In synthetic biology, it is a challenge to increase the production of target proteins by maximizing their expression levels. In order to augment expression levels, we need to focus on both homologous recombination and codon adaptation, which are estimated by three objective functions, namely HD (Hamming distance), LRCS (length of repeated or common substring) and CAI (codon adaptation index). Optimizing these objective functions simultaneously becomes a multi-objective optimization problem. The aim is to find satisfying solutions that have high codon adaptation and a low incidence of homologous recombination. However, obtaining satisfactory solutions requires calculating the objective functions multiple times with many cycles and solutions. In this paper, we propose an approach to accelerate the method of designing a set of CDSs (CoDing sequences) based on NSGA-II (non-dominated sorting genetic algorithm II) on NVIDIA GPUs. The implementation accelerated by GPUs improves overall performance by 187.5× using 100 cycles and 128 solutions. Our implementation allows us to use larger solutions and more cycles, leading to outstanding solution quality. The improved implementation provides much better solutions in a similar amount of time compared to other available methods by 1.22× improvements in hypervolume. Furthermore, our approach on GPUs also suggests how to efficiently utilize the latest computational resources in bioinformatics. Finally, we discuss the impacts of the number of cycles and the number of solutions on designing a set of CDSs.
Citation: Donghyeon Kim, Jinsung Kim. Optimization of designing multiple genes encoding the same protein based on NSGA-II for efficient execution on GPUs[J]. Electronic Research Archive, 2023, 31(9): 5313-5339. doi: 10.3934/era.2023270
[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 |
In synthetic biology, it is a challenge to increase the production of target proteins by maximizing their expression levels. In order to augment expression levels, we need to focus on both homologous recombination and codon adaptation, which are estimated by three objective functions, namely HD (Hamming distance), LRCS (length of repeated or common substring) and CAI (codon adaptation index). Optimizing these objective functions simultaneously becomes a multi-objective optimization problem. The aim is to find satisfying solutions that have high codon adaptation and a low incidence of homologous recombination. However, obtaining satisfactory solutions requires calculating the objective functions multiple times with many cycles and solutions. In this paper, we propose an approach to accelerate the method of designing a set of CDSs (CoDing sequences) based on NSGA-II (non-dominated sorting genetic algorithm II) on NVIDIA GPUs. The implementation accelerated by GPUs improves overall performance by 187.5× using 100 cycles and 128 solutions. Our implementation allows us to use larger solutions and more cycles, leading to outstanding solution quality. The improved implementation provides much better solutions in a similar amount of time compared to other available methods by 1.22× improvements in hypervolume. Furthermore, our approach on GPUs also suggests how to efficiently utilize the latest computational resources in bioinformatics. Finally, we discuss the impacts of the number of cycles and the number of solutions on designing a set of CDSs.
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] |
S. Fields, O. Song, A novel genetic system to detect protein–protein interactions, Nature, 340 (1989), 245–246. https://doi.org/10.1038/340245a0 doi: 10.1038/340245a0
![]() |
[2] |
S. Varambally, S. M. Dhanasekaran, M. Zhou, T. R. Barrette, C. Kumar-Sinha, M. G. Sanda, et al., The polycomb group protein ezh2 is involved in progression of prostate cancer, Nature, 419 (2002), 624–629. https://doi.org/10.1038/nature01075 doi: 10.1038/nature01075
![]() |
[3] |
G. Blander, L. Guarente, The sir2 family of protein deacetylases, Annu. Rev. Biochem., 73 (2004), 417–435. https://doi.org/10.1146/annurev.biochem.73.011303.073651 doi: 10.1146/annurev.biochem.73.011303.073651
![]() |
[4] |
S. P. Kaur, V. Gupta, Covid-19 vaccine: A comprehensive status report, Virus Res., 288 (2020), 198114. https://doi.org/10.1016/j.virusres.2020.198114 doi: 10.1016/j.virusres.2020.198114
![]() |
[5] |
M. Ahmad, M. Hirz, H. Pichler, H. Schwab, Protein expression in pichia pastoris: Recent achievements and perspectives for heterologous protein production, Appl. Microbiol. Biotechnol., 98 (2014), 5301–5317. https://doi.org/10.1007/s00253-014-5732-5 doi: 10.1007/s00253-014-5732-5
![]() |
[6] |
D. Fouque, K. Kalantar-Zadeh, J. Kopple, N. Cano, P. Chauveau, L. Cuppari, et al., A proposed nomenclature and diagnostic criteria for protein–energy wasting in acute and chronic kidney disease, Kidney Int., 73 (2008), 391–398. https://doi.org/10.1038/sj.ki.5002585 doi: 10.1038/sj.ki.5002585
![]() |
[7] | A. D. Bandaranayake, S. C. Almo, Recent advances in mammalian protein production, FEBS Lett., 588 (2014), 253–260. |
[8] |
J. Dehghani, A. Movafeghi, E. Mathieu-Rivet, N. Mati-Baouche, S. Calbo, P. Lerouge, et al., Microalgae as an efficient vehicle for the production and targeted delivery of therapeutic glycoproteins against sars-cov-2 variants, Marine Drugs, 20 (2022), 657. https://doi.org/10.3390/md20110657 doi: 10.3390/md20110657
![]() |
[9] |
S. C. Spohner, H. Müller, H. Quitmann, P. Czermak, Expression of enzymes for the usage in food and feed industry with pichia pastoris, J. Biotechnol., 202 (2015), 118–134. https://doi.org/10.1016/j.jbiotec.2015.01.027 doi: 10.1016/j.jbiotec.2015.01.027
![]() |
[10] | A. Haldimann, B. L. Wanner, Conditional-replication, integration, excision, and retrieval plasmid-host systems for gene structure-function studies of bacteria, J. Bacteriol., 183 (2001), 6384–6393. |
[11] |
P. Gu, F. Yang, T. Su, Q. Wang, Q. Liang, Q. Qi, A rapid and reliable strategy for chromosomal integration of gene (s) with multiple copies, Sci. Rep., 5 (2015), 1–9. https://doi.org/10.1038/srep09684 doi: 10.1038/srep09684
![]() |
[12] |
C. A. Scorer, J. J. Clare, W. R. McCombie, M. A. Romanos, K. Sreekrishna, Rapid selection using g418 of high copy number transformants of pichia pastoris for high–level foreign gene expression, Nat. Biotechnol., 12 (1994), 181–184. https://doi.org/10.1038/nbt0294-181 doi: 10.1038/nbt0294-181
![]() |
[13] |
K. E. Tyo, P. K. Ajikumar, G. Stephanopoulos, Stabilized gene duplication enables long-term selection-free heterologous pathway expression, Nat. Biotechnol., 27 (2009), 760–765. https://doi.org/10.1038/nbt.1555 doi: 10.1038/nbt.1555
![]() |
[14] |
G. Terai, S. Kamegai, A. Taneda, K. Asai, Evolutionary design of multiple genes encoding the same protein, Bioinformatics, 33 (2017), 1613–1620. https://doi.org/10.1093/bioinformatics/btx030 doi: 10.1093/bioinformatics/btx030
![]() |
[15] |
A. Vassileva, D. A. Chugh, S. Swaminathan, N. Khanna, Expression of hepatitis b surface antigen in the methylotrophic yeast pichia pastoris using the gap promoter, J. Biotechnol., 88 (2001), 21–35. https://doi.org/10.1016/S0168-1656(01)00254-1 doi: 10.1016/S0168-1656(01)00254-1
![]() |
[16] |
R. Aw, K. M. Polizzi, Can too many copies spoil the broth?, Microbial cell factories, 12 (2013), 1–9. https://doi.org/10.1186/1475-2859-12-128 doi: 10.1186/1475-2859-12-128
![]() |
[17] |
J. M. Buerstedde, N. Lowndes, D. G. Schatz, Induction of homologous recombination between sequence repeats by the activation induced cytidine deaminase (aid) protein, Elife, 3 (2014), e03110. https://doi.org/10.7554/eLife.03110 doi: 10.7554/eLife.03110
![]() |
[18] |
J. Jurka, P. Klonowski, V. Dagman, P. Pelton, Censor–a program for identification and elimination of repetitive elements from dna sequences, Comput. Chem., 20 (1996), 119–121. https://doi.org/10.1016/S0097-8485(96)80013-1 doi: 10.1016/S0097-8485(96)80013-1
![]() |
[19] |
J. Athey, A. Alexaki, E. Osipova, A. Rostovtsev, L. V. Santana-Quintero, U. Katneni, et al., A new and updated resource for codon usage tables, BMC Bioinf., 18 (2017), 1–10. https://doi.org/10.1186/s12859-017-1793-7 doi: 10.1186/s12859-017-1793-7
![]() |
[20] |
J. M. Comeron, M. Aguadé, An evaluation of measures of synonymous codon usage bias, J. Mol. Evol., 47 (1998), 268–274. https://doi.org/10.1007/PL00006384 doi: 10.1007/PL00006384
![]() |
[21] |
M. Gouy, C. Gautier, Codon usage in bacteria: correlation with gene expressivity, Nucleic Acids Res., 10 (1982), 7055–7074. https://doi.org/10.1093/nar/10.22.7055 doi: 10.1093/nar/10.22.7055
![]() |
[22] |
T. Ikemura, Correlation between the abundance of escherichia coli transfer rnas and the occurrence of the respective codons in its protein genes: a proposal for a synonymous codon choice that is optimal for the E. coli translational system, J. Mol. Biol., 151 (1981), 389–409. https://doi.org/10.1016/0022-2836(81)90003-6 doi: 10.1016/0022-2836(81)90003-6
![]() |
[23] |
P. M. Sharp, W. H. Li, The codon adaptation index-a measure of directional synonymous codon usage bias, and its potential applications, Nucleic Acids Res., 15 (1987), 1281–1295. https://doi.org/10.1093/nar/15.3.1281 doi: 10.1093/nar/15.3.1281
![]() |
[24] |
K. Deb, A. Pratap, S. Agarwal, T. Meyarivan, A fast and elitist multiobjective genetic algorithm: Nsga-ii, IEEE Trans. Evol. Comput., 6 (2002), 182–197. https://doi.org/10.1109/4235.996017 doi: 10.1109/4235.996017
![]() |
[25] |
B. Gonzalez-Sanchez, M. A. Vega-Rodríguez, S. Santander-Jiménez, J. M. Granado-Criado, Multi-objective artificial bee colony for designing multiple genes encoding the same protein, Appl. Soft Comput., 74 (2019), 90–98. https://doi.org/10.1016/j.asoc.2018.10.023 doi: 10.1016/j.asoc.2018.10.023
![]() |
[26] |
L. Dagum, R. Menon, Openmp: An industry standard api for shared-memory programming, IEEE Comput. Sci. Eng., 5 (1998), 46–55. https://doi.org/10.1109/99.660313 doi: 10.1109/99.660313
![]() |
[27] | Y. Zhou, Y. Tan, Gpu-based parallel multi-objective particle swarm optimization, Int. J. Artif. Intell., 7 (2011), 125–141. |
[28] | B. Gonzalez-Sanchez, M. A. Vega-Rodríguez, S. Santander-Jiménez, Parallel multi-objective optimization approaches for protein encoding, J. Supercomput., 1–31. https://doi.org/10.1007/s11227-021-04073-z |
[29] |
F. C. Holstege, E. G. Jennings, J. J. Wyrick, T. I. Lee, C. J. Hengartner, M. R. Green, et al., Dissecting the regulatory circuitry of a eukaryotic genome, Cell, 95 (1998), 717–728. https://doi.org/10.1016/S0092-8674(00)81641-4 doi: 10.1016/S0092-8674(00)81641-4
![]() |
[30] | Z. Jia, M. Maggioni, B. Staiger, D. P. Scarpazza, Dissecting the nvidia volta gpu architecture via microbenchmarking, preprint arXiv: 1804.06826. |
[31] |
T. U. Consortium, UniProt: The universal protein knowledgebase in 2023, Nucleic Acids Res., 51 (2023), D523–D531. https://doi.org/10.1093/nar/gkac1052 doi: 10.1093/nar/gkac1052
![]() |
[32] |
J. X. Chin, B. K. S. Chung, D. Y. Lee, Codon optimization online (cool): A web-based multi-objective optimization platform for synthetic gene design, Bioinformatics, 30 (2014), 2210–2212. https://doi.org/10.1093/bioinformatics/btu192 doi: 10.1093/bioinformatics/btu192
![]() |
[33] |
J. C. Guimaraes, M. Rocha, A. P. Arkin, G. Cambray, D-tailor: Automated analysis and design of dna sequences, Bioinformatics, 30 (2014), 1087–1094. https://doi.org/10.1093/bioinformatics/btt742 doi: 10.1093/bioinformatics/btt742
![]() |
[34] |
P. Puigbo, E. Guzmán, A. Romeu and S. Garcia-Vallve, Optimizer: a web server for optimizing the codon usage of dna sequences, Nucleic Acids Res., 35 (2007), W126–W131. https://doi.org/10.1093/nar/gkm219 doi: 10.1093/nar/gkm219
![]() |
[35] |
B. Gonzalez-Sanchez, M. A. Vega-Rodríguez, S. Santander-Jiménez, A multi-objective butterfly optimization algorithm for protein encoding, Appl. Soft Comput., 139 (2023), 110269. https://doi.org/10.1016/j.asoc.2023.110269 doi: 10.1016/j.asoc.2023.110269
![]() |
[36] | M. V. Díaz-Galián, M. A. Vega-Rodríguez, Many-objective approach based on problem-aware mutation operators for protein encoding, Inf. Sci., 613 (2022), 376–400. |
[37] |
K. Deb, H. Jain, An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, part i: solving problems with box constraints, IEEE Trans. Evol. Comput., 18 (2013), 577–601. 10.1109/TEVC.2013.2281535 doi: 10.1109/TEVC.2013.2281535
![]() |
[38] | I. Das, J. E. Dennis, Normal-boundary intersection: A new method for generating the pareto surface in nonlinear multicriteria optimization problems, SIAM J. Optim., 8 (1998), 631–657. |
[39] |
M. A. Dulebenets, An adaptive polyploid memetic algorithm for scheduling trucks at a cross-docking terminal, Inf. Sci., 565 (2021), 390–421. https://doi.org/10.1016/j.ins.2021.02.039 doi: 10.1016/j.ins.2021.02.039
![]() |
[40] |
J. Pasha, A. L. Nwodu, A. M. Fathollahi-Fard, G. Tian, Z. Li, H. Wang, et al., Exact and metaheuristic algorithms for the vehicle routing problem with a factory-in-a-box in multi-objective settings, Adv. Eng. Inf., 52 (2022), 101623. https://doi.org/10.1016/j.aei.2022.101623 doi: 10.1016/j.aei.2022.101623
![]() |
[41] | H. Gholizadeh, H. Fazlollahtabar, A. M. Fathollahi-Fard, M. A. Dulebenets, Preventive maintenance for the flexible flowshop scheduling under uncertainty: A waste-to-energy system, Environ. Sci. Pollut. Res., 1–20. https://doi.org/10.1007/s11356-021-16234-x |
[42] |
M. A. Dulebenets, M. Kavoosi, O. Abioye, J. Pasha, A self-adaptive evolutionary algorithm for the berth scheduling problem: Towards efficient parameter control, Algorithms, 11 (2018), 100. https://doi.org/10.3390/a11070100 doi: 10.3390/a11070100
![]() |
[43] |
H. Zhao, C. Zhang, An online-learning-based evolutionary many-objective algorithm, Inf. Sci., 509 (2020), 1–21. https://doi.org/10.1016/j.ins.2019.08.069 doi: 10.1016/j.ins.2019.08.069
![]() |
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 |