Research article

A generalized conic domain and its applications to certain subclasses of multivalent functions associated with the basic (or q-) calculus

  • Received: 13 November 2020 Accepted: 08 April 2021 Published: 16 April 2021
  • MSC : 11M35, 30C45, 30C50

  • Citation: H. M. Srivastava, T. M. Seoudy, M. K. Aouf. A generalized conic domain and its applications to certain subclasses of multivalent functions associated with the basic (or q-) calculus[J]. AIMS Mathematics, 2021, 6(6): 6580-6602. doi: 10.3934/math.2021388

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  • 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:

    U={z:zCand|z|<1}

    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 A(p) the class of functions f(z) 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 U. In particular, we write A(1)=:A.

    A function f(z)A(p) is said to be in the class Sp(α) of p-valently starlike functions of order α in U if and only if

    (zf(z)f(z))>α   (0α<p;zU). (1.2)

    Moreover, a function f(z)A(p) is said to be in the class Cp(α) of p-valently convex functions of order α in U if and only if

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

    The p-valent function classes Sp(α) and Cp(α) 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 P denote the Carathéodory class of functions p(z), analytic in U, which are normalized by

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

    such that (p(z))>0.

    Recently, Kanas and Wiśniowska [18,19] (see also [17,31]) introduced the conic domain Ωk(k0), 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 Ωk, Kanas and Wiśniowska [18,19] also introduced and studied the class k-UCV of k-uniformly convex functions in U as well as the corresponding class k-ST of k-starlike functions in U. For fixed k, Ωk represents the conic region bounded successively by the imaginary axis when k=0. For k=1, the domain Ωk represents a parabola. For 1<k<, the domain Ωk represents the right branch of a hyperbola. And, for k>1, the domain Ωk 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(z)=zκ1κz(0<κ<1;zU),

    where κ is so chosen that

    k=cosh(πK(κ)4K(κ)).

    Here K(κ) is Legendre's complete elliptic integral of the first kind and

    K(κ)=K(1κ2),

    that is, K(κ) is the complementary integral of K(κ) (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 [λ]q is defined by

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

    so that

    limq1[λ]q=1qλ1q=λ.

    .

    Definition 2. For functions given by (1.1), the q-derivative (or the q-difference) operator Dq 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(0) 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 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 Dq given by (1.6), we introduce the subclass Sq,p(α) of p-valently q-starlike functions of order α in U and the subclass Cq,p(α) of p-valently q-convex functions of order α in U as follows (see [54]):

    f(z)Sq,p(α)(1[p]qzDqf(z)f(z))>α (1.10)
    (0<q<1;0α<1;zU)

    and

    f(z)Cq,p(α)(1[p]qDp,q(zDqf(z))Dqf(z))>α (1.11)
    (0<q<1;0α<1;zU),

    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 Sq,p and Cq,p of p-valently q-starlike functions in U and p-valently q-convex functions in U, respectively, we have write

    Sq,p(0)=:Sq,pandCq,p(0)=:Cq,p.

    Obviously, in the limit when q1, the function classes Sq,p(α) and Cq,p(α) reduce to the familiar function classes Sp(α) and Cp(α), respectively.

    Definition 3. A function fA(p) is said to belong to the class Sq,p of p-valently q-starlike functions in U if

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

    In the limit when q1, the closed disk

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

    becomes the right-half plane and the class Sq,p of p-valently q-starlike functions in U reduces to the familiar class Sp 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 Sq,1=Sq (see [12,41]).

    Definition 4. (see [50]) A function p(z) given by (1.5) is said to be in the class k-Pq if and only if

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

    where pk(z) is given by (1.6).

    Geometrically, the function pk-Pq takes on all values from the domain Ωk,q (k0) 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 Ωk,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 q1, to the conic domain Ωk studied by Kanas and Wiśniowska [18]. We note the following simpler cases.

    (1) k-PqP(2k2k+1+q), where P(2k2k+1+q) is the familiar class of functions with real part greater than 2k2k+1+q;

    (2) limq1{k-Pq}=P(pk(z)), where P(pk(z)) is the known class introduced by Kanas and Wiśniowska [18];

    (3) limq1{0-Pq}=P, where P is Carathéodory class of analytic functions with positive real part.

    Definition 5. A function fA(p) is said to be in the class k-STq,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-STq,1=k-STq, where k-STq is the function class introduced and studied by Srivastava et al. [50] and Zhang et al. [60] with γ=1;

    (2) 0-STq,p=Sq,p;

    (3) limq1{k-STq,p}=k-STp, where k-STp is the class of p-valently uniformly starlike functions;

    (4) limq1{0-STq,p}=Sp, where Sp is the class of p-valently starlike functions;

    (5) 0-STq,1=Sq, where Sq (see [12,41]);

    (6) limq1{k-STq,1}=k-ST, where k-ST is a function class introduced and studied by Kanas and Wiśniowska [19];

    (7) limq1{0-STq,1}=S, where S is the familiar class of starlike functions in U.

    Definition 6. By using the idea of Alexander's theorem [9], the class k-UCVq,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-STq,p and k-UCVq,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 0k< be fixed and let pk 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(0,1) is so chosen that

    k=cosh(πK(t)K(t)),

    K(t) being Legendre's complete elliptic integral of the first kind.

    Lemma 2. Let 0k< be fixed and suppose that

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

    where pk(z) be defined by (1.6). Then

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

    where Q1 and Q2 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+n=1cnznP

    be analytic in U and satisfy (h(z))>0 for z in 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+n=1cnznP,

    then

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

    When ν>1, the equality holds true if and only if

    h(z)=1+z1z

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

    h(z)=1+z21z2

    or one of its rotations. If ν=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 ν=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 ν=0.

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

    |c2νc21|+ν|c1|22(0ν12)

    and

    |c2νc21|+(1ν)|c1|22(12ν1).

    We assume throughout our discussion that, unless otherwise stated, 0k<, 0<q<1, pN, Q1 is given by (2.1), Q2 is given by (2.2) and zU.

    Theorem 1. If a function fA(p) 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 fk-STq,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(z)k-STq,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 fA 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 fk-STq.

    Letting q1 in Theorem 1, we obtain the following known result [29,Theorem 1] with

    α1=β1=p,αi=1(i=2,,s+1)andβj=1(j=2,,s).

    Corollary 3. If a function fA(p) is of the form (1.1) and satisfies the following condition:

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

    then the function fk-STp.

    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 fA(p) 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 fk-UCVq,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 fA 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 fk-UCVq.

    Letting q1 in Theorem 2, we obtain the following corollary (see [29,Theorem 1]) with

    α1=p+1,β1=p,α=1(=2,,s+1)andβj=1(j=2,,s).

    Corollary 5. If a function fA(p) is of the form (1.1) and satisfies the following condition:

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

    then the function fk-UCVp.

    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 fA 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 fk-UCV.

    Theorem 3. If fk-STq,p, then

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

    and, for all n=3,4,5,,

    |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 zn+p1 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]):

    |cn|12(1+q)Q1(nN),

    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(tN), 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(nN{1}). This completes the proof of Theorem 3.

    Similarly, we can prove the following result.

    Theorem 4. If fk-UCVq,p and is of form (1.1), then

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

    and, for all n=3,4,5,,

    |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 fk-STq, then

    |a2|(1+q)Q12q

    and, for all n=3,4,5,,

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

    Corollary 8. If fk-UCVq, then

    |a2|Q12q

    and, for all n=3,4,5,,

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

    Theorem 5. Let fk-STq,p. Then f(U) contains an open disk of the radius given by

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

    Proof. Let w00 be a complex number such that f(z)w0 for zU. Then

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

    Since f1 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 fk-STq,p be of the form (1.1). Then, for a complex number μ,

    |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 fk-STq,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(z) is a Schwarz function with w(0)=0 and |w(z)|<1 for zU.

    Let hP 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 μ, 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 fk-UCVq,p be of the form (1.1). Then, for a complex number μ,

    |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 fk-STq be of the form (1.1) (with p=1). Then, for a complex number μ,

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

    The result is sharp.

    Corollary 10. Let the function fk-UCVq be of the form (1.1) (with p=1). Then, for a complex number μ,

    |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-STq,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 σ1μσ3, 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 σ3μσ2, 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-STq, 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 σ4μσ6, then

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

    If σ3μσ2, 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-UCVq,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 η1μη3, 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 η3μη2, 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-UCVq, 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 η4μη6, then

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

    If η3μη2, 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:

    U={z:zCand|z|<1}

    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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