Research article Special Issues

Evaluating toxic endpoints of exposure to a commercial PCB mixture: an in vivo laboratory study

  • Received: 28 January 2015 Accepted: 19 April 2015 Published: 26 April 2015
  • Polychlorinated biphenyls (PCBs) are ubiquitous environmental contaminants that produce a wide range of toxic effects. To determine sensitive endpoints in various organ systems, the effects of Aroclor 1260 on immune, endocrine, and hepatic systems were evaluated in a dose-response study. Nine-week old male rats were treated with Aroclor 1260 by oral gavage at dosages ranging from 0.025 to 156 mg/kg/day for 10 consecutive days and killed two days after the last treatment. Eight days prior to sacrifice, rats were injected i.v. with sheep red blood cells (SRBC) for determination of humoral immunity. No observable adverse effect level (NOAEL) and lowest observable adverse effect level (LOAEL) were determined for liver, thymus and genital organ weights, body weight, serum luteinizing hormone (LH), testosterone, thyroxine and thyroid-stimulating hormone (TSH) concentrations, hepatic microsomal testosterone hydroxylase activities, and hepatic microsomal cytochrome P450 (CYP) 1A1, CYP1A2, CYP2B1 and CYP2B2 protein levels. Treatment with Aroclor 1260, at all dosages, had no effect on testis, seminal vesicle or ventral prostate weights, on thymus weight or on serum LH or testosterone levels. Among the endpoints altered by Aroclor 1260, the most sensitive, with a LOAEL of 1.25 mg/kg/day, were increased testosterone 16β-hydroxylase activity and androstenedione formation. The LOAEL for increased liver weight, testosterone 16α-hydroxylase activity and CYP2B1 protein content was 3.13 mg/kg/day, while the LOAEL for decreased serum thyroxine levels and anti-SRBC IgM titer was 6.25 mg/kg/day. Less sensitive responses, as reflected by larger LOAEL values, included CYP1A enzyme induction and decreased body weight. In summary, comparison of NOAEL and LOAEL values indicated that hepatic CYP2B-mediated activities were a more sensitive response to Aroclor 1260 exposure in male rats than immune or endocrine endpoints.

    Citation: Herve Aloysius, Daniel Desaulniers, Stelvio M. Bandiera. Evaluating toxic endpoints of exposure to a commercial PCB mixture: an in vivo laboratory study[J]. AIMS Environmental Science, 2015, 2(2): 314-332. doi: 10.3934/environsci.2015.2.314

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  • Polychlorinated biphenyls (PCBs) are ubiquitous environmental contaminants that produce a wide range of toxic effects. To determine sensitive endpoints in various organ systems, the effects of Aroclor 1260 on immune, endocrine, and hepatic systems were evaluated in a dose-response study. Nine-week old male rats were treated with Aroclor 1260 by oral gavage at dosages ranging from 0.025 to 156 mg/kg/day for 10 consecutive days and killed two days after the last treatment. Eight days prior to sacrifice, rats were injected i.v. with sheep red blood cells (SRBC) for determination of humoral immunity. No observable adverse effect level (NOAEL) and lowest observable adverse effect level (LOAEL) were determined for liver, thymus and genital organ weights, body weight, serum luteinizing hormone (LH), testosterone, thyroxine and thyroid-stimulating hormone (TSH) concentrations, hepatic microsomal testosterone hydroxylase activities, and hepatic microsomal cytochrome P450 (CYP) 1A1, CYP1A2, CYP2B1 and CYP2B2 protein levels. Treatment with Aroclor 1260, at all dosages, had no effect on testis, seminal vesicle or ventral prostate weights, on thymus weight or on serum LH or testosterone levels. Among the endpoints altered by Aroclor 1260, the most sensitive, with a LOAEL of 1.25 mg/kg/day, were increased testosterone 16β-hydroxylase activity and androstenedione formation. The LOAEL for increased liver weight, testosterone 16α-hydroxylase activity and CYP2B1 protein content was 3.13 mg/kg/day, while the LOAEL for decreased serum thyroxine levels and anti-SRBC IgM titer was 6.25 mg/kg/day. Less sensitive responses, as reflected by larger LOAEL values, included CYP1A enzyme induction and decreased body weight. In summary, comparison of NOAEL and LOAEL values indicated that hepatic CYP2B-mediated activities were a more sensitive response to Aroclor 1260 exposure in male rats than immune or endocrine endpoints.


    The theory of the basic and the fractional quantum calculus, that is, the basic (or $ q $-) calculus and the fractional basic (or $ q $-) calculus, play important roles in many diverse areas of the mathematical, physical and engineering sciences (see, for example, [10,15,33,45]). Our main objective in this paper is to introduce and study some subclasses of the class of the normalized $ p $-valently analytic functions in the open unit disk:

    $ \mathbb{U} = \left\{z: z\in \mathbb{C}\qquad \text{and} \qquad \left\vert z\right\vert < 1\right\} $

    by applying the $ q $-derivative operator in conjunction with the principle of subordination between analytic functions (see, for details, [8,30]).

    We begin by denoting by $ \mathcal{A}\left(p\right) $ the class of functions $ f\left(z\right) $ of the form:

    $ f(z)=zp+n=p+1anzn   (pN:={1,2,3,}),
    $
    (1.1)

    which are analytic and $ p $-valent in the open unit disk $ \mathbb{U} $. In particular, we write $ \mathcal{A}\left(1\right) = :\mathcal{A} $.

    A function $ f\left(z\right) \in \mathcal{A}\left(p\right) $ is said to be in the class $ \mathcal{S}_{p}^{\ast}\left(\alpha \right) $ of $ p $-valently starlike functions of order $ \alpha $ in $ \mathbb{U} $ if and only if

    $ (zf(z)f(z))>α   (0α<p;zU).
    $
    (1.2)

    Moreover, a function $ f\left(z\right) \in \mathcal{A}\left(p\right) $ is said to be in the class $ \mathcal{C}_{p}\left(\alpha \right) $ of $ p $-valently convex functions of order $ \alpha $ in $ \mathbb{U} $ if and only if

    $ (1+zf(z)f(z))>α   (0α<p;zU).
    $
    (1.3)

    The $ p $-valent function classes $ \mathcal{S}_{p}^{\ast}\left(\alpha \right) $ and $ \mathcal{C}_{p}\left(\alpha \right) $ were studied by Owa [32], Aouf [2,3] and Aouf et al. [4,5]. From (1.2) and (1.3), it follows that

    $ f(z)Cp(α)zf(z)pSp(α).
    $
    (1.4)

    Let $ \mathcal{P} $ denote the Carathéodory class of functions $ \mathfrak p\left(z\right) $, analytic in $ \mathbb{U} $, which are normalized by

    $ p(z)=1+n=1cnzn,
    $
    (1.5)

    such that $ \Re\big(p\left(z\right)\big) > 0 $.

    Recently, Kanas and Wiśniowska [18,19] (see also [17,31]) introduced the conic domain $ \Omega _{k}\left(k\geqq 0\right) $, which we recall here as follows:

    $ Ωk={u+iv:u>k(u1)2+v2}
    $

    or, equivalently,

    $ Ωk={w:wCand(w)>k|w1|}.
    $

    By using the conic domain $ \Omega_{k} $, Kanas and Wiśniowska [18,19] also introduced and studied the class $ k $-$ \mathcal{UCV} $ of $ k $-uniformly convex functions in $ \mathbb{U} $ as well as the corresponding class $ k $-$ \mathcal{ST} $ of $ k $-starlike functions in $ \mathbb{U} $. For fixed $ k $, $ \Omega _{k} $ represents the conic region bounded successively by the imaginary axis when $ k = 0 $. For $ k = 1 $, the domain $ \Omega_{k} $ represents a parabola. For $ 1 < k < \infty $, the domain $ \Omega_{k} $ represents the right branch of a hyperbola. And, for $ k > 1 $, the domain $ \Omega_{k} $ represents an ellipse. For these conic regions, the following function plays the role of the extremal function:

    $ pk(z)={1+z1z(k=0)1+2π2[log(1+z1z)]2(k=1)1+11k2cos(2iπ(arccosk)log(1+z1z))(0<k<1)1+1k21sin(π2K(κ)u(z)κ0dt1t21κ2t2)+k2k21(1<k<)
    $
    (1.6)

    with

    $ u\left(z\right) = \dfrac{z-\sqrt{\kappa}}{1-\sqrt{\kappa z}} \qquad (0 < \kappa < 1;\; z\in \mathbb{U}), $

    where $ \kappa $ is so chosen that

    $ k=cosh(πK(κ)4K(κ)).
    $

    Here $ {\text K}(\kappa) $ is Legendre's complete elliptic integral of the first kind and

    $ K(κ)=K(1κ2),
    $

    that is, $ {\text K}^{\prime }\left(\kappa\right) $ is the complementary integral of $ {\text K}\left(\kappa\right) $ (see, for example, [48,p. 326,Eq 9.4 (209)]).

    We now recall the definitions and concept details of the basic (or $ q $-) calculus, which are used in this paper (see, for details, [13,14,45]; see also [1,6,7,11,34,38,39,42,54,59]). Throughout the paper, unless otherwise mentioned, we suppose that $ 0 < q < 1 $ and

    $ N={1,2,3}=N0{0}         (N0:={0,1,2,}).
    $

    Definition 1. The $ q $-number $ \left[\lambda\right]_{q} $ is defined by

    $ [λ]q={1qλ1q(λC)n1k=0qk=1+q+q2+qn1(λ=nN),
    $
    (1.7)

    so that

    $ \lim\limits_{q\rightarrow 1{-}}\left[\lambda \right]_{q} = \dfrac{1-q^{\lambda }}{1-q} = \lambda. $

    .

    Definition 2. For functions given by (1.1), the $ q $-derivative (or the $ q $-difference) operator $ D_{q} $ of a function $ f $ is defined by

    $ Dqf(z)={f(z)f(qz)(1q)z(z0)f(0)(z=0),
    $
    (1.8)

    provided that $ f^{\prime}\left(0\right) $ exists.

    We note from Definition 2 that

    $ limq1Dqf(z)=limq1f(z)f(qz)(1q)z=f(z)
    $

    for a function $ f $ which is differentiable in a given subset of $ \mathbb{C} $. It is readily deduced from (1.1) and (1.8) that

    $ Dqf(z)=[p]qzp1+n=p+1[n]qanzn1.
    $
    (1.9)

    We remark in passing that, in the above-cited recently-published survey-cum-expository review article, the so-called $ (p, q) $-calculus was exposed to be a rather trivial and inconsequential variation of the classical $ q $-calculus, the additional parameter $ p $ being redundant or superfluous (see, for details, [42,p. 340]).

    Making use of the $ q $-derivative operator $ D_{q} $ given by (1.6), we introduce the subclass $ \mathcal{S}_{q, p}^{\ast}\left(\alpha \right) $ of $ p $-valently $ q $-starlike functions of order $ \alpha $ in $ \mathbb{U} $ and the subclass $ \mathcal{C}_{q, p}\left(\alpha\right) $ of $ p $-valently $ q $-convex functions of order $ \alpha $ in $ \mathbb{U} $ as follows (see [54]):

    $ f(z)Sq,p(α)(1[p]qzDqf(z)f(z))>α
    $
    (1.10)
    $ \left(0 < q < 1;\; 0\leqq \alpha < 1;\;z\in \mathbb{U}\right) $

    and

    $ f(z)Cq,p(α)(1[p]qDp,q(zDqf(z))Dqf(z))>α
    $
    (1.11)
    $ \left(0 < q < 1;\; 0\leqq \alpha < 1;\; z\in \mathbb{U}\right), $

    respectively. From (1.10) and (1.11), it follows that

    $ f(z)Cq,p(α)zDqf(z)[p]qSq,p(α).
    $
    (1.12)

    For the simpler classes $ \mathcal{S}_{q, p}^{\ast} $ and $ \mathcal{C}_{q, p}^{\ast} $ of $ p $-valently $ q $-starlike functions in $ \mathbb{U} $ and $ p $-valently $ q $-convex functions in $ \mathbb{U} $, respectively, we have write

    $ \mathcal{S}_{q, p}^{\ast}\left(0\right) = :\mathcal{S}_{q, p}^{\ast} \qquad \text{and} \qquad \mathcal{C}_{q, p}\left( 0\right) = :\mathcal{C}_{q, p}. $

    Obviously, in the limit when $ q\rightarrow 1{-} $, the function classes $ \mathcal{S}_{q, p}^{\ast}\left(\alpha\right) $ and $ \mathcal{C}_{q, p}\left(\alpha \right) $ reduce to the familiar function classes $ \mathcal{S}_{p}^{\ast}\left(\alpha \right) $ and $ \mathcal{C}_{p}\left(\alpha \right) $, respectively.

    Definition 3. A function $ f\in \mathcal{A}\left(p\right) $ is said to belong to the class $ \mathcal{S}_{q, p}^{\ast} $ of $ p $-valently $ q $-starlike functions in $ \mathbb{U} $ if

    $ |zDqf(z)[p]qf(z)11q|11q(zU).
    $
    (1.13)

    In the limit when $ q\rightarrow 1{-} $, the closed disk

    $ |w11q|11q(0<q<1)
    $

    becomes the right-half plane and the class $ \mathcal{S}_{q, p}^{\ast} $ of $ p $-valently $ q $-starlike functions in $ \mathbb{U} $ reduces to the familiar class $ \mathcal{S}_{p}^{\ast} $ of $ p $-valently starlike functions with respect to the origin ($ z = 0 $). Equivalently, by using the principle of subordination between analytic functions, we can rewrite the condition (1.13) as follows (see [58]):

    $ zDqf(z)[p]qf(z)ˆp(z)   (ˆp(z)=1+z1qz).
    $
    (1.14)

    We note that $ \mathcal{S}_{q, 1}^{\ast} = \mathcal{S}_{q}^{\ast} $ (see [12,41]).

    Definition 4. (see [50]) A function $ \mathfrak p\left(z\right) $ given by (1.5) is said to be in the class $ k $-$ \mathcal{P}_{q} $ if and only if

    $ p(z)2pk(z)(1+q)+(1q)pk(z),
    $

    where $ p_{k}\left(z\right) $ is given by (1.6).

    Geometrically, the function $ p\in k $-$ \mathcal{P}_{q} $ takes on all values from the domain $ \Omega_{k, q} $ ($ k\geqq 0 $) which is defined as follows:

    $ Ωk,q={w:((1+q)w(q1)w+2)>k|(1+q)w(q1)w+21|}.
    $
    (1.15)

    The domain $ \Omega_{k, q} $ represents a generalized conic region which was introduced and studied earlier by Srivastava et al. (see, for example, [43,50]). It reduces, in the limit when $ q\rightarrow 1{-} $, to the conic domain $ \Omega_{k} $ studied by Kanas and Wiśniowska [18]. We note the following simpler cases.

    (1) $ k $-$ \mathcal{P}_{q}\subseteq \mathcal{P}\left(\frac{2k}{2k+1+q}\right) $, where $ \mathcal{P}\left(\frac{2k}{2k+1+q}\right) $ is the familiar class of functions with real part greater than $ \frac{2k}{2k+1+q} $;

    (2) $ \lim_{q\rightarrow 1{-}}\{k $-$ \mathcal{P}_{q}\} = \mathcal{P}\left(p_{k}\left(z\right) \right) $, where $ \mathcal{P}\big(p_{k}(z)\big) $ is the known class introduced by Kanas and Wiśniowska [18];

    (3) $ \lim_{q\rightarrow 1{-}}\{0 $-$ \mathcal{P}_{q}\} = \mathcal{P} $, where $ \mathcal{P} $ is Carathéodory class of analytic functions with positive real part.

    Definition 5. A function $ f\in \mathcal{A}\left(p\right) $ is said to be in the class $ k $-$ \mathcal{ST}_{q, p} $ if and only if

    $ ((1+q)zDqf(z)[p]qf(z)(q1)zDqf(z)[p]qf(z)+2)>k|(1+q)zDqf(z)[p]qf(z)(q1)zDqf(z)[p]qf(z)+21|(zU)
    $
    (1.16)

    or, equivalently,

    $ zDqf(z)[p]qf(z)k-Pq.
    $
    (1.17)

    The folowing special cases are worth mentioning here.

    (1) $ k $-$ \mathcal{ST}_{q, 1} = k $-$ \mathcal{ST}_{q} $, where $ k $-$ \mathcal{ST}_{q} $ is the function class introduced and studied by Srivastava et al. [50] and Zhang et al. [60] with $ \gamma = 1 $;

    (2) $ 0 $-$ \mathcal{ST}_{q, p} = \mathcal{S}_{q, p} $;

    (3) $ \lim_{q\rightarrow 1}\{k $-$ \mathcal{ST}_{q, p}\} = k $-$ \mathcal{ST}_{p} $, where $ k $-$ \mathcal{ST}_{p} $ is the class of $ p $-valently uniformly starlike functions;

    (4) $ \lim_{q\rightarrow 1}\{0 $-$ \mathcal{ST}_{q, p}\} = \mathcal{S}_{p} $, where $ \mathcal{S}_{p}^{\ast } $ is the class of $ p $-valently starlike functions;

    (5) $ 0 $-$ \mathcal{ST}_{q, 1} = \mathcal{S}_{q}^{\ast } $, where $ \mathcal{S}_{q}^{\ast} $ (see [12,41]);

    (6) $ \lim_{q\rightarrow 1}\{k $-$ \mathcal{ST}_{q, 1}\} = k $-$ \mathcal{ST} $, where $ k $-$ \mathcal{ST} $ is a function class introduced and studied by Kanas and Wiśniowska [19];

    (7) $ \lim_{q\rightarrow 1}\{0 $-$ \mathcal{ST}_{q, 1}\} = \mathcal{S}^{\ast } $, where $ \mathcal{S}^{\ast} $ is the familiar class of starlike functions in $ \mathbb{U} $.

    Definition 6. By using the idea of Alexander's theorem [9], the class $ k $-$ \mathcal{UCV}_{q, p} $ can be defined in the following way:

    $ f(z)k-UCVq,pzDqf(z)[p]qk-STq,p.
    $
    (1.18)

    In this paper, we investigate a number of useful properties including coefficient estimates and the Fekete-Szegö inequalities for the function classes $ k $-$ \mathcal{ST}_{q, p} $ and $ k $-$ \mathcal{UCV}_{q, p} $, which are introduced above. Various corollaries and consequences of most of our results are connected with earlier works related to the field of investigation here.

    In order to establish our main results, we need the following lemmas.

    Lemma 1. (see [16]) Let $ 0\leqq k < \infty $ be fixed and let $ p_{k} $ be defined by $ (1.6) $. If

    $ pk(z)=1+Q1z+Q2z2+,
    $

    then

    $ Q1={2A21k2(0k<1)8π2(k=1)π24t(k21)[K(t)]2(1+t)(1<k<)
    $
    (2.1)

    and

    $ Q2={A2+23Q1(0k<1)23Q1(k=1)4[K(t)]2(t2+6t+1)π224t[K(t)]2(1+t)Q1(1<k<),
    $
    (2.2)

    where

    $ A=2arccoskπ,
    $

    and $ t\in \left(0, 1\right) $ is so chosen that

    $ k = \cosh \left(\frac{\pi {\text K}^{{\prime}}\left( t\right)}{{\text K}\left(t\right)}\right), $

    $ {\text K}\left(t\right) $ being Legendre's complete elliptic integral of the first kind.

    Lemma 2. Let $ 0\leqq k < \infty $ be fixed and suppose that

    $ pk,q(z)=2pk(z)(1+q)+(1q)pk(z),
    $
    (2.3)

    where $ p_{k}\left(z\right) $ be defined by $ (1.6) $. Then

    $ pk,q(z)=1+12(1+q)Q1z+12(1+q)[Q212(1q)Q21]z2+ ,
    $
    (2.4)

    where $ Q_{1} $ and $ Q_{2} $ are given by $ (2.1) $ and $ (2.2), $ respectively.

    Proof. By using (1.6) in (2.3), we can easily derive (2.4).

    Lemma 3. (see [26]) Let the function $ h $ given by

    $ h(z) = 1+\sum\limits_{n = 1}^{\infty }c_{n}z^{n}\in \mathcal{P} $

    be analytic in $ \mathbb{U} $ and satisfy $ \Re\big(h(z)\big) > 0 $ for $ z $ in $ \mathbb{U} $. Then the following sharp estimate holds true$ : $

    $ |c2vc21|2max{1,|2v1|}(vC).
    $

    The result is sharp for the functions given by

    $ g(z)=1+z21z2org(z)=1+z1z.
    $
    (2.5)

    Lemma 4. (see [26]) If the function $ h $ is given by

    $ h(z) = 1+\sum\limits_{n = 1}^{\infty }c_{n}z^{n}\in \mathcal{P}, $

    then

    $ |c2νc21|{4ν+2(ν0)2(0ν1)4ν2(ν1).
    $
    (2.6)

    When $ \nu > 1, $ the equality holds true if and only if

    $ h(z) = \frac{1+z}{1-z} $

    or one of its rotations. If $ 0 < \nu < 1, $ then the equality holds true if and only if

    $ h(z) = \frac{1+z^{2}}{1-z^{2}} $

    or one of its rotations. If $ \nu = 0, $ the equality holds true if and only if

    $ h(z)=(1+λ2)(1+z1z)+(1λ2)(1z1+z)(0λ1)
    $

    or one of its rotations. If $ \nu = 1, $ the equality holds true if and only if the function $ h $ is the reciprocal of one of the functions such that equality holds true in the case when $ \nu = 0 $.

    The above upper bound is sharp and it can be improved as follows when $ 0 < \nu < 1 $$ : $

    $ |c2νc21|+ν|c1|22(0ν12)
    $

    and

    $ |c2νc21|+(1ν)|c1|22(12ν1).
    $

    We assume throughout our discussion that, unless otherwise stated, $ 0\leqq k < \infty $, $ 0 < q < 1 $, $ p\in \mathbb{N} $, $ Q_{1} $ is given by (2.1), $ Q_{2} $ is given by (2.2) and $ z\in \mathbb{U} $.

    Theorem 1. If a function $ f\in \mathcal{A}\left(p\right) $ is of the form $ (1.1) $ and satisfies the following condition$ : $

    $ n=p+1{2(k+1)([n]q[p]q)+qn+2[p]q1}|an|<(1+q)[p]q,
    $
    (3.1)

    then the function $ f\in k $-$ \mathcal{ST}_{q, p} $.

    Proof. Suppose that the inequality (3.1) holds true. Then it suffices to show that

    $ k|(1+q)zDqf(z)[p]qf(z)(q1)zDqf(z)[p]qf(z)+21|((1+q)zDqf(z)[p]qf(z)(q1)zDqf(z)[p]qf(z)+21)<1.
    $

    In fact, we have

    $ k|(1+q)zDqf(z)[p]qf(z)(q1)zDqf(z)[p]qf(z)+21|((1+q)zDqf(z)[p]qf(z)(q1)zDqf(z)[p]qf(z)+21)(k+1)|(1+q)zDqf(z)[p]qf(z)(q1)zDqf(z)[p]qf(z)+21|=2(k+1)|zDqf(z)[p]qf(z)(q1)zDqf(z)+2[p]qf(z)|=2(k+1)|n=p+1([n]q[p]q)anznp(1+q)[p]q+n=p+1((q1)[n]q+2[p]q)anznp|2(k+1)n=p+1([n]q[p]q)|an|(1+q)[p]qn=p+1(qn+2[p]q1)|an|.
    $

    The last expression is bounded by $ 1 $ if (3.1) holds true. This completes the proof of Theorem 1.

    Corollary 1. If $ f\left(z\right) \in k $-$ \mathcal{ST}_{q, p}, $ then

    $ |an|(1+q)[p]q{2(k+1)([n]q[p]q)+qn+2[p]q1}(np+1).
    $

    The result is sharp for the function $ f(z) $ given by

    $ f(z)=zp+(1+q)[p]q{2(k+1)([n]q[p]q)+qn+2[p]q1}zn(np+1).
    $

    Remark 1. Putting $ p = 1 $ Theorem 1, we obtain the following result which corrects a result of Srivastava et al. [50,Theorem 3.1].

    Corollary 2. (see Srivastava et al. [50,Theorem 3.1]) If a function $ f\in \mathcal{A} $ is of the form $ (1.1) $ $ ($with $ p = 1) $ and satisfies the following condition$ : $

    $ n=2{2(k+1)([n]q1)+qn+1}|an|<(1+q)
    $

    then the function $ f\in k $-$ \mathcal{ST}_{q} $.

    Letting $ q\rightarrow 1{-} $ in Theorem 1, we obtain the following known result [29,Theorem 1] with

    $ \alpha _{1} = \beta _{1} = p, \;\; \alpha _{i} = 1(i = 2, \cdots, s+1)\quad \text{and} \quad \beta _{j} = 1(j = 2, \cdots, s). $

    Corollary 3. If a function $ f\in \mathcal{A}\left(p\right) $ is of the form $ (1.1) $ and satisfies the following condition$ : $

    $ n=p+1{(k+1)(np)+p}|an|<p,
    $

    then the function $ f\in k $-$ \mathcal{ST}_{p} $.

    Remark 2. Putting $ p = 1 $ in Corollary 3, we obtain the result obtained by Kanas and Wiśniowska [19,Theorem 2.3].

    By using Theorem 1 and (1.18), we obtain the following result.

    Theorem 2. If a function $ f\in \mathcal{A}\left(p\right) $ is of the form $ (1.1) $ and satisfies the following condition$ : $

    $ n=p+1([n]q[p]q){2(k+1)([n]q[p]q)+qn+2[p]q1}|an|<(1+q)[p]q,
    $

    then the function $ f\in k $-$ \mathcal{UCV}_{q, p} $.

    Remark 3. Putting $ p = 1 $ Theorem 1, we obtain the following result which corrects the result of Srivastava et al. [50,Theorem 3.3].

    Corollary 4. (see Srivastava et al. [50,Theorem 3.3]) If a function $ f\in \mathcal{A} $ is of the form (1.1) (with $ p = 1 $) and satisfies the following condition$ : $

    $ n=2[n]q{2(k+1)([n]q1)+qn+1}|an|<(1+q),
    $

    then the function $ f\in k $-$ \mathcal{UCV}_{q} $.

    Letting $ q\rightarrow 1{-} $ in Theorem 2, we obtain the following corollary (see [29,Theorem 1]) with

    $ \alpha_{1} = p+1, \;\; \beta _{1} = p, \;\; \alpha_{\ell} = 1\;\; (\ell = 2, \cdots, s+1) \quad \text{and} \quad \beta_{j} = 1(j = 2, \cdots, s). $

    Corollary 5. If a function $ f\in \mathcal{A}\left(p\right) $ is of the form $ (1.1) $ and satisfies the following condition$ : $

    $ n=p+1(np){(k+1)(np)+p}|an|<p,
    $

    then the function $ f\in k $-$ \mathcal{UCV}_{p} $.

    Remark 4. Putting $ p = 1 $ in Corollary 5, we obtain the following corollary which corrects the result of Kanas and Wiśniowska [18,Theorem 3.3].

    Corollary 6. If a function $ f\in \mathcal{A} $ is of the form $ (1.1) $ $ ($with $ p = 1) $ and satisfies the following condition$ : $

    $ n=2n{n(k+1)k}|an|<1,
    $

    then the function $ f\in k $-$ \mathcal{UCV} $.

    Theorem 3. If $ f\in k $-$ \mathcal{ST}_{q, p}, $ then

    $ |ap+1|(1+q)[p]qQ12qp[1]q
    $
    (3.2)

    and$, $ for all $ n = 3, 4, 5, \cdots, $

    $ |an+p1|(1+q)[p]qQ12qp[n1]qn2j=1(1+(1+q)[p]qQ12qp[j]q).
    $
    (3.3)

    Proof. Suppose that

    $ zDqf(z)[p]qf(z)=p(z),
    $
    (3.4)

    where

    $ p(z)=1+n=1cnznk-Pq.
    $

    Eq (3.4) can be written as follows:

    $ zDqf(z)=[p]qf(z)p(z),
    $

    which implies that

    $ n=p+1([n]q[p]q)anzn=[p]q(zp+n=p+1anzn)(n=1cnzn).
    $
    (3.5)

    Comparing the coefficients of $ z^{n+p-1} $ on both sides of (3.5), we obtain

    $ ([n+p1]q[p]q)an+p1=[p]q{cn1+ap+1cn2++an+p2c1}.
    $

    By taking the moduli on both sides and then applying the following coefficient estimates (see [50]):

    $ \left\vert c_{n}\right\vert \leqq \frac{1}{2}\left( 1+q\right) Q_{1} \qquad (n\in \mathbb{N}), $

    we find that

    $ |an+p1|(1+q)[p]qQ12qp[n1]q{1+|ap+1|++|an+p2|}.
    $
    (3.6)

    We now apply the principle of mathematical induction on (3.6). Indeed, for $ n = 2 $, we have

    $ |ap+1|(1+q)[p]qQ12qp[1]q,
    $
    (3.7)

    which shows that the result is true for $ n = 2 $. Next, for $ n = 3 $ in (3.7), we get

    $ |ap+2|(1+q)[p]qQ12qp[2]q{1+|ap+1|}.
    $

    By using (3.7), we obtain

    $ |ap+2|(1+q)[p]qQ12qp[2]q(1+(1+q)[p]qQ12qp[1]q),
    $

    which is true for $ n = 3 $. Let us assume that (3.3) is true for $ n = t\; \; (t\in\mathbb{N}) $, that is,

    $ |at+p1|(1+q)[p]qQ12qp[t1]qt2j=1(1+(1+q)[p]qQ12qp[j]q).
    $

    Let us consider

    $ |at+p|(1+q)[p]qQ12qp[t]q{1+|ap+1|+|ap+2|++|at+p1|}(1+q)[p]qQ12qp[t]q{1+(1+q)[p]qQ12qp[1]q+(1+q)[p]qQ12qp[2]q(1+(1+q)[p]qQ12qp[1]q)++(1+q)[p]qQ12qp[t1]qt2j=1(1+(1+q)[p]qQ12qp[j]q)}=(1+q)[p]qQ12qp[t]q{(1+(1+q)[p]qQ12qp[1]q)(1+(1+q)[p]qQ12qp[2]q)(1+(1+q)[p]qQ12qp[t1]q)}=(1+q)[p]qQ12qp[t]qt1j=1(1+(1+q)[p]qQ12qp[j]q)
    $

    Therefore, the result is true for $ n = t+1 $. Consequently, by the principle of mathematical induction, we have proved that the result holds true for all $ n\; \; \left(n\in \mathbb{N}\setminus\{1\}\right) $. This completes the proof of Theorem 3.

    Similarly, we can prove the following result.

    Theorem 4. If $ f\in k $-$ \mathcal{UCV}_{q, p} $ and is of form $ (1.1), $ then

    $ |ap+1|(1+q)[p]2qQ12qp[p+1]q
    $

    and$, $ for all $ n = 3, 4, 5, \cdots, $

    $ |an+p1|(1+q)[p]2qQ12qp[n+p1]q[n1]qn2j=1(1+(1+q)[p]qQ12qp[j]q).
    $

    Putting $ p = 1 $ in Theorems 3 and 4, we obtain the following corollaries.

    Corollary 7. If $ f\in k $-$ \mathcal{ST}_{q}, $ then

    $ |a2|(1+q)Q12q
    $

    and$, $ for all $ n = 3, 4, 5, \cdots, $

    $ |an|(1+q)Q12q[n1]qn2j=1(1+(1+q)Q12q[j]q).
    $

    Corollary 8. If $ f\in k $-$ \mathcal{UCV}_{q}, $ then

    $ |a2|Q12q
    $

    and$, $ for all $ n = 3, 4, 5, \cdots, $

    $ |an|(1+q)Q12q[n]q[n1]qn2j=1(1+(1+q)Q12q[j]q).
    $

    Theorem 5. Let $ f\in k $-$ \mathcal{ST}_{q, p} $. Then $ f\left(\mathbb{U}\right) $ contains an open disk of the radius given by

    $ r=2qp2(p+1)qp+(1+q)[p]qQ1.
    $

    Proof. Let $ w_{0}\neq 0 $ be a complex number such that $ f\left(z\right) \neq w_{0} $ for $ z\in \mathbb{U} $. Then

    $ f1(z)=w0f(z)w0f(z)=zp+1+(ap+1+1w0)zp+1+.
    $

    Since $ f_{1} $ is univalent, so

    $ |ap+1+1w0|p+1.
    $

    Now, using Theorem 3, we have

    $ |1w0|p+1+(1+q)[p]qQ12qp=2qp(p+1)+(1+q)[p]qQ12qp.
    $

    Hence

    $ |w0|2qp2qp(p+1)+(1+q)[p]qQ1.
    $

    This completes the proof of Theorem 5.

    Theorem 6. Let the function $ f\in k $-$ \mathcal{ST}_{q, p} $ be of the form $ (1.1) $. Then$, $ for a complex number $ \mu, $

    $ |ap+2μa2p+1|[p]qQ12qpmax{1,|Q2Q1+([p]q(1+q)qp(1q))Q12qp(1μ(1+q)2[p]q[p]q(1+q)qp(1q))|}.
    $
    (3.8)

    The result is sharp.

    Proof. If $ f\in k $-$ \mathcal{ST}_{q, p} $, we have

    $ zDqf(z)[p]qf(z)pk,q(z)=2pk(z)(1+q)+(1q)pk(z).
    $

    From the definition of the differential subordination, we know that

    $ zDqf(z)[p]qf(z)=pk,q(w(z))(zU),
    $
    (3.9)

    where $ w\left(z\right) $ is a Schwarz function with $ w\left(0\right) = 0 $ and $ \left\vert w\left(z\right) \right\vert < 1 $ for $ z\in\mathbb{U} $.

    Let $ h\in \mathcal{P} $ be a function defined by

    $ h(z)=1+w(z)1w(z)=1+c1z+c2z2+(zU).
    $

    This gives

    $ w(z)=12c1z+12(c2c212)z2+
    $

    and

    $ pk,q(w(z))=1+1+q4c1Q1z+1+q4{Q1c2+12(Q2Q11q2Q21)c21}z2+.
    $
    (3.10)

    Using (3.10) in (3.9), we obtain

    $ ap+1=(1+q)[p]qc1Q14qp
    $

    and

    $ ap+2=[p]qQ14qp[c212(1Q2Q1[p]q(1+q)qp(1q)2qpQ1)c21]
    $

    Now, for any complex number $ \mu $, we have

    $ ap+2μa2p+1=[p]qQ14qp[c212(1Q2Q1[p]q(1+q)qp(1q)2qpQ1)c21]μ(1+q)2[p]2qQ21c2116q2p.
    $
    (3.11)

    Then (3.11) can be written as follows:

    $ ap+2μa2p+1=[p]qQ14qp{c2vc21},
    $
    (3.12)

    where

    $ v=12[1Q2Q1([p]q(1+q)qp(1q))Q12qp(1μ(1+q)2[p]q[p]q(1+q)qp(1q))].
    $
    (3.13)

    Finally, by taking the moduli on both sides and using Lemma 4, we obtain the required result. The sharpness of (3.8) follows from the sharpness of (2.5). Our demonstration of Theorem 6 is thus completed.

    Similarly, we can prove the following theorem.

    Theorem 7. Let the function $ f\in k $-$ \mathcal{UCV}_{q, p} $ be of the form $ (1.1) $. Then$, $ for a complex number $ \mu, $

    $ |ap+2μa2p+1|[p]2qQ12qp[p+2]qmax{1,|Q2Q1+((1+q)[p]q(1q)qp)Q12qp(1μ[p+2]q(1+q)2[p]2q((1+q)[p]q(1q)qp)[p+1]2q)|}.
    $

    The result is sharp.

    Putting $ p = 1 $ in Theorems 6 and 7, we obtain the following corollaries.

    Corollary 9. Let the function $ f\in k $-$ \mathcal{ST}_{q} $ be of the form $ (1.1) $ $ ($with $ p = 1) $. Then$, $ for a complex number $ \mu, $

    $ |a3μa22|Q12qmax{1,|Q2Q1+(1+q2)Q12q(1μ(1+q)21+q2)|}.
    $

    The result is sharp.

    Corollary 10. Let the function $ f\in k $-$ \mathcal{UCV}_{q} $ be of the form $ (1.1) $ $ ($with $ p = 1) $. Then$, $ for a complex number $ \mu, $

    $ |a3μa22|Q12q[3]qmax{1,|Q2Q1+(1+q2)Q12q(1μ[3]q1+q2)|}.
    $

    The result is sharp.

    Theorem 8. Let

    $ σ1=([p]q(1+q)qp(1q))Q21+2qp(Q2Q1)[p]q(1+q)2Q21,
    $
    $ σ2=([p]q(1+q)qp(1q))Q21+2qp(Q2+Q1)[p]q(1+q)2Q21
    $

    and

    $ σ3=([p]q(1+q)qp(1q))Q21+2qpQ2[p]q(1+q)2Q21.
    $

    If the function $ f $ given by $ (1.1) $ belongs to the class $ k $-$ \mathcal{ST}_{q, p}, $ then

    $ |ap+2μa2p+1|{[p]qQ12qp{Q2Q1+([p]q(1+q)qp(1q))Q12qp(1μ(1+q)2[p]q[p]q(1+q)qp(1q))}(μσ1)[p]qQ12qp(σ1μσ2),[p]qQ12qp{Q2Q1+([p]q(1+q)qp(1q))Q12qp(1μ(1+q)2[p]q[p]q(1+q)qp(1q))}(μσ2).
    $

    Furthermore$, $ if $ \sigma _{1}\leqq \mu \leqq \sigma _{3}, $ then

    $ |ap+2μa2p+1|+2qp(1+q)2[p]qQ1{1Q2Q1([p]q(1+q)qp(1q))Q12qp(1μ(1+q)2[p]q([p]q(1+q)qp(1q)))}|ap+1|2[p]qQ12qp.
    $

    If $ \sigma _{3}\leqq \mu \leqq \sigma _{2}, $ then

    $ |ap+2μa2p+1|+2qp(1+q)2[p]qQ1{1+Q2Q1+([p]q(1+q)qp(1q))Q12qp(1μ(1+q)2[p]q([p]q(1+q)qp(1q)))}|ap+1|2[p]qQ12qp.
    $

    Proof. Applying Lemma 4 to (3.12) and (3.13), respectively, we can derive the results asserted by Theorem 8.

    Putting $ p = 1 $ in Theorem 8, we obtain the following result.

    Corollary 11. Let

    $ σ4=(1+q2)Q21+2q(Q2Q1)(1+q)2Q21,
    $
    $ σ5=(1+q2)Q21+2q(Q2+Q1)(1+q)2Q21
    $

    and

    $ σ6=(1+q2)Q21+2qQ2(1+q)2Q21.
    $

    If the function $ f $ given by $ (1.1) $ $ ($with $ p = 1) $ belongs to the class $ k $-$ \mathcal{ST}_{q}, $ then

    $ |a3μa22|{Q12q{Q2Q1+(1+q2)Q12q(1μ(1+q)21+q2)}(μσ4)Q12q(σ4μσ5)Q12q{Q2Q1+(1+q2)Q12q(1μ(1+q)21+q2)}(μσ5).
    $

    Furthermore$, $ if $ \sigma _{4}\leqq \mu \leqq \sigma _{6}, $ then

    $ |a3μa22|+2q(1+q)2Q1{1Q2Q1(1+q2)Q12q(1μ(1+q)21+q2)}|a2|2Q12q.
    $

    If $ \sigma _{3}\leqq \mu \leqq \sigma _{2}, $ then

    $ |a3μa22|+2q(1+q)2Q1{1+Q2Q1+(1+q2)Q12q(1μ(1+q)21+q2)}|a2|2Q12q.
    $

    Similarly, we can prove the following result.

    Theorem 9. Let

    $ η1=[((1+q)[p]q(1q)qp)Q21+2qp(Q2Q1)][p+1]2q[p]2q[p+2]q(1+q)2Q21,
    $
    $ η2=[((1+q)[p]q(1q)qp)Q21+2qp(Q2+Q1)][p+1]2q[p]2q[p+2]q(1+q)2Q21
    $

    and

    $ η3=[((1+q)[p]q(1q)qp)Q21+2qpQ2][p+1]2q[p]2q[p+2]q(1+q)2Q21.
    $

    If the function $ f $ given by $ (1.1) $ belongs to the class $ k $-$ \mathcal{UCV}_{q, p}, $ then

    $ |ap+2μa2p+1|{[p]2qQ12qp[p+2]q{Q2Q1+((1+q)[p]q(1q)qp)Q12qp(1[p+2]q(1+q)2[p]2q μ((1+q)[p]q(1q)qp)[p+1]2q)}(μη1)[p]2qQ12qp[p+2]q(η1μη2)[p]2qQ12qp[p+2]q{Q2Q1+((1+q)[p]q(1q)qp)Q12qp(1[p+2]q(1+q)2[p]2q μ((1+q)[p]q(1q)qp)[p+1]2q)}(μη2).
    $

    Furthermore$, $ if $ \eta _{1}\leqq \mu \leqq \eta _{3}, $ then

    $ |ap+2μa2p+1|+2qp[p+1]2qQ1[p+2]q(1+q)2[p]2qQ21{1Q2Q1((1+q)[p]q(1q)qp)Q12qp(1μ[p+2]q(1+q)2[p]2q((1+q)[p]q(1q)qp)[p+1]2q)}|ap+1|2[p]2qQ12qp[p+2]q.
    $

    If $ \eta _{3}\leqq \mu \leqq \eta _{2}, $ then

    $ |ap+2μa2p+1|+2qp[p+1]2qQ1[p+2]q(1+q)2[p]2qQ21{1+Q2Q1+((1+q)[p]q(1q)qp)Q12qp(1μ[p+2]q(1+q)2[p]2q((1+q)[p]q(1q)qp)[p+1]2q)}|ap+1|2[p]2qQ12qp[p+2]q.
    $

    Putting $ p = 1 $ in Theorem 9, we obtain the following result.

    Corollary 12. Let

    $ η4=(1+q2)Q21+2q(Q2Q1)[3]qQ21,
    $
    $ η5=(1+q2)Q21+2q(Q2+Q1)[3]qQ21
    $

    and

    $ η6=(1+q2)Q21+2qQ2[3]qQ21.
    $

    If the function $ f $ given by $ (1.1) $ $ ($with $ p = 1) $ belongs to the class $ k $-$ \mathcal{UCV}_{q}, $ then

    $ |a3μa22|{Q12q[3]q{Q2Q1+(1+q2)Q12q(1μ[3]q1+q2)}(μη4)Q12q[3]q(η4μη5)Q12q[3]q{Q2Q1+(1+q2)Q12q(1μ[3]q1+q2)}(μη5).
    $

    Furthermore$, $ if $ \eta _{4}\leqq \mu \leqq \eta _{6}, $ then

    $ |a3μa22|+2q[3]qQ1{1Q2Q1(1+q2)Q12q(1μ[3]q1+q2)}|a2|2Q12q[3]q.
    $

    If $ \eta _{3}\leqq \mu \leqq \eta _{2}, $ then

    $ |a3μa22|+2q[3]qQ1{1+Q2Q1+(1+q2)Q12q(1μ[3]q1+q2)}|a2|2Q12q[3]q.
    $

    In our present investigation, we have applied the concept of the basic (or $ q $-) calculus and a generalized conic domain, which was introduced and studied earlier by Srivastava et al. (see, for example, [43,50]). By using this concept, we have defined two subclasses of normalized multivalent functions which map the open unit disk:

    $ \mathbb{U} = \left\{z: z\in \mathbb{C}\qquad \text{and} \qquad \left\vert z\right\vert < 1\right\} $

    onto this generalized conic domain. We have derived a number of useful properties including (for example) the coefficient estimates and the Fekete-Szegö inequalities for each of these multivalent function classes. Our results are connected with those in several earlier works which are related to this field of Geometric Function Theory of Complex Analysis.

    Basic (or $ q $-) series and basic (or $ q $-) polynomials, especially the basic (or $ q $-) hypergeometric functions and basic (or $ q $-) hypergeometric polynomials, are applicable particularly in several diverse areas [see, for example, [48,pp. 350-351]. Moreover, as we remarked in the introductory Section 1 above, in the recently-published survey-cum-expository review article by Srivastava [42], the so-called $ (p, q) $-calculus was exposed to be a rather trivial and inconsequential variation of the classical $ q $-calculus, the additional parameter $ p $ being redundant or superfluous (see, for details, [42,p. 340]). This observation by Srivastava [42] will indeed apply to any attempt to produce the rather straightforward $ (p, q) $-variations of the results which we have presented in this paper.

    In conclusion, with a view mainly to encouraging and motivating further researches on applications of the basic (or $ q $-) analysis and the basic (or $ q $-) calculus in Geometric Function Theory of Complex Analysis along the lines of our present investigation, we choose to cite a number of recently-published works (see, for details, [25,47,51,53,56] on the Fekete-Szegö problem; see also [20,21,22,23,24,27,28,35,36,37,40,44,46,49,52,55,57] dealing with various aspects of the usages of the $ q $-derivative operator and some other operators in Geometric Function Theory of Complex Analysis). Indeed, as it is expected, each of these publications contains references to many earlier works which would offer further incentive and motivation for considering some of these worthwhile lines of future researches.

    The authors declare no conflicts of interest.

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