Citation: Xiuying Li, Yang Gao, Boying Wu. Approximate solutions of Atangana-Baleanu variable order fractional problems[J]. AIMS Mathematics, 2020, 5(3): 2285-2294. doi: 10.3934/math.2020151
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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] |
A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Therm. Sci., 20 (2016), 763-769. doi: 10.2298/TSCI160111018A
![]() |
[2] | S. Qureshia, A. Yusuf, A. A. Shaikha, et al. Transmission dynamics of varicella zoster virus modeled by classical and novel fractional operators using real statistical data, Physica A, 534 (2019), 122149. |
[3] | A. Yusuf, S. Qureshi, S. F. Shah, Mathematical analysis for an autonomous financial dynamical system via classical and modern fractional operators, Chaos Soliton. Fract., 132 (2020), 109552. |
[4] |
S. Qureshi, A. Yusuf, Mathematical modeling for the impacts of deforestation on wildlife species using Caputo differential operator, Chaos Soliton. Fract., 126 (2019), 32-40. doi: 10.1016/j.chaos.2019.05.037
![]() |
[5] | A. Jajarmi, A. Yusuf, D. Baleanu, et al., A new fractional HRSV model and its optimal control: A non-singular operator approach, Physica A, Available from: https://doi.org/10.1016/j.physa.2019.123860. |
[6] | S. Qureshi, A. Yusuf, A new third order convergent numerical solver for continuous dynamical systems, J. King Saud Univ. Sci., Available from: https://doi.org/10.1016/j.jksus.2019.11.035. |
[7] |
A. Atangana, I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Soliton. Fract., 89 (2016), 447-454. doi: 10.1016/j.chaos.2016.02.012
![]() |
[8] | A. Atangana, On the new fractional derivative and application to nonlinear Fishers reactiondiffusion equation, Appl. Math. Comput., 273 (2016), 948-956. |
[9] |
D. Baleanu, A. Jajarmi, M. Hajipour, A new formulation of the fractional optimal control problems involving Mittag-Leffler nonsingular kernel, J. Optim. Theory Appl., 175 (2017), 718-737. doi: 10.1007/s10957-017-1186-0
![]() |
[10] |
A. Akgül, A novel method for a fractional derivative with non-local and non-singular kernel, Chaos Soliton. Fract., 114 (2018), 478-482. doi: 10.1016/j.chaos.2018.07.032
![]() |
[11] |
A. Akgül, M. Modanli, Crank-Nicholson difference method and reproducing kernel function for third order fractional differential equations in the sense of Atangana-Baleanu Caputo derivative, Chaos Soliton. Fract., 127 (2019), 10-16. doi: 10.1016/j.chaos.2019.06.011
![]() |
[12] | E. K. Akgül, Solutions of the linear and nonlinear differential equations within the generalized fractional derivatives, Chaos, 29 (2019), 023108. |
[13] |
O. Abu Arqub, B. Maayah, Modulation of reproducing kernel Hilbert space method for numerical solutions of Riccati and Bernoulli equations in the Atangana-Baleanu fractional sense, Chaos Soliton. Fract., 125 (2019), 163-170. doi: 10.1016/j.chaos.2019.05.025
![]() |
[14] |
O. Abu Arqub, M. Al-Smadi, Atangana-Baleanu fractional approach to the solutions of BagleyTorvik and Painlev equations in Hilbert space, Chaos Soliton. Fract., 117 (2018), 161-167. doi: 10.1016/j.chaos.2018.10.013
![]() |
[15] |
O. Abu Arqub, B. Maayah, Numerical solutions of integrodifferential equations of Fredholm operator type in the sense of the Atangana-Baleanu fractional operator, Chaos Soliton. Fract., 117 (2018), 117-124. doi: 10.1016/j.chaos.2018.10.007
![]() |
[16] |
O. Abu Arqub, B. Maayah, Fitted fractional reproducing kernel algorithm for the numerical solutions of ABC-Fractional Volterra integro-differential equations, Chaos Soliton. Fract., 126 (2019), 394-402. doi: 10.1016/j.chaos.2019.07.023
![]() |
[17] |
O. Abu Arqub, Numerical algorithm for the solutions of fractional order systems of Dirichlet function types with comparative analysis, Fund. Inform., 166 (2019), 111-137. doi: 10.3233/FI-2019-1796
![]() |
[18] |
S. Yadav, R. K. Pandey, A. K. Shukla, Numerical approximations of Atangana-Baleanu Caputo derivative and its application, Chaos Soliton. Fract., 118 (2019), 58-64. doi: 10.1016/j.chaos.2018.11.009
![]() |
[19] | S. Hasan, A. El-Ajou, S. Hadid, et al., Atangana-Baleanu fractional framework of reproducing kernel technique in solving fractional population dynamics system, Chaos Soliton. Fract., 133 (2020), 109624. |
[20] |
X. Y. Li, B. Y. Wu, A new reproducing kernel method for variable order fractional boundary value problems for functional differential equations, J. Comput. Appl. Math., 311 (2017), 387-393. doi: 10.1016/j.cam.2016.08.010
![]() |
[21] |
X. Y. Li, B. Y. Wu, A numerical technique for variable fractional functional boundary value problems, Appl. Math. Lett., 43 (2015), 108-113. doi: 10.1016/j.aml.2014.12.012
![]() |
[22] |
X. Y. Li, B. Y. Wu, Error estimation for the reproducing kernel method to solve linear boundary value problems, J. Comput. Appl. Math., 243 (2013), 10-15. doi: 10.1016/j.cam.2012.11.002
![]() |
[23] |
F. Z. Geng, S. P. Qian, Modified reproducing kernel method for singularly perturbed boundary value problems with a delay, Appl. Math. Model., 39 (2015), 5592-5597. doi: 10.1016/j.apm.2015.01.021
![]() |
[24] |
F. Z. Geng, S. P. Qian, Reproducing kernel method for singularly perturbed turning point problems having twin boundary layers, Appl. Math. Lett., 26 (2013), 998-1004. doi: 10.1016/j.aml.2013.05.006
![]() |
[25] |
L. C. Mei, Y. T. Jia, Y. Z. Lin, Simplified reproducing kernel method for impulsive delay differential equations, Appl. Math. Lett., 83 (2018), 123-129. doi: 10.1016/j.aml.2018.03.024
![]() |
[26] |
O. A. Arqub, Numerical solutions for the Robin time-fractional partial differential equations of heat and fluid flows based on the reproducing kernel algorithm, Int. J. Numer. Method H., 28 (2018), 828-856. doi: 10.1108/HFF-07-2016-0278
![]() |
[27] |
O. A. Arqub, B. Maayah, Solutions of Bagley-Torvik and Painlev equations of fractional order using iterative reproducing kernel algorithm with error estimates, Neural Comput. Appl., 29 (2018), 1465-1479. doi: 10.1007/s00521-016-2484-4
![]() |
[28] | M. Al-Smadi, O. Abu Arqub, Computational algorithm for solving fredholm time-fractional partial integrodifferential equations of Dirichlet functions type with error estimates, Appl. Math. Comput., 342 (2019), 280-294. |
[29] |
M. Al-Smadi, Simplified iterative reproducing kernel method for handling time-fractional BVPs with error estimation, Ain Shams Eng. J., 9 (2018), 2517-2525. doi: 10.1016/j.asej.2017.04.006
![]() |
[30] |
Z. Altawallbeh, M. Al-Smadi, I. Komashynska, et al., Numerical solutions of fractional systems of two-point BVPs by using the iterative reproducing kernel algorithm, Ukr. Math. J., 70 (2018), 687-701. doi: 10.1007/s11253-018-1526-8
![]() |
[31] |
A. Akgül, Reproducing kernel Hilbert space method based on reproducing kernel functions for investigating boundary layer flow of a Powell-Eyring non-Newtonian fluid, J. Taibah Univ. Sci., 13 (2019), 858-863. doi: 10.1080/16583655.2019.1651988
![]() |
[32] |
A. Akgül, E. K. Akgül, A novel method for solutions of fourth-order fractional boundary value problems, Fractal Fract., 3 (2019), 1-13. doi: 10.3390/fractalfract3010001
![]() |
[33] |
E. K. Akgül, Reproducing kernel Hilbert space method for nonlinear boundary-value problems, Math. Method Appl. Sci., 41 (2018), 9142-9151. doi: 10.1002/mma.5102
![]() |
[34] |
B. Boutarfa, A. Akgül, M. Inc, New approach for the Fornberg-Whitham type equations, J. Comput. Appl. Math., 312 (2017), 13-26. doi: 10.1016/j.cam.2015.09.016
![]() |
[35] |
A. Akgül, E. K. Akgül, S. Korhan, New reproducing kernel functions in the reproducing kernel Sobolev spaces, AIMS Math., 5 (2020), 482-496. doi: 10.3934/math.2020032
![]() |
[36] | N. Aronszajn, Theory of reproducing kernel, Trans. A.M.S., 168 (1950), 1-50. |
[37] | K. Diethelm, The analysis of fractional differential equations, New York: Springer, 2010. |
[38] | J. Shawe-Taylor, N. Cristianini, Kernel methods for pattern analysis, New York: Cambridge University Press, 2004. |