We consider in this paper the stochastic nonlinear Schrödinger equation forced by multiplicative noise in the Itô sense. We use two different methods as sine-cosine method and Riccati-Bernoulli sub-ODE method to obtain new rational, trigonometric and hyperbolic stochastic solutions. These stochastic solutions are of a qualitatively distinct nature based on the parameters. Moreover, the effect of the multiplicative noise on the solutions of nonlinear Schrödinger equation will be discussed. Finally, two and three-dimensional graphs for some solutions have been given to support our analysis.
Citation: Mahmoud A. E. Abdelrahman, Wael W. Mohammed, Meshari Alesemi, Sahar Albosaily. The effect of multiplicative noise on the exact solutions of nonlinear Schrödinger equation[J]. AIMS Mathematics, 2021, 6(3): 2970-2980. doi: 10.3934/math.2021180
[1] | Muhammad Ajmal, Xiwang Cao, Muhammad Salman, Jia-Bao Liu, Masood Ur Rehman . A special class of triple starlike trees characterized by Laplacian spectrum. AIMS Mathematics, 2021, 6(5): 4394-4403. doi: 10.3934/math.2021260 |
[2] | Chang Liu, Jianping Li . Sharp bounds on the zeroth-order general Randić index of trees in terms of domination number. AIMS Mathematics, 2022, 7(2): 2529-2542. doi: 10.3934/math.2022142 |
[3] | Tariq A. Alraqad, Igor Ž. Milovanović, Hicham Saber, Akbar Ali, Jaya P. Mazorodze, Adel A. Attiya . Minimum atom-bond sum-connectivity index of trees with a fixed order and/or number of pendent vertices. AIMS Mathematics, 2024, 9(2): 3707-3721. doi: 10.3934/math.2024182 |
[4] | Zhen Lin, Ting Zhou, Xiaojing Wang, Lianying Miao . The general Albertson irregularity index of graphs. AIMS Mathematics, 2022, 7(1): 25-38. doi: 10.3934/math.2022002 |
[5] | Swathi Shetty, B. R. Rakshith, N. V. Sayinath Udupa . Extremal graphs and bounds for general Gutman index. AIMS Mathematics, 2024, 9(11): 30454-30471. doi: 10.3934/math.20241470 |
[6] | Guifu Su, Yue Wu, Xiaowen Qin, Junfeng Du, Weili Guo, Zhenghang Zhang, Lifei Song . Sharp bounds for the general Randić index of graphs with fixed number of vertices and cyclomatic number. AIMS Mathematics, 2023, 8(12): 29352-29367. doi: 10.3934/math.20231502 |
[7] | Zenan Du, Lihua You, Hechao Liu, Yufei Huang . The Sombor index and coindex of two-trees. AIMS Mathematics, 2023, 8(8): 18982-18994. doi: 10.3934/math.2023967 |
[8] | Kun Wang, Wenjie Ning, Yuheng Song . Extremal values of the modified Sombor index in trees. AIMS Mathematics, 2025, 10(5): 12092-12103. doi: 10.3934/math.2025548 |
[9] | Akbar Ali, Sadia Noureen, Akhlaq A. Bhatti, Abeer M. Albalahi . On optimal molecular trees with respect to Sombor indices. AIMS Mathematics, 2023, 8(3): 5369-5390. doi: 10.3934/math.2023270 |
[10] | Shabana Anwar, Muhammad Kamran Jamil, Amal S. Alali, Mehwish Zegham, Aisha Javed . Extremal values of the first reformulated Zagreb index for molecular trees with application to octane isomers. AIMS Mathematics, 2024, 9(1): 289-301. doi: 10.3934/math.2024017 |
We consider in this paper the stochastic nonlinear Schrödinger equation forced by multiplicative noise in the Itô sense. We use two different methods as sine-cosine method and Riccati-Bernoulli sub-ODE method to obtain new rational, trigonometric and hyperbolic stochastic solutions. These stochastic solutions are of a qualitatively distinct nature based on the parameters. Moreover, the effect of the multiplicative noise on the solutions of nonlinear Schrödinger equation will be discussed. Finally, two and three-dimensional graphs for some solutions have been given to support our analysis.
Let Hp denote the family of analytic functions of the following form:
φ(x)=xp(1+∞∑j=1bp+jxj), (p∈N={1,2,3,...}), | (1.1) |
which are p-valent (multivalent of order p) in Δ={x∈C:|x|<1} with H1=H and also, the subfamily of H consisting of univalent (one-to-one) functions in Δ is denoted by U. We denote by K and S∗ the usual subclasses of U consisting of functions that are, respectively, bounded turning and starlike in Δ, and have the following geometric inequalities: ℜ{φ′(x)}>0 and ℜ{xφ′(x)/φ(x)}>0. Singh [21] introduced an important subfamily of U denoted by Bα that consists of Bazilevi č functions with the next inequality:
ℜ{xφ′(x)φ(x)[φ(x)x]α}>0, |
for a non-negative real number α. He noted in his work, that the cases α=0 and α=1 correspond to S∗ and K, respectively. In [14], Obradovic introduced and studied the well-known subfamily of non-Bazilevič functions, that is,
Nβ={φ∈H:ℜ{xφ′(x)φ(x)[xφ(x)]β}>0},0<β<1. |
Recently, several research papers have appeared on subfamilies related to Bazilevič functions, non-Bazilevič functions that are sometimes defined by linear operators, and their generalizations (see, for example, [7,20,22,23,24,27,28]).
Let Qk(σ) denote the family of analytic functions g(x) of the form:
g(x)=1+∞∑j=1djxj (x∈Δ), | (1.2) |
satisfying the following inequality
∫2π0|ℜ{g(x)}−σ1−σ|dθ≤kπ, | (1.3) |
where k≥2, 0≤σ<1, and x=reiθ∈Δ. The family Qk(σ) was introduced and studied by Padmanabhan and Parvatham [17]. For σ=0, we obtain the family Qk(0)=Qk that was introduced by Pinchuk [18].
Remark 1. For g(x)∈Qk(σ), we can write
g(x)=12π∫2π01+(1−2σ)xe−is1−xe−isdμ(s) (x∈Δ), | (1.4) |
where μ(s) is a function with bounded variation on [0,2π] such that
∫2π0dμ(s)=2π, ∫2π0|dμ(s)|<kπ. | (1.5) |
Since μ(s) has a bounded variation on [0,2π], we may put μ(s)=A(s)−B(s), where A(s) and B(s) are two non-negative increasing functions on [0,2π] satisfying (1.5). Thus, if we set A(s)=k+24μ1(s) and B(s)=k−24μ2(s), then (1.4) becomes
g(x)=(k+24)12π∫2π01+(1−2σ)xe−is1−xe−isdμ1(s)−(k−24)12π∫2π01+(1−2σ)xe−is1−xe−isdμ2(s). | (1.6) |
Now, using Herglotz-Stieltjes formula for the family Q(σ) of analytic functions with positive real part greater than σ and (1.6), we obtain
g(x)=(k+24)g1(x)−(k−24)g2(x), | (1.7) |
where g1(x),g2(x)∈Q(σ). Also, we have here Q(0)=Q, where Q is the family of analytic functions g(x) in Δ with ℜ{g(x)}>0.
Remark 2. It is well-known from [13] that the family Qk(σ) is a convex set.
Remark 3. For 0≤σ1<σ2<1, we have Qk(σ2)⊂Qk(σ1) (see [6]).
In recent years, researchers have been using the family Qk(σ) of analytic functions associated with bounded boundary rotation in various branches of mathematics very effectively, especially in geometric function theory (GFT). For further developments and discussion about this family, we can obtain selected articles produced by some mathematicians like [1,4,5,8,12,25] and many more.
Now, using the family Qk(σ), we introduce the subfamily BNα,βp,k(λ,σ) of p-valent Bazilevič and non-Bazilevič functions of Hp as the following definition:
Definition 1. A function φ∈Hp is said to be in the subfamily BNα,βp,k(λ,σ) if it satisfies the following condition:
(1−α−βα+βλ)[φ(x)xp]α−β+α−βα+βλxφ′(x)pφ(x)[φ(x)xp]α−β∈Qk(σ), |
which is equivalent to
∫2π0|ℜ{(1−α−βα+βλ)[φ(x)xp]α−β+α−βα+βλxφ′(x)pφ(x)[φ(x)xp]α−β}−σ1−σ|dθ≤kπ, |
where α,β≥0; α+β≥0; λ>0; k≥2; 0≤σ<1; x∈Δ; and all powers are principal ones.
Example 1. Let φ(x):Δ→C be an analytic function given by
φ(x)=xp(1+p(α+β)(1−σ)k[p(α+β)+λ]x)1α−β∈Hp (α≠β). |
Clearly φ(x)∈Hp (with all powers being principal ones). After some calculations, we find that
(1−α−βα+βλ)[φ(x)xp]α−β+α−βα+βλ[φ(x)xp]α−βxφ′(x)pφ(x)=1+(1−σ)kx. |
Now, if x=reiθ(0≤r<1), then
ℜ{(1−α−βα+βλ)[φ(x)xp]α−β+α−βα+βλxφ′(x)pφ(x)[φ(x)xp]α−β}−σ1−σ=1+krcosθ, |
and
∫2π0|ℜ{(1−α−βα+βλ)[φ(x)xp]α−β+α−βα+βλxφ′(x)pφ(x)[φ(x)xp]α−β}−σ1−σ|dθ=∫2π0(1+krcosθ)dθ=2π≤kπ(k≥2). |
Hence, φ(x) belongs to the subfamily BNα,βp,k(λ,σ), and it is not empty.
By specializing the parameters α, β, λ, p, k, and σ involved in Definition 1, we get the following subfamilies, which were studied in many earlier works:
(ⅰ) For k=2, β=0 and σ=1−L1−M(−1≤M<L≤1), we have BNα,02,p(λ,1−L1−M)=Bαp(λ,L,M)
={φ∈Hp:(1−λ)[φ(x)xp]α+λxφ′(x)pφ(x)[φ(x)xp]α≺1+Lx1+Mx}, |
where ≺ denotes the usual meaning of subordination, Bαp(λ,L,M) is a subfamily of multivalently Bazilevič functions introduced by Liu [10] and Bαp(1,L,M)=Bαp(L,M)
={φ∈Hp:xφ′(x)pφ(x)[φ(x)xp]α≺1+Lx1+Mx}, |
where the subfamily Bαp(L,M) was introduced by Yang [30];
(ⅱ) BNα,02,1(1,1−L1−M)=Bα(L,M)
={φ∈H:xφ′(x)φ(x)[φ(x)x]α≺1+Lx1+Mx}, |
where the subfamily Bα(L,M) was studied by Singh [21] (see also Owa and Obradovic [15]);
(ⅲ) BNα,02,1(1,σ)=Bα(σ)
={φ∈H:ℜ{xφ′(x)φ(x)[φ(x)x]α}>σ}, |
where the family Bα(σ) was considered by Owa [16];
(ⅳ) For k=2, α=0 and σ=1−L1−M(−1≤M<L≤1), we have BN0,β2,p(λ,1−L1−M)=Nβp(λ,L,M)
={φ∈Hp:(1+λ)[xpφ(x)]β−λxφ′(x)pφ(x)[xpφ(x)]β≺1+Lx1+Mx}, |
where Nβp(λ,L,M) is the family of non-Bazilevič multivalent functions introduced by Aouf and Seoudy [3], and Nβ1(λ,L,M)=Nβ(λ,L,M)
={φ∈H:(1+λ)[xφ(x)]β−λxφ′(x)φ(x)[xφ(x)]β≺1+Lx1+Mx}, |
where Nβ(λ,L,M) is the subclass of non-Bazilevič univalent functions defined by Wang et al. [29];
(ⅴ) BN0,β2,1(−1,σ)=Nβ(σ)
={φ∈H:ℜ{xφ′(x)φ(x)[xφ(x)]β}>σ}, |
where Nβ(σ) is the family of non-Bazilevič functions of order σ (see Tuneski and Daus [26]) and Nβ(0)=Nβ
={φ∈H:ℜ{xφ′(x)φ(x)[xφ(x)]β}>0}, |
where Nβ is the family of non-Bazilevič functions (see Obradovic [14]).
Also, we note that
(ⅰ) BNα,0k,p(λ,σ)=Bαk,p(λ,σ)
={φ∈Hp:(1−λ)[φ(x)xp]α+λxφ′(x)pφ(x)[φ(x)xp]α∈Qk(σ)}, |
and Bαk,1(λ,σ)=Bαk(λ,σ)
={φ∈H:(1−λ)[φ(x)x]α+λxφ′(x)φ(x)[φ(x)x]α∈Qk(σ)}; |
(ⅱ) BN0,βk,p(λ,σ)=Nβk,p(λ,σ)
={φ∈Hp:(1+λ)[xpφ(x)]β−λxφ′(x)pφ(x)[xpφ(x)]β∈Qk(σ)}, |
and Nβk,1(λ,σ)=Nβk(λ,σ)
={φ∈H:(1+λ)[xφ(x)]β−λxφ′(x)φ(x)[xφ(x)]β∈Qk(σ)}; |
(ⅲ) BN1,0k,p(λ,σ)=Bk,p(λ,σ)
={φ∈Hp:(1−λ)φ(x)xp+λφ′(x)pxp−1∈Qk(σ)}, |
and Bk,1(λ,σ)=Bk(λ,σ)
={φ∈H:(1−λ)φ(x)x+λφ′(x)∈Qk(σ)}; |
(ⅳ) BN0,1k,p(λ,σ)=Nk,p(λ,σ)
={φ∈Hp:(1+λ)xpφ(x)−λxp+1φ′(x)pφ2(x)∈Qk(σ)}, |
and Nk,1(λ,σ)=Nk(λ,σ)
={φ∈H:(1+λ)xφ(x)−λx2φ′(x)φ2(x)∈Qk(σ)}; |
(ⅴ) BNα,0k,p(1,σ)=Bαk,p(σ)
={φ∈Hp:xφ′(x)pφ(x)[φ(x)xp]α∈Qk(σ)}, |
and Bαk,1(σ)=Bαk(σ)
={φ∈H:xφ′(x)φ(x)[φ(x)x]α∈Qk(σ)}; |
(ⅵ) BN0,βk,p(−1,σ)=Nβk,p(σ)
={φ∈Hp:xφ′(x)pφ(x)[xpφ(x)]β∈Qk(σ)}, |
and Nβk,1(σ)=Nβk(σ)
={φ∈H:xφ′(x)φ(x)[xφ(x)]β∈Qk(σ)}; |
(ⅶ) B0k,p(σ)=Sk,p(σ)
={φ∈Hp:xφ′(x)pφ(x)∈Qk(σ)}, |
and Sk,1(σ)=Sk(σ)
={φ∈H:xφ′(x)φ(x)∈Qk(σ)}. |
To prove our main results, the next lemmas will be required in our investigation.
Lemma 1. [11] Let γ=γ1+iγ2 and δ=δ1+iδ2 and Θ(γ,δ) be a complex-valued function satisfying the next conditions:
(ⅰ) Θ(γ,δ) is continuous in a domain D∈C2.
(ⅱ) (0,1)∈D and Θ(1,0)>0.
(ⅲ) ℜ{Θ(iγ2,δ1)}>0 whenever (iγ2,δ1)∈D and δ1≤−1+γ222.
If g(x) given by (1.2) is analytic in Δ such that (g(x),xg′(x))∈D and ℜ{Θ(g(x),xg′(x))}>0 for x∈Δ, then ℜ{g(x)}>0 in Δ.
Lemma 2. [2, Theorem 5 with p=1] If g(x)∈Qk(σ) is given by (1.2), then
|dj|≤(1−σ)k (j∈N). | (1.8) |
This result is sharp.
Remark 4. For σ=0 in Lemma 2, we get the result for the family Qk obtained by Goswami et al. [9].
In the present article, we have combined Bazilevič and non-Bazilevič analytic functions into a new family BNα,βk,p(λ,σ) associated with a bounded boundary rotation. In the next section, several properties like inclusion results, some connections with the generalized Bernardi-Libera-Livingston integral operator, and the upper bounds for |bp+1| and |bp+2+α−β−12b2p+1| for this family BNα,βk,p(λ,σ) and its special subfamilies are investigated. The motivation of this article is to generalize and improve previously known works.
Theorem 1. If φ∈BNα,βk,p(λ,σ), then
[φ(x)xp]α−β∈Qk(σ1), | (2.1) |
where σ1 is given by
σ1=2p(α+β)σ+λ2p(α+β)+λ. | (2.2) |
Proof. Let φ∈BNα,βk,p(λ,σ) and set
[φ(x)xp]α−β=(1−σ1)g(x)+σ1 (x∈Δ)=(k+24){(1−σ1)g1(x)+σ1}−(k−24){(1−σ1)g2(x)+σ1}, | (2.3) |
where gi(x) is analytic in Δ with gi(0)=1, i=1,2. Differentiating (2.3) with respect to x, we obtain
(1−α−βα+βλ)[φ(x)xp]α−β+α−βα+βλxφ′(x)pφ(x)[φ(x)xp]α−β=(1−σ1)g(x)+σ1+λ(1−σ1)xg′(x)p(α+β)∈Qk(σ), |
this implies that
11−σ{(1−σ1)gi(x)+σ1−σ+λ(1−σ1)xg′i(x)p(α+β)}∈Q (x∈Δ;i=1,2). |
We form the functional Θ(γ,δ) by choosing γ=gi(x), δ=xg′i(x),
Θ(γ,δ)=(1−σ1)γ+σ1−σ+λ(1−σ1)δp(α+β). |
Clearly, the first two conditions of Lemma 1 are satisfied. Now, we verify the condition (ⅲ) of Lemma 1 as follows:
ℜ{Θ(iγ2,δ1)}=σ1−σ+ℜ{λ(1−σ1)δ1p(α+β)}≤σ1−σ−λ(1−σ1)(1+γ22)2p(α+β)=2p(α+β)(σ1−σ)−λ(1−σ1)−λ(1−σ1)γ222p(α+β)=A+Bγ222C, |
where
A=2p(α+β)(σ1−σ)−λ(1−σ1),B=−λ(1−σ1)<0,C=2p(α+β)>0. |
We note that ℜ{Θ(iγ2,δ1)}<0 if and only if A=0, B<0, and C>0, and this gives us
σ1=2p(α+β)σ+λ2p(α+β)+λ. |
Since B=−λ(1−σ1)<0 gives us 0≤σ1<1. Therefore, applying Lemma 1, gi(x)∈Q(i=1,2) and consequently g(x)∈Qk(σ1) for x∈Δ. This completes the proof of Theorem 1.
Putting β=0 in Theorem 1, we obtain the next result.
Corollary 1. If φ∈Bαk,p(λ,σ), then
[φ(x)xp]α∈Qk(σ2), |
where σ2 is given by
σ2=2pαρ+λ2pα+λ. |
Putting α=0 in Theorem 1, we get the following result.
Corollary 2. If φ∈Nβk,p(λ,σ), then
[xpφ(x)]β∈Qk(σ3), |
where σ3 is given by
σ3=2pβρ+λ2pβ+λ. |
Theorem 2. If φ∈BNα,βk,p(λ,σ), then
[φ(x)xp]α−β2∈Qk(σ4), | (2.4) |
where σ4 is given by
σ4=λ+√λ2+4[p(α+β)+λ]p(α+β)σ2[p(α+β)+λ]. | (2.5) |
Proof. Let φ∈BNα,βk,p(λ,σ) and let
[φ(x)xp]α−β=(k4+12)[(1−σ4)g1(x)+σ4]2−(k4+12)[(1−σ4)g1(x)+σ4]2=[(1−σ4)g(x)+σ4]2, | (2.6) |
where gi(x) is analytic in Δ with gi(0)=1, i=1,2. Differentiating both sides of (2.6) with respect to x, we obtain
(1−α−βα+βλ)[φ(x)xp]α−β+α−βα+βλxφ′(x)pφ(x)[φ(x)xp]α−β={[(1−σ4)g(x)+σ4]2+[(1−σ4)g(x)+σ4]2λ(1−σ4)xg′(x)p(α+β)}∈Qk(σ), |
this implies that
11−σ{[(1−σ4)gi(x)+σ4]2+[(1−σ4)gi(x)+σ4]2λ(1−σ4)xg′i(x)p(α+β)−σ}∈Q (i=1,2). |
We form the functional Θ(γ,δ) by choosing γ=gi(x), δ=xg′i(x),
Θ(γ,δ)=[(1−σ4)γ+σ4]2+[(1−σ4)γ+σ4]2λ(1−σ4)δp(α+β)−σ. |
Clearly, the conditions (ⅰ) and (ⅱ) of Lemma 1 are satisfied. Now, we verify the condition (ⅲ) of Lemma 1 as follows:
ℜ{Θ(iγ2,δ1)}=σ24−(1−σ4)2γ22+2λσ4(1−σ4)δ1p(α+β)−σ≤σ24−σ−(1−σ4)2γ22−λσ4(1−σ4)(1+γ22)p(α+β)=A+Bγ222C, |
where
A=p(α+β)(σ24−σ)−λσ4(1−σ4),B=−(1−σ4)(1−σ4+λσ4)<0,C=p(α+β)2>0. |
We note that ℜ{Θ(iγ2,δ1)}<0 if and only if A=0, B<0, and C>0, and this gives us σ4 as given by (2.5), and B<0 gives us 0≤σ4<1. Therefore, applying Lemma 1, gi(x)∈Q (i=1,2), and consequently g(x)∈Qk(σ4) for x∈Δ. This completes the proof of Theorem 2.
Putting β=0 in Theorem 2, we obtain the following.
Corollary 3. If φ∈Bαk,p(λ,σ), then
[φ(x)xp]α2∈Qk(σ5), |
where σ5 is given by
σ5=λ+√λ2+4(pα+λ)ρpα2(pα+λ). |
Putting α=0 in Theorem 2, we obtain the following.
Corollary 4. If φ∈Nβk,p(λ,σ), then
[xpφ(x)]β2∈Qk(σ6), |
where σ6 is given by
σ6=λ+√λ2+4(pβ+λ)ρpβ2(pβ+λ). |
For a function φ∈Hp, the generalized Bernardi-Libera-Livingston integral operator Φp,μ:Hp→Hp, with μ>−p, is given by (see [19])
Φp,μ(φ(x))=μ+pxμx∫0ωμ−1φ(ω)dω (μ>−p). | (2.7) |
It is easy to verify that for all φ∈Hp given by (1.2), we have
x(Φp,μ(φ(x)))′=(μ+p)φ(x)−μΦp,μ(φ(x)). | (2.8) |
Theorem 3. If the function φ∈Hp satisfies the next condition
(1−α−βα+βλ)[Φp,μ(φ(x))xp]α−β +α−βα+βλφ(x)Φp,μ(φ(x))[Φp,μ(φ(x))xp]α−β∈Qk(σ), | (2.9) |
with Φp,μ is the integral operator defined by (2.7), then
[Φp,μ(φ(x))xp]α−β∈Qk(σ7), |
where σ7 is given by
σ7=2(p+μ)(α+β)σ+λ2(p+μ)(α+β)+λ. | (2.10) |
Proof. Let
[Φp,μ(φ(x))xp]α−β=(k4+12){(1−σ7)g1(x)+σ7}−(k4+12){(1−σ7)g2(x)+σ7}=(1−σ7)g(x)+σ7 (x∈Δ), | (2.11) |
then where gi(x) is analytic in Δ with gi(0)=1, i=1,2. Differentiating (2.11) with respect to x and using (2.8) in the resulting relation, we get
(1−α−βα+βλ)[Φp,μ(φ(x))xp]α−β +α−βα+βλφ(x)Φp,μ(φ(x))[Φp,μ(φ(x))xp]α−β=(1−σ7)g(x)+λ(1−σ7)xg′(x)(p+μ)(α+β)∈Qk(σ) (x∈Δ). |
Using the same method we used to prove Theorem 1, the remaining part of this theorem can be derived in a similar way.
Putting β=0 in Theorem 3, we obtain the following.
Corollary 5. If the function φ∈Hp satisfies the next condition
(1−λ)[Φp,μ(φ(x))xp]α +λφ(x)Φp,μ(φ(x))[Φp,μ(φ(x))xp]α∈Qk(σ), |
with Φp,μ is defined by (2.7), then
[Φp,μ(φ(x))xp]α∈Qk(σ8), |
where σ8 is given by
σ8=2(p+μ)αρ+λ2(p+μ)α+λ. |
Putting α=0 in Theorem 3, we obtain the following.
Corollary 6. If the function φ∈Hp satisfies the next condition
(1+λ)[xpΦp,μ(φ(x))]β −λφ(x)Φp,μ(φ(x))[xpΦp,μ(φ(x))]β∈Qk(σ), |
with Φp,μ is defined by (2.7), then
[xpΦp,μ(φ(x))]β∈Qk(σ9), |
where σ9 is given by
σ9=2(p+μ)(α+β)σ+λ2(p+μ)(α+β)+λ. |
Theorem 4. If 0≤λ1<λ2, then
BNα,βk,p(λ2,σ)⊂BNα,βk,p(λ1,σ). |
Proof. If we consider an arbitrary function φ∈BNα,βk,p(λ2,σ), then
φ2(x)=(1−α−βα+βλ2)[φ(x)xp]α−β+α−βα+βλ2xφ′(x)pφ(x)[φ(x)xp]α−β∈Qk(σ). |
According to Theorem 1, we have
φ1(x)=[φ(x)xp]α−β∈Qk(σ1), |
where σ1 is given by (2.2). From (2.2), it follows that σ1≥σ, and from Remark 3, we conclude that Qk(σ1)⊂Qk(σ); hence, φ1(x)∈Qk(σ).
A simple computation shows that
(1−α−βα+βλ1)[φ(x)xp]α−β+α−βα+βλ1xφ′(x)pφ(x)[φ(x)xp]α−β=(1−λ1λ2)φ1(x)+λ1λ2φ2(x). | (2.12) |
Since the class Qk(σ) is a convex set (see Remark 2), it follows that the right-hand side of (2.12) belongs to Qk(σ) for 0≤λ1<λ2, which implies that φ∈BNα,βk,p(λ1,σ).
Putting β=0 in Theorem 4, we obtain the following.
Corollary 7. If 0≤λ1<λ2, then
Bαk,p(λ2,σ)⊂Bαk,p(λ1,σ). |
Putting α=0 in Theorem 4, we get the following.
Corollary 8. If 0≤λ1<λ2, then
Nβk,p(λ2,σ)⊂Nβk,p(λ1,σ). |
Theorem 5. If φ∈BNα,βk,p(λ,σ) given by (1.1) with α≠β, p(α+β)+λ≠0 and p(α+β)+2λ≠0, then
|bp+1|≤p|α+β|(1−σ)k|α−β||p(α+β)+λ|, | (2.13) |
and
|bp+2+α−β−12b2p+1|≤p|α+β|(1−σ)k|α−β||p(α+β)+2λ|. | (2.14) |
Proof. If φ∈BNα,βk,p(λ,σ), from Definition 1, we have
(1−α−βα+βλ)[φ(x)xp]α−β+α−βα+βλxφ′(x)pφ(x)[φ(x)xp]α−β=G(x), | (2.15) |
where G(x)∈Qk(σ) is given by
G(x)=1+d1x+d2x2+d3x3+... . | (2.16) |
Since
(1−α−βα+βλ)[φ(x)xp]α−β+α−βα+βλxφ′(x)pφ(x)[φ(x)xp]α−β=1+(α−β)[p(α+β)+λ]p(α+β)bp+1x+(α−β)[p(α+β)+2λ]p(α+β)(bp+2+α−β−12b2p+1)x2+..., | (2.17) |
Comparing the coefficients in (2.15) by using (2.16) and (2.17), we obtain
(α−β)[p(α+β)+λ]p(α+β)bp+1=d1, | (2.18) |
(α−β)[p(α+β)+2λ]p(α+β)(bp+2+α−β−12b2p+1)=d2. | (2.19) |
Therefore,
bp+1=p(α+β)(α−β)[p(α+β)+λ]d1, |
and
bp+2+α−β−12b2p+1=p(α+β)(α−β)[p(α+β)+2λ]d2. |
Our result now follows by an application of Lemma 2. This completes the proof of Theorem 5.
Putting β=0 in Theorem 5, we obtain the following.
Corollary 9. If φ∈Bαk,p(λ,σ) is given by (1.1) with pα+λ≠0 and pα+2λ≠0, then
|bp+1|≤p(1−σ)k|pα+λ|, |
and
|bp+2+α−12b2p+1|≤p(1−σ)k|pα+2λ|. |
Putting k=2 and σ=1−L1−M(−1≤M<L≤1) in Corollary 9, we obtain the following corollary, which improves the result of Liu [10, Theorem 4 with n=1].
Corollary 10. If φ∈Bαp(λ,L,M) is given by (1.1) with pα+λ≠0 and pα+2λ≠0, then
|bp+1|≤2p(L−M)|pα+λ|(1−M), |
and
|bp+2+α−12b2p+1|≤2p(L−M)|pα+2λ|(1−M). |
Putting α=0 in Theorem 5, we get the following.
Corollary 11. If φ∈Nβk(λ,σ) given by (1.1) with pβ+λ≠0 and pβ+2λ≠0, then
|bp+1|≤p(1−σ)k|pβ+λ|, |
and
|bp+2−β+12b2p+1|≤p(1−σ)k|pβ+2λ|. |
Putting k=2 and σ=1−L1−M(−1≤M<L≤1) in Corollary 11, we obtain the following corollary, which improves the result of Aouf and Seoudy [3, Theorem 8 with n=1].
Corollary 12. If φ∈Nβk(λ,L,M) given by (1.1) with pβ+λ≠0 and pβ+2λ≠0, then
|bp+1|≤2p(L−M)|pβ+λ|(1−M), |
and
|bp+2−β+12b2p+1|≤2p(L−M)|pβ+2λ|(1−M). |
In this investigation, we have presented the subfamily BNα,βk,p(λ,σ) of multivalent Bazilevič and non-Bazilevič functions related to bounded boundary rotation. Also, we have computed a number of important properties, including the inclusion results and the upper bounds for the first two Taylor-Maclaurin coefficients for this function subfamily. For different choices of the parameters α, β, λ, p, k, and σ in the above results, we can get the corresponding results for each of the next subfamilies: Bαp(λ,L,M), Bαp(L,M), Bα(L,M), Bα(L,M), Bα(σ), Nβp(λ,L,M), Nβ(λ,L,M), Nβ(σ), Nβ, Bαk,p(λ,σ), Bαk(λ,σ), Nβk,p(λ,σ), Nβk(λ,σ), Bk,p(λ,σ), Bk(λ,σ), Nk,p(λ,σ), Nk(λ,σ), Bαk,p(σ), Bαk(σ), Nβk,p(σ), Nβk(σ), Sk,p(σ), and Sk(σ), which are defined in an introduction section. In addition, this work lays the foundation for future research and encourages researchers to explore more Bazilevič and non-Bazilevič functions involving some linear operators in geometric function theory and related fields.
The authors contributed equally to this work. All authors have read and approved the final version of the manuscript for publication.
The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.
The authors extend their appreciation to Umm Al-Qura University, Saudi Arabia, for funding this research work through grant number: 25UQU4350561GSSR01.
This research work was funded by Umm Al-Qura University, Saudi Arabia, under grant number: 25UQU4350561GSSR01.
The authors declare that they have no competing interests.
[1] |
M. A. E. Abdelrahman, Global solutions for the ultra-relativistic Euler equations, Nonlinear Anal., 155 (2017), 140–162. doi: 10.1016/j.na.2017.01.014
![]() |
[2] |
C. O. Alves, F. Gao, M. Squassina, M. Yang, Singularly perturbed critical Choquard equations, J. Differ. Equations, 263 (2017), 3943–3988. doi: 10.1016/j.jde.2017.05.009
![]() |
[3] |
P. I. Naumkin, J. J. Perez, Higher-order derivative nonlinear Schrödinger equation in the critical case, J. Math. Phys., 59 (2018), 021506. doi: 10.1063/1.5008500
![]() |
[4] |
M. A. E. Abdelrahman, Cone-grid scheme for solving hyperbolic systems of conservation laws and one application, Comput. Appl. Math., 37 (2018), 3503–3513. doi: 10.1007/s40314-017-0527-9
![]() |
[5] |
M. A. E. Abdelrahman, G. M. Bahaa, Elementary waves, Riemann problem, Riemann invariants and new conservation laws for the pressure gradient model, Eur. Phys. J. Plus, 134 (2019), 187. doi: 10.1140/epjp/i2019-12580-7
![]() |
[6] |
M. A. E. Abdelrahman, N. F. Abdo, On the nonlinear new wave solutions in unstable dispersive environments, Phys. Scripta, 95 (2020), 045220. doi: 10.1088/1402-4896/ab62d7
![]() |
[7] |
H. G. Abdelwahed, Nonlinearity contributions on critical MKP equation, J. Taibah Univ. Sci., 14 (2020), 777–782. doi: 10.1080/16583655.2020.1774136
![]() |
[8] |
H. G. Abdelwahed, Super electron acoustic propagations in critical plasma density, J. Taibah Univ. Sci., 14 (2020), 1363–1368. doi: 10.1080/16583655.2020.1822653
![]() |
[9] |
M. K. Sharaf, E. K. El-Shewy, M. A. Zahran, Fractional anisotropic diffusion equation in cylindrical brush model, J. Taibah Univ. Sci., 14 (2020), 1416–1420. doi: 10.1080/16583655.2020.1824743
![]() |
[10] |
A. M. Wazwaz, The integrable time-dependent sine-Gordon with multiple optical kink solutions, Optik, 182 (2019), 605–610. doi: 10.1016/j.ijleo.2019.01.018
![]() |
[11] |
M. A. E. Abdelrahman, M. A. Sohaly, On the new wave solutions to the MCH equation, Indian J. Phys., 93 (2019), 903–911. doi: 10.1007/s12648-018-1354-6
![]() |
[12] |
M. Eslami, Trial solution technique to chiral nonlinear Schrödinger's equation in (1 + 2)-dimensions, Nonlinear Dyn., 85 (2016), 813–816. doi: 10.1007/s11071-016-2724-2
![]() |
[13] |
M. Mirzazadeh, M. Eslami, A. Biswas, 1-Soliton solution of KdV equation, Nonlinear Dyn., 80 (2015), 387–396. doi: 10.1007/s11071-014-1876-1
![]() |
[14] |
B. Ghanbari, C. K. Kuo, New exact wave solutions of the variable-coefficient (1 + 1)-dimensional Benjamin-Bona-Mahony and (2 + 1)-dimensional asymmetric Nizhnik-Novikov-Veselov equations via the generalized exponential rational function method, Eur. Phys. J. Plus, 134 (2019), 134. doi: 10.1140/epjp/i2019-12635-9
![]() |
[15] |
C. K. Kuo, B. Ghanbari, Resonant multi-soliton solutions to new (3 + 1)-dimensional Jimbo-Miwa equations by applying the linear superposition principle, Nonlinear Dyn., 96 (2019), 459–464. doi: 10.1007/s11071-019-04799-9
![]() |
[16] |
M. A. E. Abdelrahman, M. A. Sohaly, The development of the deterministic nonlinear PDEs in particle physics to stochastic case, Results Phys., 9 (2018), 344–350. doi: 10.1016/j.rinp.2018.02.032
![]() |
[17] | M. A. E. Abdelrahman, S. Z. Hassan, M. Inc, The coupled nonlinear Schrödinger-type equations, Mod. Phys. Lett. B, 34 (2020), 2050078. |
[18] | W. W. Mohammed, Amplitude equation with quintic nonlinearities for the generalized Swift-Hohenberg equation with additive degenerate noise, Adv. Differ. Equ., 1 (2016), 84. |
[19] |
W. W. Mohammed, Approximate solution of the Kuramoto-Shivashinsky equation on an unbounded domain, Chinese Ann. Math. B, 39 (2018), 145–162. doi: 10.1007/s11401-018-1057-5
![]() |
[20] | W. W. Mohammed, Modulation equation for the stochastic Swift–Hohenberg equation with cubic and quintic nonlinearities on the real line, Mathematics, 6 (2020), 1–12. |
[21] |
H. G. Abdelwahed, E. K. El-Shewy, M. A. E. Abdelrahman, R. Sabry, New super waveforms for modified Korteweg-de-Veries-equation, Results Phys., 19 (2020), 103420. doi: 10.1016/j.rinp.2020.103420
![]() |
[22] | N. W. Ashcroft, N. D. Mermin, Solid state physics, New York: Cengage Learning, 1976. |
[23] |
H. T. Chu, Eigen energies and eigen states of conduction electrons in pure bismithunder size and magnetic fields quatizations, J. Phys. Chem. Solids, 50 (1989), 319–324. doi: 10.1016/0022-3697(89)90494-0
![]() |
[24] |
P. I. Kelley, Self-focusing of optical beams, Phys. Rev. Lett., 15 (1965), 1005–1008. doi: 10.1103/PhysRevLett.15.1005
![]() |
[25] |
M. Blencowe, Quantum electromechanical systems, Phys. Rep., 395 (2004), 159–222. doi: 10.1016/j.physrep.2003.12.005
![]() |
[26] |
W. Grecksch, H. Lisei, Stochastic nonlinear equations of Schrödinger type, Stoch. Anal. Appl., 29 (2011), 631–653. doi: 10.1080/07362994.2011.581091
![]() |
[27] |
C. H. Bruneau, L. Di Menza, T. Lehner, Numerical resolution of some nonlinear Schrödinger-like equations in plasmas, Numer. Meth. Part. D. E., 15 (1999), 672–696. doi: 10.1002/(SICI)1098-2426(199911)15:6<672::AID-NUM5>3.0.CO;2-J
![]() |
[28] |
V. Barbu, M. Röckner, D. Zhang, Stochastic nonlinear Schrödinger equations with linear multiplicative noise: rescaling approach, J. Nonlin. Sci., 24 (2014), 383–409. doi: 10.1007/s00332-014-9193-x
![]() |
[29] |
M. A. E. Abdelrahman, W. W. Mohammed, The impact of multiplicative noise on the solution of the Chiral nonlinear Schrödinger equation, Phys. Scripta, 95 (2020), 085222. doi: 10.1088/1402-4896/aba3ac
![]() |
[30] |
S. Albosaily, W. W. Mohammed, M. A. Aiyashi, M. A. E. Abdelrahman, Exact solutions of the (2 + 1)-dimensional stochastic chiral nonlinear Schrödinger equation, Symmetry, 12 (2020), 1874. doi: 10.3390/sym12111840
![]() |
[31] |
A. Debussche, C. Odasso, Ergodicity for a weakly damped stochastic nonlinear Schrödinger equation, J. Evol. Equ., 5 (2005), 317–356. doi: 10.1007/s00028-005-0195-x
![]() |
[32] | G. E. Falkovich, I. Kolokolov, V. Lebedev, S. K. Turitsyn, Statistics of soliton-bearing systems with additive noise, Phys. Rev. E, 63 (2001), 025601. |
[33] |
A. Debussche, L. Di Menzab, Numerical simulation of focusing stochastic nonlinear Schrödinger equations, Physica D, 162 (2002), 131–154. doi: 10.1016/S0167-2789(01)00379-7
![]() |
[34] | K. Cheung, R. Mosincat, Stochastic nonlinear Schrö dinger equations on tori, Stoch. Partial Differ., 7 (2019), 169–208. |
[35] |
A. De Bouard, A. Debussche, A semidiscrete scheme for the stochastic nonlinear Schrödinger equation, Numer. Math., 96 (2004), 733–770. doi: 10.1007/s00211-003-0494-5
![]() |
[36] |
J. Cui, J. Hong, Z. Liu, Strong convergence rate of finite difference approximations for stochastic cubic Schrödinger equations, J. Differ. Equations, 263 (2017), 3687–3713. doi: 10.1016/j.jde.2017.05.002
![]() |
[37] |
J. Cui, J. Hong, Z. Liu, W. Zhou, Strong convergence rate of splitting schemes for stochastic nonlinear Schrödinger equation, J. Differ. Equations, 266 (2019), 5625–5663. doi: 10.1016/j.jde.2018.10.034
![]() |
[38] | X. F. Yang, Z. C. Deng, Y. Wei, A Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application, Adv. Differ. Equ., 1 (2015), 117–133. |
[39] |
A. M. Wazwaz, A sine-cosine method for handling nonlinear wave equations, Math. Comput. Model., 40 (2004), 499–508. doi: 10.1016/j.mcm.2003.12.010
![]() |
[40] | A. M. Wazwaz, The sine-cosine method for obtaining solutions with compact and noncompact structures, Appl. Math. Comput., 159 (2004), 559–576 |
[41] |
E. Yusufoglu, A. Bekir, Solitons and periodic solutions of coupled nonlinear evolution equations by using sine-cosine method, Int. J. Comput. Math., 83 (2006), 915–924. doi: 10.1080/00207160601138756
![]() |