Research article

Majorization problem for two subclasses of meromorphic functions associated with a convolution operator

  • Received: 03 December 2019 Accepted: 25 May 2020 Published: 15 June 2020
  • MSC : 30C45, 30C50

  • In the present paper, we investigate a majorization problem for the class of meromorphic functions and the class Nν,jα,β(θ,b;A,B) of meromorphic spirllike functions related with a convolution operator. We extend the results existing in literature for higher order derivative. Several consequences of the main results in the form of corollaries are also pointed out.

    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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  • In the present paper, we investigate a majorization problem for the class of meromorphic functions and the class Nν,jα,β(θ,b;A,B) of meromorphic spirllike functions related with a convolution operator. We extend the results existing in literature for higher order derivative. Several consequences of the main results in the form of corollaries are also pointed out.


    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:ωC 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:

    (λδ)(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)ωU, if there exists a function φ(ω), analytic in U, satisfying

    |φ(ω)|1,  and  λ(ω)=φ(ω)δ(ω), ωU. (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)  ( ωU),

    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  ωU,

    satisfying

      λ(ω)=φ(ω)δ(k(ω))  ωU. (1.5)

    (ⅰ) For φ(ω)=1 in (1.5), we have

      λ(ω)=δ(k(ω))  ωU,

    and we say that the λ function is subordinate to δ in U, denoted by (see [20])

    λ(ω)δ(ω)  ( ωU).

    (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)+[βω2ν2+(1α)]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=(ν1+α+12)(Lν1,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 1B<A1,ηC{0},jW 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 η0 in U if and only if

    11η(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, λMν,jα,β(η,ϰ;A,B) if and only if

    [11η(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).

    [11η(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 b0 in U, if and only if

    1eiθbcosθ(ω(Lν,α,βλ(ω))j+1(Lν,α,βλ(ω))j+j+1)1+Aω1+Bω, (1.11)

    where,

    (π2<θ<π2, 1β<A1,ηC{0}, jW, ν,α,β>0andωU ).

    (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:

    [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:

    [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 δMν,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(ν1+α+12)[(AB)|η|1ϰ(α+12)|B|]r3(ν1+α+12)[ρ(α+12)+ρ2|B||B|]r2(ν1+α+12)[(AB)|η|1ϰ(α+12)|B|+ρ2|B|1]r+ρ(ν1+α+12)(α+12)=0. (2.2)

    Proof. Since δMν,jα,β(η,ϰ;A,B), we have

    11η(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|  (ωU). (2.4)

    Taking

    §(ω)=11η(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+(ABϰeiθ1ϰeiθ)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+[(AB)η1ϰeiθ+(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=(ν1+α+12)(Lν1,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

    (Lν1,α,βδ(ω))j(Lν,α,βδ(ω))j=α+12[(AB)η1ϰeiθ(α+12)B]k(ω)(1+Bk(ω))(ν1+α+12). (2.9)

    Or, equivalently

    (Lν,α,βδ(ω))j=(1+Bk(ω))(ν1+α+12)α+12[(AB)η1ϰeiθ(α+12)B]k(ω)(Lν1,α,βδ(ω))j. (2.10)

    Since |k(ω)||ω|, (2.10) gives us

    |(Lν,α,βδ(ω))j|[1+|B||ω|](ν1+α+12)α+12|(AB)η1ϰeiθ(α+12)B||ω||(Lν1,α,βδ(ω))j|[1+|B||ω|](ν1+α+12)α+12[(AB)|η|1ϰ(α+12)|B|]|ω||(Lν1,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(ν1+α+12)ωφ(ω)(Lν,α,βδ(ω))j+φ(ω)(Lν1,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   (ωU). (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|](ν1+α+12)(ν1+α+12)(1|ω|2)(α+12[(AB)|η|1ϰ(α+12)B]|ω|)]×|(Lν1,α,βδ(ω))j|,

    By taking

    |ω|=r,  |φ(ω)|=ρ    (0ρ1),

    reduces to the inequality

    |(Lν,α,βλ(ω))j|Φ1(ρ)(ν1+α+12)(1r2)(α+12[(AB)|η|1ϰ(α+12)B]r)|(Lν1,α,βδ(ω))j|,

    where

    Φ1(ρ)=[ρ(ν1+α+12)(1r2)(α+12[(AB)|η|1ϰ(α+12)B]r)+r(1ρ2)[1+|B|r](ν1+α+12)]=r[1+|B|r](ν1+α+12)ρ2+ρ(ν1+α+12)(1r2)(α+12[(AB)|η|1ϰ(α+12)B]r)+r[1+|B|r](ν1+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 0ξ0r0(η,ϰ,ν,α,β,A,B), then the function ψ1(ρ) defined by

    ψ1(ρ)=ξ0[1+|B|ξ0](ν1+α+12)ρ2+ρ(ν1+α+12)(1ξ20)(α+12[(AB)|η|1ϰ(α+12)B]ξ0)+ξ0[1+|B|ξ0](ν1+α+12),          (2.17)

    is an increasing function on the interval (0ρ1), so that

    ψ1(ρ)ψ1(1)=(ν1+α+12)(1ξ20)[α+12((AB)|η|1ϰ(α+12)B)ξ0](0ρ1, 0ξ0r0(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 δMν,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(ν1+α+12)[2|η|1ϰ(α+12)]r3(ν1+α+12)[ρ(α+12)+ρ21]r2(ν1+α+12)[ρ{2|η|1ϰ(α+12)}+ρ21]r+ρ(ν1+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=k0k204ρ2(ν1+α+12)[2|η|1ϰ(α+12)](ν1+α+12)(α+12)2ρ(ν1+α+12)[2|η|1ϰ(α+12)],

    where

    k0=(ν1+α+12)[ρ{2|η|1ϰ2(α+12)}+ρ21].

    Taking ϰ=0 in corollary 1, we state the following:

    Corollary 2. Let the function λ and suppose that δMν,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(ν1+α+12)[2|η|(α+12)]r3(ν1+α+12)[ρ(α+12)+ρ21]r2(ν1+α+12)[ρ{2|η|(α+12)}+ρ21]r+ρ(ν1+α+12)(α+12)=0, (2.19)

    given by

    r2=k1k214ρ2(ν1+α+12)[2|η|(α+12)](ν1+α+12)(α+12)2ρ(ν1+α+12)[2|η|(α+12)],

    where

    k1=(ν1+α+12)[ρ{2|η|2(α+12)}+ρ21].

    Taking α=β=1 in corollary 2, we get the following:

    Corollary 3. Let the function λ and suppose that δMν,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]r3ν[ρ+ρ21]r2ν[ρ(2|η|1)+ρ21]r+ρν=0, (2.20)

    given by

    r3=k2k224ρ2ν[2|η|1]ν2ρν[2|η|1],

    where

    k2=ν[ρ{2|η|2}+ρ21].

    Secondly, we exam majorization property for the class Nν,jα,β(θ,b;A,B).

    Theorem 2. Let the function λ and suppose that δNν,jα,β(θ,b;A,B). If

    (Lν,α,βλ(ω))j(Lν,α,βδ(ω))j,(j0,1,2...),

    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[|(BA)bcosθ+(ν+α+121)|B||]r3[ρ{ν+α+121}|B|(1ρ2)(ν1+α+12)]r2+[ρ{|(BA)bcosθ+(ν+α+121)|B||}+(1ρ2)(ν1+α+12)]r+ρ[ν+α+121]=0,(π2<θ<π2,1β<A1,ηC{0},ν,α,β>0,andωU). (3.2)

    Proof. Since δNν,jα,β(θ,b;A,B), so

    1eiθbcosθ(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=[(BA)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

    (Lν1,α,βδ(ω))j(Lν,α,βδ(ω))j=[(BA)bcosθ+(ν+α+121)Beiθ]k(ω)+[(ν+j+α+12)1]eiθeiθ(1+Bk(ω))(ν1+α+12), (3.5)

    which, in view of  |k(ω)||ω| (ωU), immediately yields the following inequality

    |(Lν,α,βδ(ω))j||eiθ|(1+|B||k(ω)|)(ν1+α+12)[|(BA)bcosθ+(ν+α+121)Beiθ|]|k(ω)|+[(ν+α+12)1]|eiθ|×|(Lν1,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

    |(Lν1,α,βλ(ω))j|[|φ(ω)|+|ω|(1|φ(ω)|2)(1+|B||ω|)(ν1+α+12)(1|ω|2)[{|(BA)bcosθ+(ν+α+121)B|}|ω|+[(ν+α+12)1]]]×|(Lν1,α,βδ(ω))j|.

    By setting |ω|=r,|φ(ω)|=ρ(0ρ1), leads us to the inequality

    |(Lν1,α,βλ(ω))j|[Φ2(ρ)(1r2)[{|(BA)bcosθ+(ν+α+121)B|}r+(ν+α+12)1]]×|(Lν1,α,βδ(ω))j|, (3.7)

    where the function Φ2(ρ) is given by

    Φ2(ρ)=ρ(1r2)[{|(BA)bcosθ+(ν+α+121)B|}r+(ν+α+12)1]+r(1ρ2)(1+Br)(ν1+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 0ξ1r4(θ,b,ν,α,β,A,B), then the function.

    ψ2(ρ)=ρ(1ξ21)[{|(BA)bcosθ+(ν+α+121)B|}ξ1+(ν+α+12)1]+ξ1(1ρ2)(1+Bξ1)(ν1+α+12),

    increasing on the interval 0ρ1, so that ψ2(ρ) does not exceed

    ψ2(1)=(1ξ21)[{|(BA)bcosθ+(ν+α+121)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 δNν,jα,β(θ,b;A,B). If

    (Lν,α,βλ(ω))j(Lν,α,βδ(ω))j,(j0,1,2,...),

    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+121)|]r3[ρ{ν+α+121}(1ρ2)(ν1+α+12)]r2+[ρ{|2bcosθ+(ν+α+121)|}+(1ρ2)(ν1+α+12)]r+ρ[ν+α+121]=0. (3.8)

    Where r=1 is first roots and the other two roots are given by

    r5=κ0κ20+4ρ2[|2bcosθ+(ν+α+121)|][ν+α+121]2ρ[|2bcosθ+(ν+α+121)|],

    and

    κ0=[(1ρ2)(ν1+α+12)ρ{|2bcosθ+2(ν+α+121)|}].

    Which reduces to Corollary 4 for θ=0.

    Corollary 5. Let the function λ and suppose that δNν,jα,β(θ,b;A,B). If

    (Lν,α,βλ(ω))j(Lν,α,βδ(ω))j,(j0,1,2,...),

    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+121)|]r3[ρ{ν+α+121}(1ρ2)(ν1+α+12)]r2+[ρ{|2b+(ν+α+121)|}+(1ρ2)(ν1+α+12)]r+ρ[ν+α+121]=0, (3.9)

    given by

    r6=κ1κ21+4ρ2[|2b+(ν+α+121)|][ν+α+121]2ρ[|2b+(ν+α+121)|],

    and

    κ1=[(1ρ2)(ν1+α+12)ρ{|2b+2(ν+α+121)|}].

    Taking α=β=1 in corollary 5, we get.

    Corollary 6. Let the function λ and suppose that δNν,jα,β(θ,b;A,B). If

    (Lν,α,βλ(ω))j(Lν,α,βδ(ω))j,(j0,1,2,...),

    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(1ρ2)ν]r2+[ρ|2b+ν|+(1ρ2)ν]r+ρ[ν]=0, (3.10)

    given by

    r7=κ2κ22+4ρ2[|2b+ν|][ν]2ρ[|2b+ν|],

    and

    κ2=[(1ρ2)νρ{|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.



    [1] A. A. A. Abubaker, M. Darus, D. Breaz, Majorization for a subclass of β-spiral functions of order α involving a generalized linear operator, Adv. Decis. Sci., 2011 (2011), 1-9.
    [2] M. K. Aouf, A. O. Mostafa, H. M. Zayed, Convolution properties for some subclasses of meromorphic functions of complex order, Abstr. Appl. Anal., 2015 (2015), 1-6.
    [3] O. Altintas, H. M. Srivastava, Some majorization problems associated with p-valently starlike and convex functions of complex order, East Asian Math. J., 17 (2001), 175-183.
    [4] O. Altintas, O. Ozkan, H. M. Srivastava, Majorization by starlike functions of complex order, Complex Var., 46 (2001), 207-218.
    [5] A. Baricz, E. Deniz, M. Caglar, et al. Differential subordinations involving the generalized Bessel functions, Bull. Malays. Math. Sci. Soc., 38 (2015), 1255-1280. doi: 10.1007/s40840-014-0079-8
    [6] A. Baricz, Generalized Bessel Functions of the First Kind, Springer-Verlag, Berlin, 2010.
    [7] E. Deniz, Differential subordination and superordination results for an operator associated with the generalized Bessel function, Preprint, 2012.
    [8] E. A. Eljamal, M. Darus, Majorization for certain classes of analytic functions defined by a new operator, CUBO, 14 (2012), 119-125. doi: 10.4067/S0719-06462012000100010
    [9] S. P. Goyal, P. Goswami, Majorization for certain classes of meromorphic functions defined by integral operator, Ann. Univ. Mariae Curie Sklodowska Lublin-Polonia., 66 (2016), 57-62.
    [10] P. Goswami, M. K. Aouf, Majorization properties for certain classes of analytic functions using the Salagean operator, Appl. Math. Lett., 23 (2010), 1351-1354. doi: 10.1016/j.aml.2010.06.030
    [11] P. Goswami, Z. G. Wang, Majorization for certain classes of analytic functions, Acta Univ. Apulensis Math. Inform., 21 (2009), 97-104.
    [12] S. P. Goyal, S. K. Bansal, P. Goswami, Majorization for certain classes of analytic functions defined by linear operator using differential subordination, J. Appl. Math. Stat. Inform., 6 (2010), 45-50.
    [13] S. P. Goyal, P. Goswami, Majorization for certain classes of analytic functions defined by fractional derivatives, Appl. Math. Lett., 22 (2009), 1855-1858. doi: 10.1016/j.aml.2009.07.009
    [14] S. H. Li, H. Tang, E. Ao, Majorization properties for certain new classes of analytic functions using the Salagean operator, J. Inequal. Appl., 2013 (2013), 1-8. doi: 10.1186/1029-242X-2013-1
    [15] A. O. Mostafa, M. K. Aouf, H. M. Zayed, Convolution properties for some subclasses of meromorphic bounded functions of complex order, Int. J. Open Probl. Complex Anal., 236 (2016), 1-8.
    [16] S. S. Miller, P. T. Mocanu, Differential Subordinations Theory and Applications, Marcel Dekker, New York, 2000.
    [17] T. H. MacGregor, Majorization by univalent functions, Duke Math. J., 34 (1967), 95-102. doi: 10.1215/S0012-7094-67-03411-4
    [18] J. K. Prajapat, M. K. Aouf, Majorization problem for certain class of p-valently analytic function defined by generalized fractional differintegral operator, Comput. Math. Appl., 63 (2012), 42-47. doi: 10.1016/j.camwa.2011.10.065
    [19] M. S. Robertson, Quasi-subordination and coefficient conjectures, Bull. Am. Math. Soc., 76 (1970), 1-9. doi: 10.1090/S0002-9904-1970-12356-4
    [20] H. M. Srivastava, S. Owa, Current Topics in Analytic Function Theory, World Scientific, Singapore, 1992.
    [21] H. Tang, G. T. Deng, S. H. Li, Majorization properties for certain classes of analytic functions involving a generalized differential operator, J. Math. Res. Appl., 33 (2013), 578-586.
    [22] H. Tang, S. H. Li, G. T. Deng, Majorization properties for a new subclass of θ- spiral functions of order γ, Math. Slovaca, 64 (2014), 1-12. doi: 10.2478/s12175-013-0181-7
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