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:z∈Cand|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 (p∈N:={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 S∗p(α) of p-valently starlike functions of order α in U if and only if
ℜ(zf′(z)f(z))>α (0≦α<p;z∈U). | (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
ℜ(1+zf′′(z)f′(z))>α (0≦α<p;z∈U). | (1.3) |
The p-valent function classes S∗p(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)p∈S∗p(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 ℜ(p(z))>0.
Recently, Kanas and Wiśniowska [18,19] (see also [17,31]) introduced the conic domain Ωk(k≧0), which we recall here as follows:
Ωk={u+iv:u>k√(u−1)2+v2} |
or, equivalently,
Ωk={w:w∈Candℜ(w)>k|w−1|}. |
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+z1−z(k=0)1+2π2[log(1+√z1−√z)]2(k=1)1+11−k2cos(2iπ(arccosk)log(1+√z1−√z))(0<k<1)1+1k2−1sin(π2K(κ)∫u(z)√κ0dt√1−t2√1−κ2t2)+k2k2−1(1<k<∞) | (1.6) |
with
u(z)=z−√κ1−√κz(0<κ<1;z∈U), |
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(√1−κ2), |
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={1−qλ1−q(λ∈C)n−1∑k=0qk=1+q+q2⋯+qn−1(λ=n∈N), | (1.7) |
so that
limq→1−[λ]q=1−qλ1−q=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)(1−q)z(z≠0)f′(0)(z=0), | (1.8) |
provided that f′(0) exists.
We note from Definition 2 that
limq→1−Dqf(z)=limq→1−f(z)−f(qz)(1−q)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]qzp−1+∞∑n=p+1[n]qanzn−1. | (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 S∗q,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)∈S∗q,p(α)⟺ℜ(1[p]qzDqf(z)f(z))>α | (1.10) |
(0<q<1;0≦α<1;z∈U) |
and
f(z)∈Cq,p(α)⟺ℜ(1[p]qDp,q(zDqf(z))Dqf(z))>α | (1.11) |
(0<q<1;0≦α<1;z∈U), |
respectively. From (1.10) and (1.11), it follows that
f(z)∈Cq,p(α)⟺zDqf(z)[p]q∈S∗q,p(α). | (1.12) |
For the simpler classes S∗q,p and C∗q,p of p-valently q-starlike functions in U and p-valently q-convex functions in U, respectively, we have write
S∗q,p(0)=:S∗q,pandCq,p(0)=:Cq,p. |
Obviously, in the limit when q→1−, the function classes S∗q,p(α) and Cq,p(α) reduce to the familiar function classes S∗p(α) and Cp(α), respectively.
Definition 3. A function f∈A(p) is said to belong to the class S∗q,p of p-valently q-starlike functions in U if
|zDqf(z)[p]qf(z)−11−q|≤11−q(z∈U). | (1.13) |
In the limit when q→1−, the closed disk
|w−11−q|≦11−q(0<q<1) |
becomes the right-half plane and the class S∗q,p of p-valently q-starlike functions in U reduces to the familiar class S∗p 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+z1−qz). | (1.14) |
We note that S∗q,1=S∗q (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)+(1−q)pk(z), |
where pk(z) is given by (1.6).
Geometrically, the function p∈k-Pq takes on all values from the domain Ωk,q (k≧0) which is defined as follows:
Ωk,q={w:ℜ((1+q)w(q−1)w+2)>k|(1+q)w(q−1)w+2−1|}. | (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 q→1−, to the conic domain Ωk studied by Kanas and Wiśniowska [18]. We note the following simpler cases.
(1) k-Pq⊆P(2k2k+1+q), where P(2k2k+1+q) is the familiar class of functions with real part greater than 2k2k+1+q;
(2) limq→1−{k-Pq}=P(pk(z)), where P(pk(z)) is the known class introduced by Kanas and Wiśniowska [18];
(3) limq→1−{0-Pq}=P, where P is Carathéodory class of analytic functions with positive real part.
Definition 5. A function f∈A(p) is said to be in the class k-STq,p if and only if
ℜ((1+q)zDqf(z)[p]qf(z)(q−1)zDqf(z)[p]qf(z)+2)>k|(1+q)zDqf(z)[p]qf(z)(q−1)zDqf(z)[p]qf(z)+2−1|(z∈U) | (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) limq→1{k-STq,p}=k-STp, where k-STp is the class of p-valently uniformly starlike functions;
(4) limq→1{0-STq,p}=Sp, where S∗p is the class of p-valently starlike functions;
(5) 0-STq,1=S∗q, where S∗q (see [12,41]);
(6) limq→1{k-STq,1}=k-ST, where k-ST is a function class introduced and studied by Kanas and Wiśniowska [19];
(7) limq→1{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,p⟺zDqf(z)[p]q∈k-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 0≦k<∞ be fixed and let pk be defined by (1.6). If
pk(z)=1+Q1z+Q2z2+⋯, |
then
Q1={2A21−k2(0≦k<1)8π2(k=1)π24√t(k2−1)[K(t)]2(1+t)(1<k<∞) | (2.1) |
and
Q2={A2+23Q1(0≦k<1)23Q1(k=1)4[K(t)]2(t2+6t+1)−π224√t[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 0≦k<∞ be fixed and suppose that
pk,q(z)=2pk(z)(1+q)+(1−q)pk(z), | (2.3) |
where pk(z) be defined by (1.6). Then
pk,q(z)=1+12(1+q)Q1z+12(1+q)[Q2−12(1−q)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=1cnzn∈P |
be analytic in U and satisfy ℜ(h(z))>0 for z in U. Then the following sharp estimate holds true:
|c2−vc21|≦2max{1,|2v−1|}(∀v∈C). |
The result is sharp for the functions given by
g(z)=1+z21−z2org(z)=1+z1−z. | (2.5) |
Lemma 4. (see [26]) If the function h is given by
h(z)=1+∞∑n=1cnzn∈P, |
then
|c2−νc21|≦{−4ν+2(ν≦0)2(0≦ν≦1)4ν−2(ν≧1). | (2.6) |
When ν>1, the equality holds true if and only if
h(z)=1+z1−z |
or one of its rotations. If 0<ν<1, then the equality holds true if and only if
h(z)=1+z21−z2 |
or one of its rotations. If ν=0, the equality holds true if and only if
h(z)=(1+λ2)(1+z1−z)+(1−λ2)(1−z1+z)(0≦λ≦1) |
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:
|c2−νc21|+ν|c1|2≦2(0≦ν≦12) |
and
|c2−νc21|+(1−ν)|c1|2≦2(12≤ν≦1). |
We assume throughout our discussion that, unless otherwise stated, 0≦k<∞, 0<q<1, p∈N, Q1 is given by (2.1), Q2 is given by (2.2) and z∈U.
Theorem 1. If a function f∈A(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]q−1}|an|<(1+q)[p]q, | (3.1) |
then the function f∈k-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)(q−1)zDqf(z)[p]qf(z)+2−1|−ℜ((1+q)zDqf(z)[p]qf(z)(q−1)zDqf(z)[p]qf(z)+2−1)<1. |
In fact, we have
k|(1+q)zDqf(z)[p]qf(z)(q−1)zDqf(z)[p]qf(z)+2−1|−ℜ((1+q)zDqf(z)[p]qf(z)(q−1)zDqf(z)[p]qf(z)+2−1)≦(k+1)|(1+q)zDqf(z)[p]qf(z)(q−1)zDqf(z)[p]qf(z)+2−1|=2(k+1)|zDqf(z)−[p]qf(z)(q−1)zDqf(z)+2[p]qf(z)|=2(k+1)|∞∑n=p+1([n]q−[p]q)anzn−p(1+q)[p]q+∞∑n=p+1((q−1)[n]q+2[p]q)anzn−p|≦2(k+1)∞∑n=p+1([n]q−[p]q)|an|(1+q)[p]q−∞∑n=p+1(qn+2[p]q−1)|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]q−1}(n≧p+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]q−1}zn(n≧p+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∈A is of the form (1.1) (with p=1) and satisfies the following condition:
∞∑n=2{2(k+1)([n]q−1)+qn+1}|an|<(1+q) |
then the function f∈k-STq.
Letting q→1− 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 f∈A(p) is of the form (1.1) and satisfies the following condition:
∞∑n=p+1{(k+1)(n−p)+p}|an|<p, |
then the function f∈k-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 f∈A(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]q−1}|an|<(1+q)[p]q, |
then the function f∈k-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 f∈A is of the form (1.1) (with p=1) and satisfies the following condition:
∞∑n=2[n]q{2(k+1)([n]q−1)+qn+1}|an|<(1+q), |
then the function f∈k-UCVq.
Letting q→1− in Theorem 2, we obtain the following corollary (see [29,Theorem 1]) with
α1=p+1,β1=p,αℓ=1(ℓ=2,⋯,s+1)andβj=1(j=2,⋯,s). |
Corollary 5. If a function f∈A(p) is of the form (1.1) and satisfies the following condition:
∞∑n=p+1(np){(k+1)(n−p)+p}|an|<p, |
then the function f∈k-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 f∈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∈k-UCV.
Theorem 3. If f∈k-STq,p, then
|ap+1|≦(1+q)[p]qQ12qp[1]q | (3.2) |
and, for all n=3,4,5,⋯,
|an+p−1|≦(1+q)[p]qQ12qp[n−1]qn−2∏j=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=1cnzn∈k-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+p−1 on both sides of (3.5), we obtain
([n+p−1]q−[p]q)an+p−1=[p]q{cn−1+ap+1cn−2+⋯+an+p−2c1}. |
By taking the moduli on both sides and then applying the following coefficient estimates (see [50]):
|cn|≦12(1+q)Q1(n∈N), |
we find that
|an+p−1|≦(1+q)[p]qQ12qp[n−1]q{1+|ap+1|+⋯+|an+p−2|}. | (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∈N), that is,
|at+p−1|≦(1+q)[p]qQ12qp[t−1]qt−2∏j=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+p−1|}≦(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[t−1]qt−2∏j=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[t−1]q)}=(1+q)[p]qQ12qp[t]qt−1∏j=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(n∈N∖{1}). This completes the proof of Theorem 3.
Similarly, we can prove the following result.
Theorem 4. If f∈k-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+p−1|≦(1+q)[p]2qQ12qp[n+p−1]q[n−1]qn−2∏j=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∈k-STq, then
|a2|≦(1+q)Q12q |
and, for all n=3,4,5,⋯,
|an|≦(1+q)Q12q[n−1]qn−2∏j=1(1+(1+q)Q12q[j]q). |
Corollary 8. If f∈k-UCVq, then
|a2|≦Q12q |
and, for all n=3,4,5,⋯,
|an|≦(1+q)Q12q[n]q[n−1]qn−2∏j=1(1+(1+q)Q12q[j]q). |
Theorem 5. Let f∈k-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 w0≠0 be a complex number such that f(z)≠w0 for z∈U. Then
f1(z)=w0f(z)w0−f(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 f∈k-STq,p be of the form (1.1). Then, for a complex number μ,
|ap+2−μa2p+1|≦[p]qQ12qpmax{1,|Q2Q1+([p]q(1+q)−qp(1−q))Q12qp⋅(1−μ(1+q)2[p]q[p]q(1+q)−qp(1−q))|}. | (3.8) |
The result is sharp.
Proof. If f∈k-STq,p, we have
zDqf(z)[p]qf(z)≺pk,q(z)=2pk(z)(1+q)+(1−q)pk(z). |
From the definition of the differential subordination, we know that
zDqf(z)[p]qf(z)=pk,q(w(z))(z∈U), | (3.9) |
where w(z) is a Schwarz function with w(0)=0 and |w(z)|<1 for z∈U.
Let h∈P be a function defined by
h(z)=1+w(z)1−w(z)=1+c1z+c2z2+⋯(z∈U). |
This gives
w(z)=12c1z+12(c2−c212)z2+⋯ |
and
pk,q(w(z))=1+1+q4c1Q1z+1+q4{Q1c2+12(Q2−Q1−1−q2Q21)c21}z2+⋯. | (3.10) |
Using (3.10) in (3.9), we obtain
ap+1=(1+q)[p]qc1Q14qp |
and
ap+2=[p]qQ14qp[c2−12(1−Q2Q1−[p]q(1+q)−qp(1−q)2qpQ1)c21] |
Now, for any complex number μ, we have
ap+2−μa2p+1=[p]qQ14qp[c2−12(1−Q2Q1−[p]q(1+q)−qp(1−q)2qpQ1)c21]−μ(1+q)2[p]2qQ21c2116q2p. | (3.11) |
Then (3.11) can be written as follows:
ap+2−μa2p+1=[p]qQ14qp{c2−vc21}, | (3.12) |
where
v=12[1−Q2Q1−([p]q(1+q)−qp(1−q))Q12qp⋅(1−μ(1+q)2[p]q[p]q(1+q)−qp(1−q))]. | (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∈k-UCVq,p be of the form (1.1). Then, for a complex number μ,
|ap+2−μa2p+1|≦[p]2qQ12qp[p+2]qmax{1,|Q2Q1+((1+q)[p]q−(1−q)qp)Q12qp⋅(1−μ[p+2]q(1+q)2[p]2q((1+q)[p]q−(1−q)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∈k-STq be of the form (1.1) (with p=1). Then, for a complex number μ,
|a3−μa22|≦Q12qmax{1,|Q2Q1+(1+q2)Q12q(1−μ(1+q)21+q2)|}. |
The result is sharp.
Corollary 10. Let the function f∈k-UCVq be of the form (1.1) (with p=1). Then, for a complex number μ,
|a3−μa22|≦Q12q[3]qmax{1,|Q2Q1+(1+q2)Q12q(1−μ[3]q1+q2)|}. |
The result is sharp.
Theorem 8. Let
σ1=([p]q(1+q)−qp(1−q))Q21+2qp(Q2−Q1)[p]q(1+q)2Q21, |
σ2=([p]q(1+q)−qp(1−q))Q21+2qp(Q2+Q1)[p]q(1+q)2Q21 |
and
σ3=([p]q(1+q)−qp(1−q))Q21+2qpQ2[p]q(1+q)2Q21. |
If the function f given by (1.1) belongs to the class k-STq,p, then
|ap+2−μa2p+1|≦{[p]qQ12qp{Q2Q1+([p]q(1+q)−qp(1−q))Q12qp(1−μ(1+q)2[p]q[p]q(1+q)−qp(1−q))}(μ≦σ1)[p]qQ12qp(σ1≦μ≦σ2),−[p]qQ12qp{Q2Q1+([p]q(1+q)−qp(1−q))Q12qp(1−μ(1+q)2[p]q[p]q(1+q)−qp(1−q))}(μ≧σ2). |
Furthermore, if σ1≦μ≦σ3, then
|ap+2−μa2p+1|+2qp(1+q)2[p]qQ1{1−Q2Q1−([p]q(1+q)−qp(1−q))Q12qp⋅(1−μ(1+q)2[p]q([p]q(1+q)−qp(1−q)))}|ap+1|2≦[p]qQ12qp. |
If σ3≦μ≦σ2, then
|ap+2−μa2p+1|+2qp(1+q)2[p]qQ1{1+Q2Q1+([p]q(1+q)−qp(1−q))Q12qp⋅(1−μ(1+q)2[p]q([p]q(1+q)−qp(1−q)))}|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(Q2−Q1)(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
|a3−μa22|≦{Q12q{Q2Q1+(1+q2)Q12q(1−μ(1+q)21+q2)}(μ≦σ4)Q12q(σ4≦μ≦σ5)−Q12q{Q2Q1+(1+q2)Q12q(1−μ(1+q)21+q2)}(μ≧σ5). |
Furthermore, if σ4≦μ≦σ6, then
|a3−μa22|+2q(1+q)2Q1{1−Q2Q1−(1+q2)Q12q(1−μ(1+q)21+q2)}|a2|2≦Q12q. |
If σ3≦μ≦σ2, then
|a3−μa22|+2q(1+q)2Q1{1+Q2Q1+(1+q2)Q12q(1−μ(1+q)21+q2)}|a2|2≦Q12q. |
Similarly, we can prove the following result.
Theorem 9. Let
η1=[((1+q)[p]q−(1−q)qp)Q21+2qp(Q2−Q1)][p+1]2q[p]2q[p+2]q(1+q)2Q21, |
η2=[((1+q)[p]q−(1−q)qp)Q21+2qp(Q2+Q1)][p+1]2q[p]2q[p+2]q(1+q)2Q21 |
and
η3=[((1+q)[p]q−(1−q)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+2−μa2p+1|≦{[p]2qQ12qp[p+2]q{Q2Q1+((1+q)[p]q−(1−q)qp)Q12qp(1−[p+2]q(1+q)2[p]2q μ((1+q)[p]q−(1−q)qp)[p+1]2q)}(μ≦η1)[p]2qQ12qp[p+2]q(η1≦μ≦η2)−[p]2qQ12qp[p+2]q{Q2Q1+((1+q)[p]q−(1−q)qp)Q12qp(1−[p+2]q(1+q)2[p]2q μ((1+q)[p]q−(1−q)qp)[p+1]2q)}(μ≧η2). |
Furthermore, if η1≦μ≦η3, then
|ap+2−μa2p+1|+2qp[p+1]2qQ1[p+2]q(1+q)2[p]2qQ21{1−Q2Q1−((1+q)[p]q−(1−q)qp)Q12qp⋅(1−μ[p+2]q(1+q)2[p]2q((1+q)[p]q−(1−q)qp)[p+1]2q)}|ap+1|2≦[p]2qQ12qp[p+2]q. |
If η3≦μ≦η2, then
|ap+2−μa2p+1|+2qp[p+1]2qQ1[p+2]q(1+q)2[p]2qQ21{1+Q2Q1+((1+q)[p]q−(1−q)qp)Q12qp⋅(1−μ[p+2]q(1+q)2[p]2q((1+q)[p]q−(1−q)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(Q2−Q1)[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
|a3−μa22|≦{Q12q[3]q{Q2Q1+(1+q2)Q12q(1−μ[3]q1+q2)}(μ≦η4)Q12q[3]q(η4≦μ≦η5)−Q12q[3]q{Q2Q1+(1+q2)Q12q(1−μ[3]q1+q2)}(μ≧η5). |
Furthermore, if η4≦μ≦η6, then
|a3−μa22|+2q[3]qQ1{1−Q2Q1−(1+q2)Q12q(1−μ[3]q1+q2)}|a2|2≦Q12q[3]q. |
If η3≦μ≦η2, then
|a3−μa22|+2q[3]qQ1{1+Q2Q1+(1+q2)Q12q(1−μ[3]q1+q2)}|a2|2≦Q12q[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:z∈Cand|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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