Research article

Cost and performance of carbon risk in socially responsible mutual funds

  • Received: 31 January 2023 Revised: 22 February 2023 Accepted: 24 February 2023 Published: 06 March 2023
  • JEL Codes: G11, G17, G23, G2, N20, Q56

  • Investors and other financial actors are attracted by the role of socially responsible (SR) mutual funds in the transition to a low-carbon economy. In response to the demand for more information, Morningstar reported the level of carbon risk of funds by using the following indicators: Carbon Risk, Carbon Management, Carbon Operations risk and Carbon Exposure. Dealing with a sample of 3370 equity SR mutual funds worldwide from 2017 to 2021, this study analyzes the relationships between these indicators and the expense ratio and performance of the funds. In general, the results point to funds with lower carbon scores that have lower fees and perform better than those with higher scores. Considering the effects of the COVID-19 crisis, this evidence holds true for most of the sample period analyzed. With a spatial analysis, although the evidence generally holds, regional differences are found. Thus, funds that invest in the USA and Canada are on average cheaper and show lower carbon scores, while funds that are oriented to other areas, such as emerging markets, are more expensive and show higher scores. In summary, there is good news for the utility function of the investor and the planet: Green investing is cheaper and better.

    Citation: Juan Carlos Matallín-Sáez, Amparo Soler-Domínguez. Cost and performance of carbon risk in socially responsible mutual funds[J]. Quantitative Finance and Economics, 2023, 7(1): 50-73. doi: 10.3934/QFE.2023003

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  • Investors and other financial actors are attracted by the role of socially responsible (SR) mutual funds in the transition to a low-carbon economy. In response to the demand for more information, Morningstar reported the level of carbon risk of funds by using the following indicators: Carbon Risk, Carbon Management, Carbon Operations risk and Carbon Exposure. Dealing with a sample of 3370 equity SR mutual funds worldwide from 2017 to 2021, this study analyzes the relationships between these indicators and the expense ratio and performance of the funds. In general, the results point to funds with lower carbon scores that have lower fees and perform better than those with higher scores. Considering the effects of the COVID-19 crisis, this evidence holds true for most of the sample period analyzed. With a spatial analysis, although the evidence generally holds, regional differences are found. Thus, funds that invest in the USA and Canada are on average cheaper and show lower carbon scores, while funds that are oriented to other areas, such as emerging markets, are more expensive and show higher scores. In summary, there is good news for the utility function of the investor and the planet: Green investing is cheaper and better.



    The q-calculus (Quantum Calculus) is a branch of mathematics related to calculus in which the concept of limit is replaced by the parameter q. This field of study has motivated the researchers in the recent past with its numerous applications in applied sciences like Physics and Mathematics, e.g., optimal control problems, the field of ordinary fractional calculus, q-transform analysis, q-difference and q-integral equations. The applications of q-generalization in special functions and quantum physics are of high value which makes the study pertinent and interesting in these fields. While the q-difference operator has a vital importance in the theory of special functions and quantum theory, number theory, statistical mechanics, etc. The q-generalization of the concepts of differentiation and integrations were introduced and studied by Jackson [1]. Similarly, Aral and Gupta [2,3] used some what similar concept by introducing the q-analogue of operator of Baskakov Durrmeyer by using q-beta function. Later, Aral and Anastassiu et. al. in [4,5] generalized some complex operators, q-Gauss-Weierstrass singular integral and q-Picard operators. For more details on the topic one can see, for example [6,7,8,9,10,11,12,13,14,15,16,17]. Some of latest inovations in the field can be seen in the work of Arif et al. [18] in which they investigated the q-generalization of Harmonic starlike functions. While Srivastava with his co-authors in [19,20] investigated some general families in q-analogue related to Janowski functions and obatained some interesting results. Later, Shafiq et al. [21] extended this idea of generalization to close to convex functions. Recently, more research seem to have diversified this field with the introduction of operator theory. Some of the details of such work can be seen in the work of Shi and co-authors [22]. Also some new domains have been explored such as Sine domain in the recent work [23]. Motivated from the discussion above we utilize the concepts of q-calculus and introduce a subclass of p-valent meromorphic functions and investigate some of their nice geometric properties.

    Before going into our main results we give some basic concepts relating to our work.

    Let Mp represents the class of meromorphic multivalent functions which are analytic in D={zC:0<|z|<1} with the representation

    f(z)=1zp+k=p+1akzk, (zD). (1.1)

    Let f(z) and g(z) be analytic in D={zC:|z|<1}. Then the function f(z) is subordinated to g(z) in D, written as f(z)g(z), zD, if there exist a Schwarz function ω(z) such that f(z)=g(ω(z)), where ω(z) is analytic in D, with w(0)=0 and w(z)<1, zD.

    Let P denote the class of analytic function l(z) normalized by

    l(z)=1+n=1pnzn (1.2)

    such that Re(z)>0.

    We now consider a class of functions in the domain of lemniscate of Bernoulli. All functions l(z) will belong to such a class if it satisfy;

    h(z)1+z. (1.3)

    These functions lie in the right-half of the lemniscate of Bernoulli and with this geometrical representation is the reason behind this name.

    With simple calculations the above can be written as

    |(h(z))21|<1.

    Similarly SL, in parallel comparison to starlike functions, for analytic functions is

    SL={f(z)A:zf(z)f(z)1+z} (1.4)

    where A represents the class of analytic functions and zD. Alternatively

    SL={f(z)A:|(zf(z)f(z))21|<1},

    Sokol and Stankiewicz [24] introduced this alongwith some properties. Further study on this was made by different authors in [25,26,27]. Upper bounds for the coefficients of this class are evaluated in [28].

    An important problem in the field of analytic functions is to study a functional |a3va22| called the Fekete-Szegö functional. Where a2 and a3 the coefficients of the original function with a parameter v over which the extremal value of the functional is evaluated. The problem of obtaining the upper bound of this functional for subclasses of normalized functions is called the Fekete-Szegö problem or inequality. M. Fekete and G. Szegö [29], were the first to estimate this classical functional for the class S. While Pfluger [30] utilized Jenkin's method to prove that this result holds for complex μ such that Reμ1μ0. For other related material on the topic reader is reffered to [31,32,33].

    Similarly the class of Janowski functions is defined for the function J(z) with 1B<A1

    J(z)1+Az1+Bz

    equivalently the functions of this class satisfies

    |J(z)1ABJ(z)|<1

    more details on Janowski functions can be seen in [34].

    The q-derivative, also known as the q-difference operator, for a function is

    Dqf(z)=f(qz)f(z)z(q1), (1.5)

    with z0 and 0<q<1. With simple calculations for nN and zD, one can see that

    Dq{n=1anzn}=n=1[n]qanzn1, (1.6)

    with

    [n]q=1qn1q=1+n1l=1ql and  [0]q=0.

    Now we define our new class and we discuss the problem of Fekete-Szegö for this class. Some geometric properties of this class related to subordinations are discussed in connection with Janowski functions.

    We introduce MSLp,q, a family of meromorphic multivalent functions associated with the domain of lemniscate of Bernoulli in q-analogue as:

    If f(z)Mp, then it will be in the class MSLp,q if the following holds

    qpzDqf(z)[p]qf(z)1+z, (1.7)

    we note that limq1MSLp,q=MSLp, where

    MSLp={f(z)Mp:zf(z)pf(z)1+zzD}.

    In this research article we investigate some properties of meromorphic multivalent functions in association with lemniscate of Bernoulli in q-analogue. The important inequality of Fekete-Szegö is evaluated in the beginning of main results. Then we evaluate some bounds of ξ which associate 1+ξzp+1Dqf(z)[p]q,1+ξzDqf(z)[p]qf(z),1+ξz1pDqf(z)[p]q(f(z))2 and 1+ξz12pDqf(z)[p]q(f(z))3 with Janowski functions and zpf(z)1+z. Utilizing these theorems along with some conditions we prove that a function may be a member of MSLp,q.

    The following Lemmas are important as they help in our main results.

    [35]. If l(z) is in P given by (1.2), then

    |p2λp21|2 max{1;|2λ1|}νC.

    [35]. If l(z) is in P given by (1.2), then

    |p2νp21|{4ν+2 (ν0),2            (0ν1)4ν2     (ν1).

    [36]. (q-Jack's lemma) For an analytic function ω(z) in U={zC:|z|<1} with ω(0)=0. If |ω(z)| attains its maximum value on the circle |z|=r at a point z0=reiθ, for θ[π,π], we can write that for 0<q<1

    z0Dqω(z0)=mω(z0),

    with m is real and m1.

    In this section we start with Fekete-Szegö problem in the first two theorems. Then some important results relating to subordination are proved using q-Jack's Lemma and with the help of these results the functions are shown to be in the class of MSLp,q in the form of some corollaries.

    Let fMSLp,q and are of the form (1.1), then

    |ap+2λa2p+1|[p]q2([p+2]q[p]q)max{1,|μ|},

    where

    μ=(qp([p+1]q)2+3qp([p]q)22λ([p]q)2+2λ[p+2]q[p]q4qp[p+1]q[p]q)4qp([p+2]q[p]q).

    Proof. Let f MSLp,q, then we have

    qpzDqf(z)[p]qf(z)=1+ω(z),  (3.1)

    where  |ω(z)|1 and ω(0)=0.Let

    Φ(ω(z))=1+ω(z).

    Thus for

    l(z)=1+p1z+p2z2+=1+ω(z)1ω(z), (3.2)

    we have l(z)  is in P and

    ω(z)=p1z+p2z2+p3z3+2+p1z+p2z2+p3z3+=l(z)1l(z)+1.

    Now as

    2l(z)l(z)+1=1+14p1z+(14p2532p21)z2+.

    So from (3.1), we get

    qpzDqf(z)=[p]q2l(z)l(z)+1 f(z),

    thus

    [p]qzpqpk=p+1[k]qakzk=
         [p]q(1+14p1z+(14p2532p21)z2+)([p]qzp+k=p+1akzk)

    By comparing of coefficients of zk+p, we get

    ap+1=[p]q4qp([p+1]q[p]q)p1, (3.3)
    ap+2=[p]qqp([p+2]q[p]q)(14p25[p+1]q7[p]q32([p+1]q[p]q)p21). (3.4)

    Form (3.3) and (3.4)

    |ap+2λa2p+1|=
    [p]q4qp([p+2]q[p]q)|p25qp([p+1]q)212qp[p+1]q[p]q+7qp([p]q)22λ([p]q)2+2[p+2]q[p]qλ8qp([p+1]q[p]q)2p21|,

    Using Lemma 2.1

    |ap+2λa2p+1|[p]q2([p+2]q[p]q)max{1,|μ|},

    with μ is defined as above.

    If fMSLp,q and of the form (1.1), then

    |ap+2λa2p+1|
    {γ4qp(αγ)qpβ2+4qpβγ3qpγ22qp(βγ)2+γ4qp(αγ)2αγ2γ22qp(βγ)2λ,     5qpβ212qpβγ+7qpγ22λγ2+2αγλ8qp(βγ)20γ2qp(αγ),                                            05qpβ212qpβγ+7qpγ22λγ2+2αγλ8qp(βγ)21γ4qp(αγ)qpβ24qpβγ+3qpγ22qp(βγ)2γ4qp(αγ)2αγ2γ22qp(βγ)2λ,       5qpβ212qpβγ+7qpγ22λγ2+2αγλ8qp(βγ)21,

    where λR,α=[p+2]q,β=[p+1]q and γ=[p]q.

    Proof. From (3.3) and (3.4) it follows that

    ap+2λa2p+1=[p]q4qp([p+2]q[p]q)(p2
    (5qp([p+1]q)212qp[p+1]q[p]q+7qp([p]q)22λ([p]q)2+2[p+2]q[p]qλ8qp([p+1]q[p]q)2)p21),

    using above notations, we get

    ap+2λa2p+1=γ4qp(αγ)(p25qpβ212qpβγ+7qpγ22λγ2+2αγλ8qp(βγ)2p21).

    Let v=5qpβ212qpβγ+7qpγ22λγ2+2αγλ8qp(βγ)20, using Lemma 2.2, we have

    |ap+2λa2p+1|γ4qp(αγ)[4(5qpβ212qpβγ+7qpγ22λγ2+2αγλ8qp(βγ)2)+2]γ4qp(αγ)qpβ2+4qpβγ3qpγ22qp(βγ)2+γ4(αγ)2αγ2γ22qp(βγ)2λ.

    Let v=5qpβ212qpβγ+7qpγ22λγ2+2αγλ8qp(βγ)2, where v[0,1] using Lemma 2.2, we get the second inequality. Now for v=5qpβ212qpβγ+7qpγ22λγ2+2αγλ8qp(βγ)21, using Lemma 2.2, we have

    |ap+2λa2p+1|γ4qp(αγ)[4(5β212βγ+7γ2λγ28(βγ)2)2]
    γ4qp(αγ)qpβ24qpβγ+3qpγ22qp(βγ)2γ4qp(αγ)2αγ2γ22qp(βγ)2λ,

    and hence the proof.

    If f(z)Mp, then for 1B<A1 with

    |ξ|232(AB)[p]q1|B|4p(1+|B|), (3.5)

    and if

    1+ξzp+1Dqf(z)[p]q1+Az1+Bz, (3.6)

    holds, then

    zpf(z)1+z.

    Proof. Suppose that

    J(z)=1+ξzp+1Dqf(z)[p]q (3.7)

    and consider

    zpf(z)=1+ω(z). (3.8)

    Now to prove the required result it will be enough if we prove that |ω(z)|<1.

    Using (3.7) and (3.8)

    J(z)=1+ξ[p]q(zDqω(z)21+ω(z)p1+ω(z))

    and so

    |J(z)1ABJ(z)|=|ξ[p]q(zDqω(z)21+ω(z)p1+ω(z))AB(1+ξ[p]q(zDqω(z)21+ω(z)p1+ω(z)))|
    =|ξzDqω(z)2pξ(1+ω(z))2[p]q(AB)1+ω(z)B(ξzDqω(z)2pξ(1+ω(z)))|

    Now if ω(z) attains its maximum value at some z=z0, which is |ω(z0)|=1. Then by Lemma 2.3, with m1 we have,ω(z0)=eiθ and z0Dqω(z0)=mω(z0), with θ[π,π] so

    |J(z0)1ABJ(z0)|=|ξ(mω(z0)2p(1+ω(z0)))2(AB)[p]q1+ω(z0)B(ξ(mω(z0)2p(1+ω(z0))))||ξ|(m2p(|1+eiθ|))2(AB)[p]q|1+eiθ|+|B|(|ξ|(m2p(|1+eiθ|)))=|ξ|(m2p2+2cosθ)2(AB)[p]q(2+2cosθ)14+|B|(|ξ|(m2p2+2cosθ))|ξ|(m4p)|B||ξ|(m+4p)+232(AB)[p]q.

    Consider

    ϕ(m)=|ξ|(m4p)|B||ξ|(m+4p)+232(AB)[p]qϕ(m)=8p|ξ|2|B|+232|ξ|(AB)[p]q(|B||ξ|(m+4p)+232(AB)[p]q)2>0,

    showing the increasing behavior of ϕ(m) so minimum of ϕ(m) will be at m=1 with

    |J(z0)1ABJ(z0)||ξ|(14p)232(AB)[p]q+|B||ξ|(1+4p),

    so from(3.5)

    |J(z0)1ABJ(z0)|1

    contradicting (3.6), thus |ω(z)|<1 and so we get the desired result.

    Let 1B<A1 and f(z)Mp. If

    |ξ|232[p]q(AB)1|B|4(1+|B|)p,

    and

    1(1p+zD2qf(z)Dqf(z)zDqf(z)f(z))ξzDqf(z)[p]2qf(z)1+Az1+Bz, (3.9)

    then f(z)MSLp,q.

    Proof. Suppose that

    l(z)=qpz1pDqf(z)[p]qf(z). (3.10)

    From (3.10) it follows that

    zp+1Dql(z)=(1p+zD2qf(z)Dqf(z)zDqf(z)f(z))zDqf(z)[p]2qf(z),

    Using the condition (3.9),we have

    1ξzp+1Dql(z)1+Az1+Bz.

    Now using Theorem 3.3, we get

    zpl(z)=qpzDqf(z)[p]qf(z)1+z,

    thus f(z)MSLp.

    Let 1B<A1 and f(z)Mp. If

    |ξ|4[p]q(AB)1|B|4(1+|B|)p (3.11)

    and

    1+ξzDqf(z)[p]qf(z)1+Az1+Bz, (3.12)

    then

    zpf(z)1+z.

    Proof. We define a function

    J(z)=1+ξzDqf(z)[p]qf(z). (3.13)

    Now as

    zpf(z)=1+ω(z) (3.14)

    Using Logarithmic differentiation on (3.14), from (3.13) we obtain that

    J(z)=1+ξ[p]q(zDqω(z)2(1+ω(z))p)

    and so

    |J(z)1ABJ(z)|=|ξ[p]q(zDqω(z)2(1+ω(z))p)AB(1+ξ[p]q(zDqω(z)2(1+ω(z))p))|
    =|ξ(zDqω(z)2p(1+ω(z)))2(AB)[p]q(1+ω(z))B(ξzDqω(z)2pξ(1+ω(z)))|.

    If at some z=z0,ω(z) attains its maximum value i.e. |ω(z0)|=1. Then using Lemma 2.3, we have

    |J(z0)1ABJ(z0)|=|ξ(mω(z0)2p(1+ω(z0)))2(AB)[p]q(1+ω(z0))B(ξmω(z0)2pξ(1+ω(z0)))||ξ|(m2p|1+eiθ|)2(AB)[p]q|1+eiθ|+|ξ||B|(m+2p|1+eiθ|)=|ξ|m2p|ξ|2+2cosθ2((AB)[p]q+p|B||ξ|)2+2cosθ+|ξ||B|m|ξ|(m4p)4((AB)[p]q+p|B||ξ|)+|ξ||B|m.

    Now let

    ϕ(m)=|ξ|(m4p)4((AB)[p]q+p |ξ||B|)+|B||ξ|mϕ(m)=|ξ|(8p|B|+4(AB)[p]q)(4((AB)[p]q+p|B||ξ|)+|B||ξ|m)2>0,

    which shows that the increasing nature of ϕ(m) and so its minimum value will be at m=1 thus

    |J(z0)1ABJ(z0)|(14p)|ξ|4(p|ξ||B|+(AB)[p]q)+|B||ξ|,

    hence by(3.11)

    |J(z0)1ABJ(z0)|1,

    which contradicts (3.12), therefore |ω(z)|<1 and so the desired result.

    Let 1B<A1 and f(z)Mp. If

    |ξ|4[p]q(AB)1|B|4(1+|B|p),

    and

    1(1p+zD2qf(z)Dqf(z)zDqf(z)f(z))ξ[p]q1+Az1+Bz,

    then f(z)MSLp,q.

    Let 1B<A1 and f(z)Mp. If

    |ξ|252[p]q(AB)1|B|4(1+|B|)p (3.15)

    and

    1+ξz1pDqf(z)[p]q(f(z))21+Az1+Bz,

    then zpf(z)1+z.

    Proof. Here we define a function

    J(z)=1+ξz1pDqf(z)[p]qf2(z).

    So if

    zpf(z)=1+ω(z),

    using some simplification we obtain that

    J(z)=1+ξ[p]q(zDqω(z)2(1+ω(z))32p1+ω(z)),

    and so

    |J(z)1ABJ(z)|=|ξ[p]q(zDqω(z)2(1+ω(z))32p1+ω(z))AB(1+ξ[p]q(zDqω(z)2(1+ω(z))32p1+ω(z)))|
    =|ξ(zDqω(z)2p(1+ω(z)))2(AB)[p]q(1+ω(z))32+2pξB(1+ω(z))BξzDqω(z)|.

    Now if ω(z) attains, at some z=z0, its maximum value which is |ω(z0)|=1. Then by Lemma 2.3, with m1 we have,ω(z0)=eiθ and z0Dqω(z0)=mω(z0), with θ[π,π] so

    |J(z0)1ABJ(z0)|=|ξ(mω(z0)2p(1+ω(z0)))2(AB)[p]q(1+ω(z0))32+2pξB(1+ω(z0))Bξmω(z0)||ξ|m2p|ξ||1+eiθ|2(AB)[p]q|1+eiθ|32+|B||ξ|m+2p|ξ||B||1+eiθ|=|ξ|m2p|ξ|2+2cosθ2(AB)[p]q(2+2cosθ)34+|B||ξ|m+2p|ξ||B|2+2cosθ(m4p)|ξ|252(AB)[p]q+4p|ξ||B|+|B||ξ|m.

    Now let

    ϕ(m)=|ξ|(m4p)252(AB)[p]q+|B||ξ|(m+4p)ϕ(m)=|ξ|(252(AB)+8p|ξ||B|)(252(AB)[p]q+4p|ξ||B|+|B||ξ|m)2>0,

    this shows ϕ(m) an increasing function which implies that at m=1 it will have its minimum value and

    |J(z0)1ABJ(z0)|(14p)|ξ|252(AB)[p]q+|B||ξ|+4p|ξ||B|,

    now by (3.15) we have

    |J(z0)1ABJ(z0)|1,

    this is a contradiction as J(z)1+Az1+Bz, thus |ω(z)|<1 and so the result.

    Let 1B<A1 and f(z)Mp. If

    |ξ|252[p]q(AB)1|B|4(1+|B|)p,

    and

    1(1p+zD2qf(z)Dqf(z)zDqf(z)f(z))ξf(z)zDqf(z)1+Az1+Bz,

    then f(z)MSLp,q.

    Let 1B<A1 and f(z)Mp. If

    |ξ|8[p]q(AB)1|B|4(1+|B|)p (3.16)

    and

    1+ξz12pDqf(z)[p]q(f(z))31+Az1+Bz, (3.17)

    then zpf(z)1+z.

    Proof. Suppose that

    J(z)=1+ξz12pDqf(z)[p]q(f(z))3.

    Now if

    zpf(z)=1+ω(z),

    with simple calculations we can easily obtain

    J(z)=1+ξ[p]q(1+ω(z))(zDqω(z)2[p]q(1+ω(z))p),

    and so

    |J(z)1ABJ(z)|=|ξ[p]q(1+ω(z))(zDqω(z)2[p]q(1+ω(z))p)AB(1+ξ[p]q(1+ω(z))(zDqω(z)2[p]q(1+ω(z))p))|
    =|ξ(zDqω(z)2p(1+ω(z)))2(AB)[p]q(1+ω(z))2+2pξB(1+ω(z))BξzDqω(z)|,

    if at some z=z0,ω(z) attains its maximum value i.e. |ω(z0)|=1. Then using Lemma 2.3,

    |J(z0)1ABJ(z0)|=|ξ(mω(z0)2p(1+ω(z0)))2(AB)[p]q(1+ω(z0))22pξB(1+ω(z0))+Bξmω(z0)||ξ|(m2p|1+eiθ|)2(AB)[p]q|1+eiθ|2+2p|ξ||B||1+eiθ|+|B||ξ|m=|ξ|(m2p2+2cosθ)2(AB)[p]q(2+2cosθ)+2p|ξ||B|2+2cosθ+|B||ξ|m(m4p)|ξ|8(AB)[p]q+|B||ξ|(m+4p).

    Now let

    ϕ(m)=(m4p)|ξ|8(AB)[p]q+|B||ξ|(m+4p)ϕ(m)=8|ξ|(AB)[p]q+8p|ξ|2|B|(8(AB)[p]q+|B||ξ|m+4p|ξ||B|)2>0

    which shows that the increasing nature of ϕ(m) and so its minimum value will be at m=1 thus

    |J(z0)1ABJ(z0)|(14p)|ξ|8(AB)[p]q+|B||ξ|(1+8p),

    and hence

    |J(z0)1ABJ(z0)|1,

    thus a contradiction by (3.17), so |ω(z)|<1 and so we get the desired proof.

    Let 1B<A1 and f(z)Mp. If

    |ξ|8(AB)[p]q1|B|4p(1+|B|)

    and

    1ξ[p]q(1p+zD2qf(z)Dqf(z)zDqf(z)f(z))(f(z)zDqf(z))21+Az1+Bz,

    then f(z)MSLp,q.

    Letting q1 in our results we obtain results for the class MSLp.

    The main purpose of this article is to seek some applications of the q-calculus in Geometric Function theory, which is the recent attraction for many researchers these days. The methods and ideas of q-calculus are used in the introduction of a new subclass of p-valent meromorphic functions with the help of subordinations. The domain of lemniscate of Bernoulli is considered in defining this class. Working on the coefficients of these functions we obtained a very important result of Fekete-Szegö for this class. Furthermore the functionals 1+ξzp+1Dqf(z)[p]q,1+ξzDqf(z)[p]qf(z),1+ξz1pDqf(z)[p]q(f(z))2 and 1+ξz12pDqf(z)[p]q(f(z))3 are connected with Janowski functions with the help of some conditions on ξ which ensures that a function to be a member of the class MSLp,q.

    The authors are grateful to the editor and anonymous referees for their comments and remarks to improve this manuscript. The author Thabet Abdeljawad would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Method in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.

    The authors declare that they have no competing interests.



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