Citation: Yu Yuan, Chen Chen. Fault detection of rolling bearing based on principal component analysis and empirical mode decomposition[J]. AIMS Mathematics, 2020, 5(6): 5916-5938. doi: 10.3934/math.2020379
[1] | Xiaoli Zhang, Shahid Khan, Saqib Hussain, Huo Tang, Zahid Shareef . New subclass of q-starlike functions associated with generalized conic domain. AIMS Mathematics, 2020, 5(5): 4830-4848. doi: 10.3934/math.2020308 |
[2] | Syed Ghoos Ali Shah, Shahbaz Khan, Saqib Hussain, Maslina Darus . q-Noor integral operator associated with starlike functions and q-conic domains. AIMS Mathematics, 2022, 7(6): 10842-10859. doi: 10.3934/math.2022606 |
[3] | Saqib Hussain, Shahid Khan, Muhammad Asad Zaighum, Maslina Darus . Certain subclass of analytic functions related with conic domains and associated with Salagean q-differential operator. AIMS Mathematics, 2017, 2(4): 622-634. doi: 10.3934/Math.2017.4.622 |
[4] | K. R. Karthikeyan, G. Murugusundaramoorthy, N. E. Cho . Some inequalities on Bazilevič class of functions involving quasi-subordination. AIMS Mathematics, 2021, 6(7): 7111-7124. doi: 10.3934/math.2021417 |
[5] | Nazar Khan, Bilal Khan, Qazi Zahoor Ahmad, Sarfraz Ahmad . Some Convolution Properties of Multivalent Analytic Functions. AIMS Mathematics, 2017, 2(2): 260-268. doi: 10.3934/Math.2017.2.260 |
[6] | Huo Tang, Kadhavoor Ragavan Karthikeyan, Gangadharan Murugusundaramoorthy . Certain subclass of analytic functions with respect to symmetric points associated with conic region. AIMS Mathematics, 2021, 6(11): 12863-12877. doi: 10.3934/math.2021742 |
[7] | Shahid Khan, Saqib Hussain, Maslina Darus . Inclusion relations of q-Bessel functions associated with generalized conic domain. AIMS Mathematics, 2021, 6(4): 3624-3640. doi: 10.3934/math.2021216 |
[8] | Jianhua Gong, Muhammad Ghaffar Khan, Hala Alaqad, Bilal Khan . Sharp inequalities for q-starlike functions associated with differential subordination and q-calculus. AIMS Mathematics, 2024, 9(10): 28421-28446. doi: 10.3934/math.20241379 |
[9] | Mohammad Faisal Khan, Ahmad A. Abubaker, Suha B. Al-Shaikh, Khaled Matarneh . Some new applications of the quantum-difference operator on subclasses of multivalent q-starlike and q-convex functions associated with the Cardioid domain. AIMS Mathematics, 2023, 8(9): 21246-21269. doi: 10.3934/math.20231083 |
[10] | Huo Tang, Shahid Khan, Saqib Hussain, Nasir Khan . Hankel and Toeplitz determinant for a subclass of multivalent q-starlike functions of order α. AIMS Mathematics, 2021, 6(6): 5421-5439. doi: 10.3934/math.2021320 |
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.
[1] |
H. Zhao, H. Liu, J. Xu, et al. Performance prediction using high-order differential mathematical morphology gradient spectrum entropy and extreme learning machine, IEEE T. Instrum. Meas., 69 (2020), 4165-4172. doi: 10.1109/TIM.2019.2948414
![]() |
[2] |
S. Lu, P. Zheng, Y. Liu, et al. Sound-aided vibration weak signal enhancement for bearing fault detection by using adaptive stochastic resonance, J. Sound Vib., 449 (2019), 18-29. doi: 10.1016/j.jsv.2019.02.028
![]() |
[3] | T. Li, J. Shi, X. Li, et al. Image encryption based on pixel-level diffusion with dynamic filtering and DNA-level permutation with 3D Latin cubes, Entropy, 21 (2019), 1-21. |
[4] |
Y. Xu, H. Chen, J. Luo, et al. Enhanced Moth-flame optimizer with mutation strategy for global optimization, Inform. Sciences, 492 (2019), 181-203. doi: 10.1016/j.ins.2019.04.022
![]() |
[5] |
H. Zhao, J. Zheng, J. Xu, et al. Fault diagnosis method based on principal component analysis and broad learning system, IEEE Access, 7 (2019), 99263-99272. doi: 10.1109/ACCESS.2019.2929094
![]() |
[6] | H. Pan, Y. Yang, J. Zheng, et al. A noise reduction method of symplectic singular mode decomposition based on Lagrange multiplier, Mech. Syst. Signal Pr., 133 (2019), 1-21. |
[7] | Q. Shu, S. Lu, M. Xia, et al. Enhanced feature extraction method for motor fault diagnosis using low-quality vibration data from wireless sensor networks, Meas. Sci. Technol., 31 (2020), 045016 |
[8] |
R. Chen, S. K. Guo, X. Z. Wang, et al. Fusion of multi-RSMOTE with fuzzy integral to classify bug reports with an imbalanced distribution, IEEE T. Fuzzy Syst., 27 (2019), 2406-2420. doi: 10.1109/TFUZZ.2019.2899809
![]() |
[9] |
H. Zhao, S. Zuo, M. Hou, et al. A novel adaptive signal processing method based on enhanced empirical wavelet transform technology, Sensors, 18 (2018), 1-17. doi: 10.1109/JSEN.2018.2870228
![]() |
[10] |
Y. Liu, X. Wang, Z. Zhai, et al. Timely daily activity recognition from headmost sensor events, ISA T., 94 (2019), 379-390. doi: 10.1016/j.isatra.2019.04.026
![]() |
[11] | H. Zhao, J. Zheng, W. Deng, et al. Semi-supervised broad learning system based on manifold regularization and broad network, IEEE T. Circuits-I., 67 (2020), 983-994. |
[12] | T. Li, Z. Qian, T. He, Short-term load forecasting with improved CEEMDAN and GWO-based multiple kernel ELM, Complexity, 2020 (2020), 1-20. |
[13] |
J. Zheng, Z. Dong, H. Pan, et al. Composite multi-scale weighted permutation entropy and extreme learning machine based intelligent fault diagnosis for rolling bearing, Measurement, 143 (2019), 69-80. doi: 10.1016/j.measurement.2019.05.002
![]() |
[14] | G. Xu, D. M. Hou, H. Qi, et al. High-speed train wheel set bearing fault diagnosis and prognostics: A new prognostic model based on extendable useful life, Mech. Syst. Signal Pr., 146 (2020), 1-23. |
[15] | F. Zhang, J. Yan, P. Fu, et al. Ensemble sparse supervised model for bearing fault diagnosis in smart manufacturing, Mech. Syst. Signal Pr., 65 (2020),1-11. |
[16] |
R. Liu, B. Yang, E. Zio, et al. Artificial intelligence for fault diagnosis of rotating machinery: A review, Mech. Syst. Signal Pr., 108 (2018), 33-47. doi: 10.1016/j.ymssp.2018.02.016
![]() |
[17] |
R. A. Carmona, W. L. Hwang, B. Torresani, Characterization of signals by the ridges of their wavelet transforms, IEEE T. Signal Proces., 45 (1997), 2586-2590. doi: 10.1109/78.640725
![]() |
[18] | W. Deng, H. Liu, J. Xu, et al. An improved quantum-inspired differential evolution algorithm for deep belief network, IEEE T. Instrum. Meas., 2020 (2020), 1-8. |
[19] |
H. Li, Y. Zhang, H. Zheng, Hilbert-Huang transform and marginal spectrum for detection and diagnosis of localized defects in roller bearing, J. Mech. Sci. Technol., 23 (2009), 291-301. doi: 10.1007/s12206-008-1110-5
![]() |
[20] |
Y. Lei, J. Lin, Z. He, et al. A review on empirical mode decomposition in fault diagnosis of rotating machinery, Mech. Syst. Signal Pr., 35 (2013), 108-126. doi: 10.1016/j.ymssp.2012.09.015
![]() |
[21] |
Y. Wang, F. Liu, Z. Jiang, et al. Complex variational mode decomposition for signal processing applications, Mech. Syst. Signal Pr., 86 (2017), 75-85. doi: 10.1016/j.ymssp.2016.09.032
![]() |
[22] | N. Lu, Z. Xiao, O. P. Malik, Feature extraction using adaptive multiwavelets and synthetic detection index for rotor fault diagnosis of rotating machinery, Mech. Syst. Signal Pr., 52 (2015), 393-415. |
[23] |
J. Zheng, H. Pan, S. Yang, et al. Adaptive parameterless empirical wavelet transform based time-frequency analysis method and its application to rotor rubbing fault diagnosis, Signal Process., 130 (2017), 305-314. doi: 10.1016/j.sigpro.2016.07.023
![]() |
[24] | F. Cheng, J. Wang, L. Qu, et al. Rotor-current-based fault diagnosis for DFIG wind turbine drivetrain gearboxes using frequency analysis and a deep classifier, IEEE T. Ind. Appl., 54 (2017), 1062-1071. |
[25] |
H. Ocak, K. A. Loparo, HMM-based fault detection and diagnosis scheme for rolling element bearings, J. Vib. Acoust., 127 (2005), 299-306. doi: 10.1115/1.1924636
![]() |
[26] |
N. Gebraeel, M. Lawley, R. Liu, et al. Residual life predictions from vibration-based degradation signals: A neural network approach, IEEE T. Ind. Electron., 51 (2004), 694-700. doi: 10.1109/TIE.2004.824875
![]() |
[27] | C. Sun, Z. Zhang, Z. He, Rescarch on bearing life prediction based on support vector machine and its application, J. Phys. Conf. Ser., 305 (2011), 1-9. |
[28] |
P. J. Vlok, M. Wnek, M. Zygmunt, Utilising statistical residual life estimates of bearings to quantify the influence of preventive maintenance actions, Mech. Syst. Signal Pr., 18 (2004), 833-847. doi: 10.1016/j.ymssp.2003.09.003
![]() |
[29] | X. Zhang, R. Xu, C. Kwan, et al. An integrated approach to bearing fault diagnostics and prognostics, In: Proceedings of the 2005, American Control Conference, 2005, 2750-2755. |
[30] |
S. Lu, Q. He, J. Wang, A review of stochastic resonance in rotating machine fault detection, Mech. Syst. Signal Pr., 116 (2019), 230-260. doi: 10.1016/j.ymssp.2018.06.032
![]() |
[31] |
H. Zhao, D. Li, W. Deng, et al. Research on vibration suppression method of alternating current motor based on fractional order control strategy, P. I. Mech. Eng. E-J. Pro., 231 (2017), 786-799. doi: 10.1177/0954408916637380
![]() |
[32] | Y. Shao, K. Nezu, Prognosis of remaining bearing life using neural networks, P. I. Mech. Eng-I. Sys., 214 (2000), 217-230. |
[33] | W. Deng, J. Xu, Y. Song, et al. An effective improved co-evolution ant colony optimization algorithm with multi-strategies and its application, Int. J. Bio-Inspired Comput., 2019 (2019),1-10. |
[34] | H. Shao, J. Cheng, H. Jiang, et al. Enhanced deep gated recurrent unit and complex wavelet packet energy moment entropy for early fault prognosis of bearing, Knowl-Based Syst., 188 (2020), 1-14. |
[35] | H. Chen, Q. Zhang, J. Luo, et al. An enhanced Bacterial Foraging Optimization and its application for training kernel extreme learning machine, Appl. Soft Comput., 86 (2020), 1-24. |
[36] | W. Yang, M. Qian, D. Huang, Detection of exons with deletions and insertions by hidden markov models, Prog. Biochem. Biophys., 29 (2002), 56-59. |
[37] |
W. Deng, J. Xu, H. Zhao, An improved ant colony optimization algorithm based on hybrid strategies for scheduling problem, IEEE Access, 7 (2019), 20281-20292. doi: 10.1109/ACCESS.2019.2897580
![]() |
[38] |
Y. Liu, Y. Mu, K. Chen, et al. Daily activity feature selection in smart homes based on pearson correlation coefficient, Neural Process. Lett., 51 (2020), 1771-1787. doi: 10.1007/s11063-019-10185-8
![]() |
[39] |
Z. He, H. Shao, X. Zhang, et al. Improved deep transfer auto-encoder for fault diagnosis of gearbox under variable working conditions with small training samples, IEEE Access, 7 (2019), 115368-115377. doi: 10.1109/ACCESS.2019.2936243
![]() |
[40] |
W. Deng, W. Li, X. Yang, A novel hybrid optimization algorithm of computational intelligence techniques for highway passenger volume prediction, Expert Syst. Appl., 38 (2011), 4198-4205. doi: 10.1016/j.eswa.2010.09.083
![]() |
[41] | J. B. Ali, B. Chebel-Morello, L. Saidi, et al. Accurate bearing remaining useful life prediction based on Weibull distribution and artificial neural network, Mech. Syst. Signal Pr., 56 (2015), 150-172. |
[42] | M. Chen, Q. Li, Application of oil pressure curve monitoring in sliding bearing wear diagnosis, Machine Tool & Hydraulics, 21 (2010), 145-148. |
[43] | B. Samanta, K. R. Al-Balushi, Artificial neural network based fault diagnostics of rolling element bearings using time-domain features, Mech. Syst. Signal Pr., 17 (2003), 317-328. |
[44] |
Y. Zhang, S. Qin, Fault detection of nonlinear processes using multiway kernel independent component analysis, Ind. Eng. Chem. Res., 46 (2007), 7780-7787. doi: 10.1021/ie070381q
![]() |
[45] |
A. Hyvärinen, E. Oja, Independent component analysis: Algorithms and applications, Neural Networks, 13 (2000), 411-430. doi: 10.1016/S0893-6080(00)00026-5
![]() |
[46] |
R. B. W. Heng, M. J. M. Nor, Statistical analysis of sound and vibration signals for monitoring rolling element bearing condition, Appl. Acoust., 53 (1998), 211-226. doi: 10.1016/S0003-682X(97)00018-2
![]() |
[47] |
N. Tandon, A comparison of some vibration parameters for the condition monitoring of rolling element bearings, Measurement, 12 (1994), 285-289. doi: 10.1016/0263-2241(94)90033-7
![]() |
[48] |
R. Li, J. Chen, X. Wu, et al. Fault diagnosis of rotating machinery based on SVD, FCM and RST, Int. J. Adv. Manuf. Tech., 27 (2005), 128-135. doi: 10.1007/s00170-004-2140-5
![]() |
[49] | N. Gebraeel, M. Lawley, R. Liu, et al. Vibration-based condition monitoring of thrust bearings for maintenance management, In: Proc. ANNIE 2002 Smart Engineering System Design: Neural Networks, Fuzzy Logic, Evolutionary Programming, Artificial Life and Data Mining, 2002, 543-551. |
1. | Qiuxia Hu, Hari M. Srivastava, Bakhtiar Ahmad, Nazar Khan, Muhammad Ghaffar Khan, Wali Khan Mashwani, Bilal Khan, A Subclass of Multivalent Janowski Type q-Starlike Functions and Its Consequences, 2021, 13, 2073-8994, 1275, 10.3390/sym13071275 | |
2. | Adel A. Attiya, T. M. Seoudy, M. K. Aouf, Abeer M. Albalahi, Salah Mahmoud Boulaaras, Certain Analytic Functions Defined by Generalized Mittag-Leffler Function Associated with Conic Domain, 2022, 2022, 2314-8888, 1, 10.1155/2022/1688741 | |
3. | Neelam Khan, H. M. Srivastava, Ayesha Rafiq, Muhammad Arif, Sama Arjika, Some applications of q-difference operator involving a family of meromorphic harmonic functions, 2021, 2021, 1687-1847, 10.1186/s13662-021-03629-w | |
4. | Huo Tang, Kadhavoor Ragavan Karthikeyan, Gangadharan Murugusundaramoorthy, Certain subclass of analytic functions with respect to symmetric points associated with conic region, 2021, 6, 2473-6988, 12863, 10.3934/math.2021742 | |
5. | R.M. El-Ashwah, Subordination results for some subclasses of analytic functions using generalized q-Dziok-Srivastava-Catas operator, 2023, 37, 0354-5180, 1855, 10.2298/FIL2306855E | |
6. | Ebrahim Amini, Shrideh Al-Omari, Dayalal Suthar, Inclusion and Neighborhood on a Multivalent q-Symmetric Function with Poisson Distribution Operators, 2024, 2024, 2314-4629, 1, 10.1155/2024/3697215 | |
7. | Khadija Bano, Mohsan Raza, Starlikness associated with limacon, 2023, 37, 0354-5180, 851, 10.2298/FIL2303851B |