ω|2)[{|(B−A)bcosθ+(ν+α+12−1)B|}|ω|+[(ν+α+12)−1]]]×|(Lν−1,α,βδ(ω))j|. |
By setting |ω|=r,|φ(ω)|=ρ(0≤ρ≤1), leads us to the inequality
|(Lν−1,α,βλ(ω))j|≤[Φ2(ρ)(1−r2)[{|(B−A)bcosθ+(ν+α+12−1)B|}r+(ν+α+12)−1]]×|(Lν−1,α,βδ(ω))j|, | (3.7) |
where the function Φ2(ρ) is given by
Φ2(ρ)=ρ(1−r2)[{|(B−A)bcosθ+(ν+α+12−1)B|}r+(ν+α+12)−1]+r(1−ρ2)(1+Br)(ν−1+α+12). |
Φ2(ρ) its maximum value at ρ=1 with r4=r4(θ,b,ν,α,β,A,B) given in (3.2). Moreover if 0≤ξ1≤r4(θ,b,ν,α,β,A,B), then the function.
ψ2(ρ)=ρ(1−ξ21)[{|(B−A)bcosθ+(ν+α+12−1)B|}ξ1+(ν+α+12)−1]+ξ1(1−ρ2)(1+Bξ1)(ν−1+α+12), |
increasing on the interval 0≤ρ≤1, so that ψ2(ρ) does not exceed
ψ2(1)=(1−ξ21)[{|(B−A)bcosθ+(ν+α+12−1)B|}ξ1+(ν+α+12)−1]. |
Therefore, from this fact (3.7) gives the inequality (3.1). We complete the proof.
Special Cases: Let A=1 and B=−1 in Theorem 2, we obtain the following corollary.
Corollary 4. Let the function λ∈∑ and suppose that δ∈Nν,jα,β(θ,b;A,B). If
(Lν,α,βλ(ω))j≪(Lν,α,βδ(ω))j,(j∈0,1,2,...), |
then
|(Lν,α,βλ(ω))j+1|≤|(Lν,α,βδ(ω))j+1|,(|ω|<r5), |
where r5=r5(θ,b,ν,α,β) is the lowest positive roots of the equation
−ρ[|−2bcosθ+(ν+α+12−1)|]r3−[ρ{ν+α+12−1}−(1−ρ2)(ν−1+α+12)]r2+[ρ{|−2bcosθ+(ν+α+12−1)|}+(1−ρ2)(ν−1+α+12)]r+ρ[ν+α+12−1]=0. | (3.8) |
Where r=−1 is first roots and the other two roots are given by
r5=κ0−√κ20+4ρ2[|−2bcosθ+(ν+α+12−1)|][ν+α+12−1]−2ρ[|−2bcosθ+(ν+α+12−1)|], |
and
κ0=[(1−ρ2)(ν−1+α+12)−ρ{|−2bcosθ+2(ν+α+12−1)|}]. |
Which reduces to Corollary 4 for θ=0.
Corollary 5. Let the function λ∈∑ and suppose that δ∈Nν,jα,β(θ,b;A,B). If
(Lν,α,βλ(ω))j≪(Lν,α,βδ(ω))j,(j∈0,1,2,...), |
then
|(Lν,α,βλ(ω))j+1|≤|(Lν,α,βδ(ω))j+1|,(|ω|<r6), |
where r6=r6(b,ν,α,β) is the least positive roots of the equation
−ρ[|−2b+(ν+α+12−1)|]r3−[ρ{ν+α+12−1}−(1−ρ2)(ν−1+α+12)]r2+[ρ{|−2b+(ν+α+12−1)|}+(1−ρ2)(ν−1+α+12)]r+ρ[ν+α+12−1]=0, | (3.9) |
given by
r6=κ1−√κ21+4ρ2[|−2b+(ν+α+12−1)|][ν+α+12−1]−2ρ[|−2b+(ν+α+12−1)|], |
and
κ1=[(1−ρ2)(ν−1+α+12)−ρ{|−2b+2(ν+α+12−1)|}]. |
Taking α=β=1 in corollary 5, we get.
Corollary 6. Let the function λ∈∑ and suppose that δ∈Nν,jα,β(θ,b;A,B). If
(Lν,α,βλ(ω))j≪(Lν,α,βδ(ω))j,(j∈0,1,2,...), |
then
|(Lν,α,βλ(ω))j+1|≤|(Lν,α,βδ(ω))j+1|,(|ω|<r7), |
where r7=r7(b,ν) is the lowest positive roots of the equation
−ρ|−2b+ν|r3−[ρν−(1−ρ2)ν]r2+[ρ|−2b+ν|+(1−ρ2)ν]r+ρ[ν]=0, | (3.10) |
given by
r7=κ2−√κ22+4ρ2[|−2b+ν|][ν]−2ρ[|−2b+ν|], |
and
κ2=[(1−ρ2)ν−ρ{|−2b+2ν|}]. |
In this paper, we explore the problems of majorization for the classes Mν,jα,β(η,ϰ;A,B) and Nν,jα,β(θ,b;A,B) by using a convolution operator Lν,α,β. These results generalizes and unify the theory of majorization which is an active part of current ongoing research in Geometric Function Theory. By specializing different parameters like ν,η,ϰ,θ and b, we obtain a number of important corollaries in Geometric Function Theory.
The work here is supported by GUP-2019-032.
The authors agree with the contents of the manuscript, and there is no conflict of interest among the authors.
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