
Wave equations describing a wide variety of wave phenomena are commonly seen in mathematical physics. The inclusion of a noise term in a deterministic wave equation allows neglected degrees of freedom or fluctuations of external fields describing the environment to be considered in the equation. Moreover, adding a noise term to the deterministic equation reveals remarkable new features in the qualitative behavior of the solution. For example, noise can lead to singularities in some equations and prevent singularities in others. Taking into account the effects of the fluctuations along with a space-time white noise, we consider a relativistic wave equation with weak and strong damping terms and investigate the effect of multiplicative noise on the behavior of solutions. The existence of local and global solutions is provided, and some qualitative properties of solutions, such as continuous dependence of solutions on initial data, and blow up of solutions, are given. Moreover, an upper bound is provided for the blow up time.
Citation: Hatice Taskesen. Qualitative results for a relativistic wave equation with multiplicative noise and damping terms[J]. AIMS Mathematics, 2023, 8(7): 15232-15254. doi: 10.3934/math.2023778
[1] | Bo Wang, Rekha Srivastava, Jin-Lin Liu . Certain properties of multivalent analytic functions defined by $ q $-difference operator involving the Janowski function. AIMS Mathematics, 2021, 6(8): 8497-8508. doi: 10.3934/math.2021493 |
[2] | Huo Tang, Shahid Khan, Saqib Hussain, Nasir Khan . Hankel and Toeplitz determinant for a subclass of multivalent $ q $-starlike functions of order $ \alpha $. AIMS Mathematics, 2021, 6(6): 5421-5439. doi: 10.3934/math.2021320 |
[3] | Pinhong Long, Huo Tang, Wenshuai Wang . Functional inequalities for several classes of q-starlike and q-convex type analytic and multivalent functions using a generalized Bernardi integral operator. AIMS Mathematics, 2021, 6(2): 1191-1208. doi: 10.3934/math.2021073 |
[4] | Ahmad A. Abubaker, Khaled Matarneh, Mohammad Faisal Khan, Suha B. Al-Shaikh, Mustafa Kamal . Study of quantum calculus for a new subclass of $ q $-starlike bi-univalent functions connected with vertical strip domain. AIMS Mathematics, 2024, 9(5): 11789-11804. doi: 10.3934/math.2024577 |
[5] | 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 |
[6] | 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 |
[7] | Muhammad Sabil Ur Rehman, Qazi Zahoor Ahmad, H. M. Srivastava, Nazar Khan, Maslina Darus, Bilal Khan . Applications of higher-order q-derivatives to the subclass of q-starlike functions associated with the Janowski functions. AIMS Mathematics, 2021, 6(2): 1110-1125. doi: 10.3934/math.2021067 |
[8] | Hanen Louati, Afrah Al-Rezami, Erhan Deniz, Abdulbasit Darem, Robert Szasz . Application of $ q $-starlike and $ q $-convex functions under $ (u, v) $-symmetrical constraints. AIMS Mathematics, 2024, 9(12): 33353-33364. doi: 10.3934/math.20241591 |
[9] | Aisha M. Alqahtani, Rashid Murtaza, Saba Akmal, Adnan, Ilyas Khan . Generalized $ q $-convex functions characterized by $ q $-calculus. AIMS Mathematics, 2023, 8(4): 9385-9399. doi: 10.3934/math.2023472 |
[10] | 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 |
Wave equations describing a wide variety of wave phenomena are commonly seen in mathematical physics. The inclusion of a noise term in a deterministic wave equation allows neglected degrees of freedom or fluctuations of external fields describing the environment to be considered in the equation. Moreover, adding a noise term to the deterministic equation reveals remarkable new features in the qualitative behavior of the solution. For example, noise can lead to singularities in some equations and prevent singularities in others. Taking into account the effects of the fluctuations along with a space-time white noise, we consider a relativistic wave equation with weak and strong damping terms and investigate the effect of multiplicative noise on the behavior of solutions. The existence of local and global solutions is provided, and some qualitative properties of solutions, such as continuous dependence of solutions on initial data, and blow up of solutions, are given. Moreover, an upper bound is provided for the blow up time.
Assume that A represents the family of analytic functions in the open unit disc
U={z:z∈C and |z|<1}. |
For f1, f2∈A, we say that f1 subordinate to f2 in U, indicated by
f1(z)≺f2(z), z∈U, |
if there exists a Schwarz function w, defined by
w∈B={w:w∈A, |w(z)|<1 and w(0)=0, z∈U}, |
that satisfies the condition
f1(z)=f2(w(z)), z∈U. |
Indeed, it is known that
f1(z)≺f2(z)⟹f1(0)=f2(0) and f1(U)⊂f2(U). |
Moreover, if the function f2 is univalent in U then
f1(z)≺f2(z)⇔f1(0)=f2(0) and f1(U)⊂f2(U). |
Let the class P be defined by
P={h∈A:h(0)=1 and Re(h(z)>0}. |
The class of all functions in the normalized analytic function class A that are univalent in U is also denoted by the symbol S. The maximization of the non-linear functional |a3−μa22| or other classes and subclasses of univalent functions has been the subject of a number of established results and these results are known as Fekete-Szegö problems, see [1]. If f∈S and of the form (1.1), then
|a3−μa22|≤{3−4μ, if μ≤0,1+2exp(2μμ−1), if 0≤μ<1,4μ−3, if μ≥1,} |
and the result |a3−μa22| are sharp (see [1]). The Fekete-Szegö problems have a rich history in literature and for complex number μ.
In the area of geometric function theory (GFT), the q-calculus and fractional q-calculus have been extensively employed by scholars who have developed and investigated a number of novel subclasses of analytic, univalent and bi-univalent functions. Jackson [2,3] first proposed the concept of the q-calculus operator and gave the definition of the q -difference operator Dq in 1909. In instance, Ismail et al. were the first to define a class of q-starlike functions in open unit disc U using Dq in [4]. The most significant usages of q -calculus in the perspective of GFT was basically furnished and the basic (or q-) hypergeometric functions were first used in GFT in a book chapter by Srivastava (see, for details, [5]). See the following articles [6,7,8,9,10] for more information about q-calculus operator theory in GFT.
Now we review some fundamental definitions and ideas of the q -calculus, we utilize them to create some new subclasses in this paper.
For a non-negative integer l, the q-number [l]q, (0<q<1), is defined by
[l]q=1−ql1−q and [0]=0, |
and the q-number shift factorial is given by
[l]q!=[1]q[2]q[3]q⋯[l]q,[0]q!=1. |
For q→1−, then [l]! reduces to l!.
The q-generalized Pochhammer symbol is defined by
[l]k=Γq(l+k)Γq(l), k∈N, l∈C. |
The q-gamma function Γq is defined by
Γq(l)=(1−q)l∞∏j=01−qj+11−qj+l. |
The q-generalized Pochhammer symbol is defined by
[l]k=Γq(l+k)Γq(l), k∈N, l∈C. |
Remark 1. For q→1−, then [l]q,k reduces to (l)k=Γ(l+k)Γ(l).
Definition 1. Jackson [3] defined the q-integral of function f(z) as follows:
∫f(z)dq(z)=∞∑n=0z(1−q)f(qn(z))qn. |
Jackson [2] introduced the q-difference operator for analytic functions as follows:
Definition 2. [2] For f∈A, the q-difference operator is defined as:
Dqf(z)=f(qz)−f(z)z(q−1), z∈U. |
Note that, for n∈N, z∈U and
Dq(zn)=[n]qzn−1, Dq(∞∑n=1anzn)=∞∑n=1[n]qanzn−1. |
Let Ap stand for the class of analytic functions with the form
f(z)=zp+∞∑n=1an+pzn+p, p∈N,z∈U | (1.1) |
in the open unit disk U. More specifically, A1=A and
f(z)=z+∞∑n=1an+1zn+1, z∈U. | (1.2) |
Consider the q-difference operator for f∈Ap as follows:
Definition 3. [11] For f∈Ap, the q-difference operator is defined as:
Dqf(z)=f(qz)−f(z)z(q−1), z∈U. |
Note that, for n∈N, z∈U and
Dq(zn+p)=[n+p]qzn+p−1, Dq(∞∑n=1an+pzn+p)=∞∑n=1[n+p]qan+pzn+p−1. |
Let S∗(p) represents the class of p-valent starlike functions and every f∈S∗(p), if
Re(zf′(z)pf(z))>0, z∈U, |
and K(p) represents the class of p-valent convex functions and every f∈K(p), if
1p(1+Re(zf′′(z)f′(z)))>0, z∈U. |
These conditions are equivalent in terms of subordination as follows:
S∗(p)={f∈Ap:zf′(z)pf(z)≺1+z1−z} |
and
K(p)={f∈Ap:1p(1+zf′′(z)f′(z))≺1+z1−z}. |
The aforementioned two classes can be generalized as follows:
S∗(p,φ)={f∈Ap:zf′(z)pf(z)≺φ(z)} |
and
K(p,φ)={f∈Ap:1p(1+zf′′(z)f′(z))≺φ(z)}, |
where φ(z) is a real part function that is positive and is normalized by the rule
φ(0)=1 and φ′(0)>0, |
and φ maps U onto a space that is symmetric with regard to the real axis and starlike with respect to 1. If p=1, then
S∗(p,φ)=S∗(φ) |
and
K(p,φ)=K(φ). |
These two classes S∗(φ) and K(φ) defined by Ma [12].
A function f∈Ap, is called p-valently starlike of order α (0≤α<1) with complex order b∈C∖{0}, if it satisfies the inequality
Re{1+1b(zf′(z)pf(z)−1)>α, z∈U}. |
The class S∗p(α,b) denotes the collection of all f∈Ap functions that satisfy the aforementioned condition.
A function f∈Ap, is called p-valently convex function of order α (0≤α<1) with complex order b∈C∖{0}, if it satisfies the inequality
Re{1−1b+1bp(1+zf′′(z)f′(z))>α, z∈U}. |
The class Kp(α,b) denotes the collection of all functions f∈Ap that satisfy the aforementioned condition.
Note that
f∈Kp(α,b)⇔1pzf′∈S∗p(α,b). |
Kargar et al. [13] investigated the classes Sp(α,β) for f∈Ap and defined as follows:
f∈Sp(α,β)⇔α<Re(1pzf′(z)f(z))<β, (0≤α<1<β, z∈U). |
For 0≤α<1<β and b∈C∖{0}, then the function f∈Ap belongs to the class Kb,p(α,β) if it satisfies the inequality
α<Re(1−1b+1bp(1+zf′′(z)f′(z)))<β, (0≤α<1<β, z∈U). |
If p=1, then Kb,p(α,β)=Kb(α,β), studied by Kargar et al. in [13] and if β→∞ in above definition, Kb,p(α,β)=Kb,p(α,b).
Recently, Bult [14] used the definition of subordination and defined new subclasses of p-valent starlike and convex functions associated with vertical strip domain as follows:
S∗p,b(α,β)={f∈Ap:1+1b(1pzf′(z)f(z)−1)≺f(α,β;z)} |
and
Kp,b(α,β)={f∈Ap:1−1b+1bp(1+zf′′(z)f′(z))≺f(α,β;z)}, |
where
f(α,β;z)=1+β−απilog(1−e2πi1−αβ−αz1−z) |
and
0≤α<1<β,b∈C∖{0}, z∈U. |
Bult [14] determined the coefficient bounds for functions belonging to these new classes.
On the basis of the geometrical interpretation of their image domains, numerous subclasses of analytic functions have established using the concept of subordination. Right half plane [15], circular disc [16], oval and petal type domains [17], conic domain [18,19], leaf-like domain [20], generalized conic domains [21], and the most important one is shell-like curve [22,23,24,25] are some fascinating geometrical classes we obtain with this domain. The function
h(z)=1+τ2z21−τz−τ2z2 | (1.3) |
is essential for the shell-like shape, where
τ=1−√52. |
The image of unit circle under the function h gives the conchoid of Maclaurin's, due to the function
h(eiφ)=√52(3−2cosφ)+isinφ(4cosφ−1)2(3−2cosφ)(1+cosφ), 0≤φ<2π. |
The function given in (1.3) has the following series representation:
h(z)=1+∞∑n=1(un−1+un+1)τnzn, |
where
un=(1−τ)n−τn√5, |
and un produces a Fibonacci series of coefficient constants that are more closely related to the Fibonacci numbers.
Taking motivation from the idea of circular disc and shell-like curves, Malik et al. [26] defined new domain for analytic functions which is named as cardioid domain. A new class of analytic functions is defined associated with cardioid domain, for more detail, see [26].
Definition 4. [26] Assume that CP(L,N) represents the class of functions p that are defined as
p(z)≺¯p(L,N,z), |
where ¯p(L,N,z) is defined by
¯p(L,N,z)=2Lτ2z2+(L−1)τz+22Nτ2z2+(N−1)τz+2 | (1.4) |
with −1<N<L≤1, τ=1−√52 and z∈U.
To understand the class CP(L,N), an explanation of the function ¯p(L,N,z) in geometric terms might be helpful in this instance. If we denote
R¯p(L,N;eiθ)=u |
and
I¯p(L,N;eiθ)=v, |
then the image ¯p(L,N,eiθ) of the unit circle is a cardioid like curve defined by
{u=4+(L−1)(N−1)τ2+4LNτ4+2λcosθ+4(L+N)τ2cos2θ4+(N−1)2τ2+4N2τ4+4(N−1)(τ+Nτ3)cosθ+8Nτ2cos2θ,v=(L−N)(τ−τ3)sinθ+2τ2sin2θ4+(N−1)2τ2+4N2τ4+4(N−1)(τ+Nτ3)cosθ+8Nτ2cos2θ,} | (1.5) |
where
λ=(L+N−2)τ+(2LN−L−N)τ3, −1<N<L≤1, τ=1−√52, |
and
0≤θ≤2π. |
Moreover, we observe that
¯p(L,N,0)=1, |
and
¯p(L,N,1)=LN+9(L+N)+1+4(N−L)√5N2+18N+1. |
According to (1.5), the cusp of the cardioid-like curve is provided by
γ(L,N)=¯p(L,N;e±iarccos(14))=2LN−3(L+N)+2+(L−N)√52(N2−3N+1). |
The image of each inner circle is a nested cardioid-like curve if the open unit disc U is considered a collection of concentric circles with origin at the center. As a result, the open unit disc U is mapped onto a cardioid region by the function ¯p(L,N,z). This means that ¯p(L,N;U) is a cardioid domain. The above discussed cardioid like curve with different values of parameters can be seen in Figures 1 and 2.
The relationship N<L links the parameters L and N. The cardioid-like curve is flipped by its voilation, as seen in the figures below.
See Figure 3, if collection of concentric circles having origin as center. Thus, the function ¯p(L,N,z) maps the open unit disk U onto a cardioid region. See [26] for more details about cardioid region.
The operator theory of quantum calculus is the primary result of this research. Using standard uses in quantum calculus operator theory and the q -difference operator, we develop numerous novel q-analogous of the differential and integral operators. We construct a large number of new classes of q-starlike and q-convex functions using these operators and study some interesting characteristics of the corresponding analytic functions. In this paper, we gain inspiration from recent research by [14,26,27] and define two new classes of p-valent starlike, convex functions connected with the cardioid domain using the q-difference operator.
Influenced by recent studies [14,26,27], we defined two new classes of p-valent starlike, convex functions related with cardioid domain.
Definition 5. The function f of the form (1.1) related with cardioid domain, represented by S∗p(L,N,q,b), is defined to be the functions f such that
1+1b(1[p]qzDqf(z)f(z)−1)≺¯p(L,N;z), |
where, b∈C∖{0} and ¯p(L,N;z) is given by (1.4).
Definition 6. The function f of the form (1.1) related with cardioid domain, represented by Kp(L,N,q,b), is defined to be the functions f such that
1−1b+1b[p]q(1+zD2qf(z)Dqf(z))≺¯p(L,N;z), |
where b∈C∖{0} and ¯p(L,N;z) is given by (1.4) and Kp(L,N,q,b) is the class of convex functions of order b related with cardioid domain.
Special cases:
(i) For q→1−,b=1 and p=1, in Definition 5, we have known class S∗(L,N) of starlike functions associated with cardioid domain proved by Zainab et al. in [27].
(ii) For q→1−,L=1,N=−1, b=1 and p=1 in Definition 5, then class S∗p(L,N,q,b)=SL and this class is defined on starlike functions associated with Fibonacci numbers, introduced and studied by Sokół in [28].
(iii) For q→1−, L=1, N=−1, b=1 and p=1 in Definition 6 then class Kp(L,N,q,b)=K, and this family is referred to as a class of convex functions connected with Fibonacci numbers.
There are four parts to this article. In Section 1, we briefly reviewed some basic concepts from geometric function theory, quantum calculus, and cardioid domain, studied the q-difference operator, and finally discussed this operator to define two new subclasses of multivalent q-starlike and q-convex functions. The established lemmas are presented in Section 2. Our main results and some known corollaries will be presented in Section 3, then some concluding remarks in Section 4.
By utilizing the following lemmas, we will determine our main results.
Lemma 1. [26] Let the function ¯p(L,N;z), defined by (1.4). Then,
(i) For the disc |z|<τ2, the function ¯p(L,N;z) is univalent.
(ii) If h(z)≺¯p(L,N;z), then Reh(z)>α, where
α=2(L+N−2)τ+2(2LN−L−N)τ3+16(L+N)τ2η4(N−1)(τ+Nτ3)+32Nτ2η, |
where
η=4+τ2−N2τ2−4N2τ4−(1−Nτ2)χ(N)4τ(1+N2τ2), |
χ(N)=√5(2Nτ2−(N−1)τ+2)(2Nτ2+(N−1)τ+2), |
−1<N<L≤1 |
and
τ=1−√52. |
(iii) If
¯p(L,N;z)=1+∞∑n=1¯Qnzn, |
then
¯Qn={(L−N)τ2,for n=1,(L−N)(5−N)τ222,for n=2,1−N2τpn−1−Nτ2pn−2,for n=3,4,5,⋯, | (2.1) |
where
−1<N<L≤1. |
(iv) Let h(z)≺¯p(L,N;z) and of the form h(z)=1+∞∑n=1hnzn. Then
|h2−vh21|≤(L−N)|τ|4max{2,|τ(v(L−N)+N−5)|}, v∈C. |
Lemma 2. [29] Let h∈P, such that h(z)=1+∞∑n=1cnzn. Then
|c2−v2c21|≤max{2,2|v−1|}={2,if 0≤v≤2,2|v−1|, elsewhere,} | (2.2) |
and
|cn|≤2 for n≥1. | (2.3) |
Lemma 3. [30] Let h∈P, such that
h(z)=1+∞∑n=1cnzn. |
Then for any complex number v
|c2−vc21|≤2max{1,|2v−1|} |
and the result is sharp for
h(z)=1+z21−z2andh(z)=1+z1−z. |
Lemma 4. [31] Let the function g given by
g(z)=∞∑k=1bkzk |
be convex in U. Also let the function f given by
f(z)=∞∑k=1akzk |
be analytic in U. If
f(z)≺g(z), |
then
|ak|<|b1|, k=1,2,3,⋯. |
For the recently described classes of multivalent q-starlike (S∗p(L,N,q,b)) and multivalent q-convex (Kp(L,N,q,b)) functions, we get sharp estimates for the coefficients of Taylor series, Fekete-Szegő problems and coefficient inequalities.
In the following theorems, we investigate the functions f(z) which can be used to find the sharpness of the results of this article.
Theorem 5. A function f∈Ap given by (1.1) is in the class S∗p(L,N,q,b) if and only if there exists an analytic function S,
S(z)≺¯p(L,N,z)=2Lτ2z2+(L−1)τz+22Nτ2z2+(N−1)τz+2, |
where, ¯p(L,N,0)=1, such that
f(z)=zpexp{b[p]qz∫0S(t)−1tdt}, z∈U. | (3.1) |
Proof. Let f∈S∗p(L,N,q,b) and
1+1b(1[p]qzDqf(z)f(z)−1)=h(z)≺¯p(L,N,z). |
Then by integrating this equation we obtain (3.1). Conversely, if given by (3.1) with an analytic function S(z) such that S(z)≺¯p(L,N,z), then by logarithmic differentiation of (3.1) we obtain
1+1b(1[p]qzDqf(z)f(z)−1)=S(z). |
Therefore we have
1+1b(1[p]qzDqf(z)f(z)−1)≺¯p(L,N,z) |
and f∈S∗p(L,N,q,b).
The initial coefficient bounds |ap+1| and |ap+2| for the functions f∈S∗p(L,N,b) are investigated in Theorem 6 using the Lemma 2.
Theorem 6. Let f∈S∗p(L,N,q,b) be given by (1.1), −1≤N<L≤1. Then
|ap+1|≤[p]q|b|(L−N)τ2,|ap+2|≤[p]q|b|(L−N)|τ|28(5−N+[p]qb(L−N)). |
These bounds are sharp.
Proof. Let f∈S∗p(L,N,q,b), and of the form (1.1). Then
1+1b(1[p]qzDqf(z)f(z)−1)≺¯p(L,N;z), | (3.2) |
where
¯p(L,N,z)=2Lτ2z2+(L−1)τz+22Nτ2z2+(N−1)τz+2. |
By applying the concept of subordination, there exists a function w with
w(0)=0 and |w(z)|<1, |
such that
1+1b(1[p]qzDqf(z)f(z)−1)=¯p(L,N;w(z)). | (3.3) |
Let
w(z)=h(z)−1h(z)+1=c1z+c2z2+c3z3+⋯2+c1z+c2z2+⋯=12c1z+12(c2−12c21)z2+12(c3−c1c2+14c31)z3+⋯. | (3.4) |
Since
¯p(L,N;z)=1+∞∑n=1¯Qnzn, |
then
¯p(L,N;w(z))=1+¯Q1{12c1z+12(c2−12c21)z2⋯}+¯Q2{12c1z+12(c2−12c21)z2⋯}2+⋯=1+¯Q1c12z+(12(c2−12c21)¯Q1+¯Q2c214)z2+⋯. | (3.5) |
Also consider the function
¯p(L,N;z)=2Lτ2z2+(L−1)τz+22Nτ2z2+(N−1)τz+2. |
Let τz=α0, then
¯p(L,N,z)=2Lα20+(L−1)α0+22Nα20+(N−1)α0+2=Lα20+(L−1)2α0+1Nα20+(N−1)2α0+1=(Lα20+(L−1)2α0+1)[1+12(1−N)α0+(N2−6N+14)α20+⋯]=1+12(L−N)α0+14(L−N)(5−N)α20+⋯. |
This implies that
¯p(L,N;z)=1+12(L−N)τz+14(L−N)(5−N)τ2z2+⋯. | (3.6) |
It is simple to observe from (3.5) that
¯p(L,N;w(z))=1+14(L−N)τc1z+(14(L−N)τ(c2−12c21)+(L−N)(5−N)τ2c2116)z2+⋯. | (3.7) |
Since f∈S∗p(L,N,b), then
1+1b(1[p]qzDqf(z)f(z)−1)=1+1b[p]qap+1z+1b[p]q(2ap+2−a2p+1)z2+⋯. | (3.8) |
It is simple to show that by utilizing (3.3) and comparing the coefficients from (3.7) and (3.8), we get
ap+1=b[p]q(L−N)τc14. | (3.9) |
Applying modulus on both side, we have
|ap+1|≤[p]q|b|(L−N)τ2. |
Now again comparing the coefficients from (3.7) and (3.8), we have
2b[p]qap+2=14(L−N)τ(c2−12c21)+(L−N)(5−N)τ2c2116+1[p]qba2p+1, |
|ap+2|=b[p]q(L−N)τ8|c2−v2c21|, | (3.10) |
where
v=1−τ2(5−N+[p]qb(L−N)). |
It shows that v>2 which is satisfied by the relation L>N. Hence, by applying Lemma 2, we obtain the required result.
Result is sharp for the function
f∗(z)=zpexp(b[p]qz∫0¯p(L,N,t)−1tdt)=zp+b[p]q(L−N)τ2zp+1+b[p]q(L−N)(5−N)τ28zp+2+⋯, | (3.11) |
where ¯p(L,N,) defined in (1.4).
Letting q→1−, b=1 and p=1 in Theorem 6, we get the known corollary proved in [32] for starlike functions connected with cardioid domain.
Corollary 1. [32] Let f∈S∗(L,N) be given by (1.2), −1≤N<L≤1. Then
|a2|≤(L−N)|τ|2,|a3|≤(L−N)|τ|28{L−2N+5}. |
Fekete-Szegö problem |ap+2−μa2p+1| for the functions f∈S∗p(L,N,b) are investigated in Theorem 7.
Theorem 7. Let f∈S∗p(L,N,q,b) and of the form (1.1). Then
|ap+2−μa2p+1|≤[p]q|b|(L−N)|τ|8max{2,|τ(−(L−N)[p]qb+N−5+2[p]qb(L−N)μ)|}. |
This result is sharp.
Proof. Since f∈S∗p(L,N,q,b), we have
1+1b(1[p]qzDqf(z)f(z)−1)=¯p(L,N;w(z)), z∈U, |
where w is Schwarz function such that w(0) and |w(z)|<1 in U. Therefore
1+1b(1[p]qzDqf(z)f(z)−1)=h(z),1+1[p]qbzDqf(z)f(z)=1b+(1+h1z+h2z2+⋯),zDqf(z)=[p]qbf(z)(1b+h1z+h2z2+⋯), |
and after some simple calculation, we have
[p]qzp+[p+1]qap+1zp+1+[p+2]qap+2zp+2+⋯=[p]qb{zp+ap+1zp+1+ap+2zp+2+⋯}(1b+h1z+h2z2+⋯)=[p]q{zp+ap+1zp+1+ap+2zp+2+⋯}(1+bh1z+bh2z2+⋯). |
Comparing the coefficients of both sides, we get
ap+1=[p]qbh1, 2ap+2=[p]qb(h1ap+1+h2). |
This implies that
|ap+2−μa2p+1|=[p]q|b|2|h2+(1−2μ)[p]qbh21|=[p]q|b|2|h2−vh21|, |
where
v=(2μ−1)[p]qb. |
By using (iv) of Lemma 1 for
v=(2μ−1)[p]qb, |
we have the required result. The equality
|ap+2−μa2p+1|=[p]q|b|(L−N)|τ|28|(L−N)[p]qb−N+5−2[p]qb(L−N)μ| |
holds for f∗ given in (3.11). Consider f0: U→C defined as:
f0(z)=zpexp([p]qbz∫0¯p(L,N;t2)−1tdt)=zp+[p]qbτ2(L−N)zp+2+⋯, | (3.12) |
where, ¯p(L,N;z) is defined in (1.4). Hence
1+1b(1[p]qzDqf(z)f(z)−1)=¯p(L,N;z2). |
This demonstrates f0∈S∗p(L,N,q,b). Hence the equality
|ap+2−μa2p+1|=[p]q|b|(L−N)|τ|2 |
holds for the function f0 given in (3.12).
Letting q→1−, b=1 and p=1 in Theorem 7, we get the known corollary proved in [32] for starlike functions associated with cardioid domain.
Corollary 2. [32] Let f∈S∗(L,N) and of the form (1.2). Then
|a3−μa22|≤(L−N)|τ|8max{2,|τ(−(L−2N+5)+2(L−N)μ)|}. |
This result is sharp.
Coefficient inequality for the class S∗p(L,N,b):
Theorem 8. For function f∈Ap, given by (1.1), if f∈S∗p(L,N,q,b), then
|ap+n|≤n+1∏k=2([k−2]q+[p]qqp|b((L−N)τ2)|)[n]q!, ( p,n∈N). |
Proof. Suppose f∈S∗p(L,N,q,b) and the function S(z) define by
S(z)=1+1b(1[p]qzDqf(z)f(z)−1). | (3.13) |
Then by Definition 5, we have
S(z)≺¯p(L,N;z), |
where, b∈C∖{0} and ¯p(L,N;z) is given by (1.4). Hence, applying the Lemma 4, we get
|S(m)(0)m!|=|cm|≤|¯Q1|, m∈N, | (3.14) |
where
S(z)=1+c1z+c2z2+⋯, |
and by (2.1), we have
|¯Q1|=|(L−N)τ2|. | (3.15) |
Also from (3.13), we find
zDqf(z)=[p]q{b[q(z)−1]+1}f(z). | (3.16) |
Since ap=1, in view of (3.16), we obtain
[n+p]q−[p]qap+n=[p]qb{cn+cn−1ap+1+⋯+c1ap+n−1}=b[p]qn∑i=1ciap+n−i. | (3.17) |
Applying (3.14) into (3.17), we get
qp[n]q|ap+n|≤[p]q|b||¯Q1|n∑i=1|ap+n−i|, p,n∈N. |
For n=1,2,3, we have
|ap+1|≤[p]qqp|b¯Q1|,|ap+2|≤[p]q|b¯Q1|qp[2]q(1+|ap+1|)≤[p]q|b¯Q1|qp[2]q(1+[p]qqp|b¯Q1|) |
and
|ap+3|≤[p]q|b¯Q1|qp[3]q(1+|ap+1|+|ap+2|)≤[p]q|b¯Q1|qp[3]q(1+[p]qqp|b¯Q1|+[p]q|b¯Q1|qp[2]q(1+[p]qqp|b¯Q1|))=[p]qqp|b¯Q1|((1+[p]qqp|b¯Q1|)([2]q+[p]qqp|b¯Q1|)[3]q[2]q), |
respectively. Applying the equality (3.15) and using the mathematical induction principle, we obtain
|ap+n|≤n+1∏k=2([k−2]q+[p]qqp|b¯Q1|)[n]q!=n+1∏k=2([k−2]q+[p]qqp|b((L−N)τ2)|)[n]q!. |
This evidently completes the proof of Theorem 8.
The initial coefficient bounds |ap+1| and |ap+2| for the functions f∈Kp(L,N,q,b) are investigated in Theorem 9 using the Lemma 2.
Theorem 9. Let f∈Kp(L,N,q,b) be given by (1.1), −1≤N<L≤1. Then
|ap+1|≤[p]2q|b|(L−N)τ2[p+1]q,|ap+2|≤b2[p]q(L−N)τ8[p+2]q(5−N+[p]qb2(L−N)). |
These bounds are sharp.
Proof. Let f∈Kp(L,N,q,b), and be of the form (1.1). Then
1−1b+1b[p]q(1+zD2qf(z)Dqf(z))≺¯p(L,N;z), | (3.18) |
where
¯p(L,N,z)=2Lτ2z2+(L−1)τz+22Nτ2z2+(N−1)τz+2. |
By applying the concepts of subordination, there exists a function w with
w(0)=0 and |w(z)|<1, |
such that
1−1b+1b[p]q(1+zD2qf(z)Dqf(z))=¯p(L,N;w(z)). | (3.19) |
Let
w(z)=h(z)−1h(z)+1=c1z+c2z2+c3z3+⋯2+c1z+c2z2+⋯=12c1z+12(c2−12c21)z2+12(c3−c1c2+14c31)z3+⋯. | (3.20) |
Since
¯p(L,N;z)=1+∞∑n=1¯Qnzn, |
then
¯p(L,N;w(z))=1+¯Q1{12c1z+12(c2−12c21)z2⋯}+¯Q2{12c1z+12(c2−12c21)z2⋯}2+⋯=1+¯Q1c12z+(12(c2−12c21)¯Q1+¯Q2c214)z2+⋯. | (3.21) |
Also consider the function
¯p(L,N;z)=2Lτ2z2+(L−1)τz+22Nτ2z2+(N−1)τz+2. |
Let τz=α0. Then
¯p(L,N,z)=2Lα20+(L−1)α0+22Nα20+(N−1)α0+2=Lα20+(L−1)2α0+1Nα20+(N−1)2α0+1=(Lα20+(L−1)2α0+1)[1+12(1−N)α0+(N2−6N+14)α20+⋯]=1+12(L−N)α0+14(L−N)(5−N)α20+⋯. |
This implies that
¯p(L,N;z)=1+12(L−N)τz+14(L−N)(5−N)τ2z2+⋯. | (3.22) |
It is simple to observe from (3.21) that
¯p(L,N;w(z))=1+14(L−N)τc1z+(14(L−N)τ(c2−12c21)+(L−N)(5−N)τ2c2116)z2+⋯. | (3.23) |
Since f∈Kp(L,N,q,b), then
1−1b+1b[p]q(1+zD2qf(z)Dqf(z))=1+[p+1]qb[p]qap+1z+1b[p]q(2[p+2]qap+2−[p+1]2q[p]qa2p+1)z2+⋯. | (3.24) |
It is simple to show that by utilizing (3.19) and comparing the coefficients from (3.23) and (3.24), we get
ap+1=b[p]2q(L−N)τc14[p+1]q. |
Applying modulus on both side, we have
|ap+1|≤[p]2q|b|(L−N)τ2[p+1]q. |
Now again comparing the coefficients from (3.7) and (3.8), we have
2[p+2]qb[p]qap+2=14(L−N)τ(c2−12c21)+(L−N)(5−N)τ2c2116+(p+1)2bp2a2p+1, |
|ap+2|=b[p]q(L−N)τ8[p+2]q|c2−v2c21|, |
where
v=1−τ2(5−N+[p]qb2(L−N)), |
it shows that v>2 which is satisfied by the relation L>N. Hence, by applying Lemma 2, we obtain the required result.
Result is sharp for the function
f∗(z)=zpexp(b[p]2qz∫0¯p(L,N,t)−1tdt)=zp+b[p]2q(L−N)τ2zp+1+b[p]2q(L−N)(5−N)τ28zp+2+⋯, | (3.25) |
where ¯p(L,N,) defined in (1.4).
Fekete-Szegö problem |ap+2−μa2p+1| for the functions f∈Kp(L,N,q,b) are investigated in Theorem 10.
Theorem 10. Let f∈Kp(L,N,q,b) and be of the form (1.1). Then
|ap+2−μa2p+1|≤[p]q|b|(L−N)|τ|4([p+1]q−[p]q+1)[p+2]q×max{2,|τ(−(L−N)[p]2qb+N−5+[p]3qb[p+2]q([p+1]q−[p]q+1)(L−N)[p+1]2qμ)|}. |
This result is sharp.
Proof. Since f∈Kp(L,N,b), we have
1−1b+1b[p]q(1+zD2qf(z)Dqf(z))=¯p(L,N;w(z)), z∈U, |
where w is Schwarz function such that w(0) and |w(z)|<1 in U. Therefore
1−1b+1b[p]q(1+zD2qf(z)Dqf(z))=h(z),1+zD2qf(z)Dqf(z)=(1b−1)b[p]q+b[p]q(1+h1z+h2z2+⋯),1+zD2qf(z)Dqf(z)=([p]q+b[p]qh1z+b[p]qh2z2+⋯),1+zD2qf(z)Dqf(z)=[p]q[1+bh1z+bh2z2+⋯], |
and after some simple calculation, we have
[p]q(1+[p−1]q)zp−1+([p+1]q([p]q+1)ap+1zp+[p+2]q([p+1]q+1)ap+2zp+1+⋯=[p]q{[p]qzp−1+[p+1]qap+1zp+[p+2]qap+2zp+1+⋯}(1+bh1z+bh2z2+⋯)=[p]2qzp−1+([p]q([p+1]qap+1+[p]2qbh1)zp+{[p]q[p+2]qap+2+[p]q[p+1]qbh1ap+1+[p]2qbh2}zp+1. |
Comparing the coefficients of both sides, we get
ap+1=[p]2qbh1[p+1]q, ap+2=[p]qb[p+2]q([p+1]q−[p]q+1)([p+1]qh1ap+1+h2). |
This implies that
ap+2−μa2p+1=[p]q|b|([p+1]q−[p]q+1)[p+2]q×(h2+(1−[p+2]q[p]q([p+1]q−[p]q+1)[p+1]2qμ)[p]2qbh21)=[p]q|b|([p+1]q−[p]q+1)[p+2]q(h2−vh21), |
where
v=([p+2]q[p]q([p+1]q−[p]q+1)[p+1]2qμ−1)[p]2qb. |
By using (iv) of Lemma 1 for
v=([p+2]q[p]q([p+1]q−[p]q+1)[p+1]2qμ−1)[p]2qb, |
we have the required result. The equality
|ap+2−μa2p+1|=[p]q|b|(L−N)|τ|24([p+1]q−[p]q+1)[p+2]q×|(L−N)[p]2qb−N+5−[p]3qb([p+1]q−[p]q+1)[p+2]q(L−N)[p+1]2qμ| |
holds for f∗ given in (3.25). Consider the function f0: U→C be defined as:
f0(z)=zpexp([p]2qbz∫0¯p(L,N;t2)−1tdt)=zp+τ[p]2qb2(L−N)zp+2+⋯, |
where, ¯p(L,N;z) is defined in (1.4). Hence and
1−1b+1b[p]q(1+zD2qf(z)Dqf(z))=¯p(L,N;z2). |
Theorem 11. Let f∈Ap, be given by (1.1). If f∈Kp(L,N,b), then
|ap+n|≤[p]qn+1∏k=2([k−2]q+[2]q[p]qqp|b((L−N)τ2)|)[n]q!([p+n]q), p,n∈N. |
Proof. We can obtain Theorem 11, by using the same technique of Theorem 8.
In this article, we have used the ideas of cardioid domain, multivalent analytic functions, and q-calculus operator theory to define the new subfamilies of multivalent q-starlike and q-convex functions. In Section 1, we discussed some basic concepts from geometric functions, analytic functions, multivalent functions, q-calculus operator theory, and the idea of the cardioid domain. We also define two new classes of p-valent starlike, convex functions connected with the cardioid domain using the q -difference operator. The already known lemmas are presented in Section 2. In Section 3, for the class S∗p(L,N,q,b), we investigated sharp coefficient bounds, Fekete-Szegö functional, and coefficient inequalities. Same type of results also studied for the class S∗p(L,N,q,b). The research also demonstrated how the parameters, including some new discoveries, expand and enhance the results.
For future studies, researchers can use a number of ordinary differential and q-analogous of difference and integral operators and can define a number of new subclasses of multivalent functions. By applying the ideas of this article, many new results can be found. The idea presented in this article can be implemented on papers [33,34,35], and researchers can discuss the new properties of multivalent functions associated with the cardioid domain.
The authors declare that they did not employ any artificial intelligence in the execution of this work.
The authors extend their appreciation to the Arab Open University for funding this work through research fund No. (AOURG-2023-007).
All the authors claim to have no conflicts of interest.
[1] | J. M. Sancho, M. San Miguel, S. Katz, J. D. Gunton, Multiplicative noise in stochastic differential equations: A numerical study, Sixth international conference on noise in physical systems, 1981, 26–29. |
[2] |
V. Pata, S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495. http://dx.doi.org/10.1088/0951-7715/19/7/001 doi: 10.1088/0951-7715/19/7/001
![]() |
[3] |
W. J. Lehr, J. L. Park, A stochastic derivation of the Klein-Gordon equation, J. Math. Phys., 18 (1977), 1235–1240. http://dx.doi.org/10.1063/1.523396 doi: 10.1063/1.523396
![]() |
[4] |
A. I. Arbab, Stochastic model related to the Klein-Gordon equation revisited, EPL, 96 (2011), 20002. http://dx.doi.org/10.1209/0295-5075/96/20002 doi: 10.1209/0295-5075/96/20002
![]() |
[5] |
S. Gur, M. E. Uysal, Continuous dependence of solutions to the strongly damped nonlinear Klein-Gordon equation, Turk. J. Math., 42 (2018), 904–910. http://dx.doi.org/10.3906/mat-1706-30 doi: 10.3906/mat-1706-30
![]() |
[6] |
A. I. Komech, E. A. Kopylova, Weighted energy decay for 3D Klein-Gordon equation, J. Differ. Equations, 248 (2010), 501–520. http://dx.doi.org/10.1016/j.jde.2009.06.011 doi: 10.1016/j.jde.2009.06.011
![]() |
[7] |
W. Y. Huang, W. L. Chen, Global existence and blow up of solutions for nonlinear Klein-Gordon equation with damping term and nonnegative potentials, Abstr. Appl. Anal., 2014 (2014), 142892. http://dx.doi.org/10.1155/2014/142892 doi: 10.1155/2014/142892
![]() |
[8] |
Y. Wang, A sufficient condition for finite time blow up of the nonlinear Klein-Gordon equations with arbitrarily positive initial energy, Proc. Amer. Math. Soc., 136 (2008), 3477–3482. http://dx.doi.org/10.1090/s0002-9939-08-09514-2 doi: 10.1090/s0002-9939-08-09514-2
![]() |
[9] | N. Burq, G. Raugel, W. Schlag, Long time dynamics for damped Klein-Gordon equations, 2017, arXiv: 1505.05981. |
[10] |
T. Cazenave, Uniform estimates for solutions of nonlinear Klein-Gordon equations, J. Funct. Anal., 60 (1985), 36–55. http://dx.doi.org/10.1016/0022-1236(85)90057-6 doi: 10.1016/0022-1236(85)90057-6
![]() |
[11] |
T. G. Ha, J. Y. Park, Global existence and uniform decay of a damped Klein-Gordon equation in a noncylindrical domain, Nonlinear Anal. Theor., 74 (2011), 577–584. http://dx.doi.org/10.1016/j.na.2010.09.011 doi: 10.1016/j.na.2010.09.011
![]() |
[12] |
R. Xu, Global existence, blow up and asymptotic behaviour of solutions for nonlinear Klein-Gordon equation with dissipative term, Math. Method. Appl. Sci., 33 (2010), 831–844. http://dx.doi.org/10.1002/mma.1196 doi: 10.1002/mma.1196
![]() |
[13] |
P. L. Chow, Stochastic wave equations with polynomial nonlinearity, Ann. Appl. Probab., 12 (2002), 361–381. http://dx.doi.org/10.1214/aoap/1015961168 doi: 10.1214/aoap/1015961168
![]() |
[14] |
V. Barb, G. D. Prato, The stochastic nonlinear damped wave equation, Appl. Math. Optim., 46 (2002), 125–141. http://dx.doi.org/10.1007/s00245-002-0744-4 doi: 10.1007/s00245-002-0744-4
![]() |
[15] |
P. L. Chow, Asymptotics of solutions to semilinear stochastic wave equations, Ann. Appl. Probab., 16 (2006), 757–789. http://dx.doi.org/10.1214/105051606000000141 doi: 10.1214/105051606000000141
![]() |
[16] |
P. L. Chow, Nonlinear stochastic wave equations: Blow up of second moments in L 2-norm, Ann. Appl. Probab., 19 (2009), 2039–2046. http://dx.doi.org/10.1214/09-AAP602 doi: 10.1214/09-AAP602
![]() |
[17] |
J. U. Kim, On the stochastic wave equation with nonlinear damping, Appl. Math. Optim., 58 (2008), 29–67. http://dx.doi.org/10.1007/s00245-007-9029-2 doi: 10.1007/s00245-007-9029-2
![]() |
[18] |
M. Ondrejat, Stochastic nonlinear wave equations in local Sobolev spaces, Electron. J. Probab., 15 (2010), 1041–1091. http://dx.doi.org/10.1214/EJP.v15-789 doi: 10.1214/EJP.v15-789
![]() |
[19] |
R. D. Parshad, M. Beauregard, A. Kasimov, B. Said-Houari, Global existence and finite time blow up in a class of stochastic nonlinear wave equations, COSA, 8 (2014), 7. http://dx.doi.org/10.31390/cosa.8.3.07 doi: 10.31390/cosa.8.3.07
![]() |
[20] |
Z. Brzezniak, M. Ondrejat, J. Seidler, Invariant measures for stochastic nonlinear beam and wave equations, J. Differ. Equations, 260 (2016), 4157–4179. http://dx.doi.org/10.1016/j.jde.2015.11.007 doi: 10.1016/j.jde.2015.11.007
![]() |
[21] |
F. Liang, Explosive solutions of stochastic nonlinear beam equations with damping, J. Math. Anal. Appl., 419 (2014), 849–869. http://dx.doi.org/10.1016/j.jmaa.2014.04.065 doi: 10.1016/j.jmaa.2014.04.065
![]() |
[22] |
L. Bo, D. Tang, Y. Wang, Explosive solutions of stochastic wave equations with damping on Rd, J. Differ. Equations, 244 (2008), 170–187. http://dx.doi.org/10.1016/j.jde.2007.10.016 doi: 10.1016/j.jde.2007.10.016
![]() |
[23] | G. G. Xu, L. B. Wang, G. G. Lin, Global random attractor for the strongly damped stochastic wave equation with white noise and nonlinear term, Theor. Math. Appl., 3 (2013), 111–122. |
[24] |
P. L. Chow, Nonexistence of global solutions to nonlinear stochastic wave equations in mean Lp-norm, Stoch. Anal. Appl., 30 (2012), 543–551. http://dx.doi.org/10.1080/07362994.2012.668448 doi: 10.1080/07362994.2012.668448
![]() |
[25] |
Z. Wang, S. Zhou, Asymptotic behavior of stochastic strongly wave equation on unbounded domains, J. Appl. Math. Phys., 3 (2015), 55167. http://dx.doi.org/10.4236/jamp.2015.33046 doi: 10.4236/jamp.2015.33046
![]() |
[26] |
Z. Wang, S. Zhou, Random attractor for non-autonomous stochastic strongly damped wave equation on unbounded domains, J. Appl. Anal. Comput., 5 (2015), 363–387. http://dx.doi.org/10.11948/2015031 doi: 10.11948/2015031
![]() |
[27] |
Z. Wang, S. Zhou, A. Gu, Random attractor of the stochastic strongly damped wave equation, Commun. Nonlinear Sci., 17 (2012), 1649–1658. http://dx.doi.org/10.1016/j.cnsns.2011.09.001 doi: 10.1016/j.cnsns.2011.09.001
![]() |
[28] |
R. Jones, B. Wang, Asymptotic behavior of a class of stochastic nonlinear wave equations with dispersive and dissipative terms, Nonlinear Anal. Real, 14 (2013), 1308–1322. http://dx.doi.org/10.1016/j.nonrwa.2012.09.019 doi: 10.1016/j.nonrwa.2012.09.019
![]() |
[29] |
T. Caraballo, B. Guo, N. H. Tuan, R. Wang, Asymptotically autonomous robustness of random attractors for a class of weakly dissipative stochastic wave equations on unbounded domains, P. Roy. Soc. Edinb. A, 151 (2021), 1700–1730. http://dx.doi.org/10.1017/prm.2020.77 doi: 10.1017/prm.2020.77
![]() |
[30] |
R. Wang, B. Wang, Random dynamics of lattice wave equations driven by infinite-dimensional nonlinear noise, Discrete Cont. Dyn. B, 25 (2020), 2461–2493. http://dx.doi.org/10.3934/dcdsb.2020019 doi: 10.3934/dcdsb.2020019
![]() |
[31] |
R. Wang, Y. Li, Backward compactness and periodicity of random attractors for stochastic wave equations with varying coefficients, Discrete Cont. Dyn. B, 24 (2019), 4145–4167. http://dx.doi.org/10.3934/dcdsb.2019054 doi: 10.3934/dcdsb.2019054
![]() |
[32] | P. Groisman, J. D. Rossi, Explosion time in stochastic differential equations with small diffusion, Electron. J. Differ. Eq., 2007 (2007), 1–9. |
[33] | G. D. Prato, J. Zabczyk, Stochastic equations in infinite dimensions, London: Cambridge University Press, 2014. http://dx.doi.org/10.1017/CBO9781107295513 |
[34] | D. Bahuguna, Strongly damped semilinear equations, J. Appl. Math. Stoch. Anal., 8 (1995), 397–404. |
[35] | H. Engler, F. Neubrander, J. Sandefur, Strongly damped semilinear second order equations, In: Nonlinear semigroups, partial dfferential equations and attractors, 1987. http://dx.doi.org/10.1007/BFb0077415 |
[36] | R. A. Adams, J. J. R. Fournier, Sobolev spaces, Netherlands: Academic Press, 2003. |
[37] |
H. Yang, Y. Z. Han, Initial boundary value problem for a strongly damped wave equation with a general nonlinearity, Evol. Equ. Control The., 11 (2022), 635–648. http://dx.doi.org/10.3934/eect.2021019 doi: 10.3934/eect.2021019
![]() |