In the present paper, we introduce a certain subfamily of multivalently Bazilevič and non-Bazilevič functions associated with bounded boundary rotation to study some interesting properties, inclusion results, and the upper bounds of the initial Taylor-Maclaurin coefficients for functions in this subfamily. In addition, some special cases of this subfamily are also discussed.
Citation: Tamer M. Seoudy, Amnah E. Shammaky. Certain subfamily of multivalently Bazilevič and non-Bazilevič functions involving the bounded boundary rotation[J]. AIMS Mathematics, 2025, 10(5): 12745-12760. doi: 10.3934/math.2025574
[1] | S. M. Madian . Some properties for certain class of bi-univalent functions defined by $ q $-Cătaş operator with bounded boundary rotation. AIMS Mathematics, 2022, 7(1): 903-914. doi: 10.3934/math.2022053 |
[2] | Muhammmad Ghaffar Khan, Wali Khan Mashwani, Jong-Suk Ro, Bakhtiar Ahmad . Problems concerning sharp coefficient functionals of bounded turning functions. AIMS Mathematics, 2023, 8(11): 27396-27413. doi: 10.3934/math.20231402 |
[3] | Anandan Murugan, Sheza M. El-Deeb, Mariam Redn Almutiri, Jong-Suk-Ro, Prathviraj Sharma, Srikandan Sivasubramanian . Certain new subclasses of bi-univalent function associated with bounded boundary rotation involving sǎlǎgean derivative. AIMS Mathematics, 2024, 9(10): 27577-27592. doi: 10.3934/math.20241339 |
[4] | Prathviraj Sharma, Srikandan Sivasubramanian, Nak Eun Cho . Initial coefficient bounds for certain new subclasses of bi-univalent functions with bounded boundary rotation. AIMS Mathematics, 2023, 8(12): 29535-29554. doi: 10.3934/math.20231512 |
[5] | Muajebah Hidan, Abbas Kareem Wanas, Faiz Chaseb Khudher, Gangadharan Murugusundaramoorthy, Mohamed Abdalla . Coefficient bounds for certain families of bi-Bazilevič and bi-Ozaki-close-to-convex functions. AIMS Mathematics, 2024, 9(4): 8134-8147. doi: 10.3934/math.2024395 |
[6] | Yumao Li, K. Vijaya, G. Murugusundaramoorthy, Huo Tang . On new subclasses of bi-starlike functions with bounded boundary rotation. AIMS Mathematics, 2020, 5(4): 3346-3356. doi: 10.3934/math.2020215 |
[7] | Bushra Kanwal, Saqib Hussain, Thabet Abdeljawad . On certain inclusion relations of functions with bounded rotations associated with Mittag-Leffler functions. AIMS Mathematics, 2022, 7(5): 7866-7887. doi: 10.3934/math.2022440 |
[8] | Mohsan Raza, Khalida Inayat Noor . Subclass of Bazilevič functions of complex order. AIMS Mathematics, 2020, 5(3): 2448-2460. doi: 10.3934/math.2020162 |
[9] | Ala Amourah, B. A. Frasin, G. Murugusundaramoorthy, Tariq Al-Hawary . Bi-Bazilevič functions of order $ \vartheta +i\delta $ associated with $ (p, q)- $ Lucas polynomials. AIMS Mathematics, 2021, 6(5): 4296-4305. doi: 10.3934/math.2021254 |
[10] | Sadaf Umar, Muhammad Arif, Mohsan Raza, See Keong Lee . On a subclass related to Bazilevič functions. AIMS Mathematics, 2020, 5(3): 2040-2056. doi: 10.3934/math.2020135 |
In the present paper, we introduce a certain subfamily of multivalently Bazilevič and non-Bazilevič functions associated with bounded boundary rotation to study some interesting properties, inclusion results, and the upper bounds of the initial Taylor-Maclaurin coefficients for functions in this subfamily. In addition, some special cases of this subfamily are also discussed.
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] |
K. M. Alsager, G. Murugusundaramoorthy, A. Catas, S. M. E. Deeb, Applications of Caputo-type fractional derivatives for subclasses of bi-univalent functions with bounded boundary rotation, Fractal Fract., 8 (2024), 501. https://doi.org/10.3390/fractalfract8090501 doi: 10.3390/fractalfract8090501
![]() |
[2] | M. K. Aouf, A generalized of functions with real part bounded in the mean on the unit disc, Math. Japonica, 33 (1988), 175–182. |
[3] |
M. K. Aouf, T. M. Seoudy, On certain class of multivalent analytic functions defined by differential subordination, Rend. Circ. Mat. Palermo, 60 (2011), 191–201. https://doi.org/10.1007/s12215-011-0042-6 doi: 10.1007/s12215-011-0042-6
![]() |
[4] |
M. K. Aouf, T. M. Seoudy, Some properties of certain subclasses of p-valent Bazilevič functions associated with the generalized operator, Appl. Math. Lett., 24 (2011), 1953–1958. https://doi.org/10.1016/j.aml.2011.05.029 doi: 10.1016/j.aml.2011.05.029
![]() |
[5] |
M. K. Aouf, T. M. Seoudy, Certain class of bi-Bazilevič functions with bounded boundary rotation involving Salagean operator, Constr. Math. Anal., 3 (2020), 139–149. https://doi.org/10.33205/cma.781936 doi: 10.33205/cma.781936
![]() |
[6] |
M. K. Aouf, T. Bulboaca, T. M. Seoudy, Subclasses of meromorphic functions associated with a convolution operator, Filomat, 33 (2019), 2211–2218. https://doi.org/10.2298/FIL1908211A doi: 10.2298/FIL1908211A
![]() |
[7] |
M. K. Aouf, T. Bulboaca, T. M. Seoudy, Subclasses of multivalent non-Bazilevič functions defined with higher order derivatives, Bull. Transilvania Univ. Brasov, Series Ⅲ Math. Comput. Sci., 13 (2020), 411–422. https://doi.org/10.31926/but.mif.2020.13.62.2.4 doi: 10.31926/but.mif.2020.13.62.2.4
![]() |
[8] |
D. Breaz, P. Sharma, S. Sivasubramanian, S. M. E. Deeb, On a new class of Bi-close-to-convex functions with bounded boundary rotation, Mathematics, 11 (2023), 4376. https://doi.org/10.3390/math11204376 doi: 10.3390/math11204376
![]() |
[9] |
P. Goswami, B. S. Alkahtani, T. Bulboacă, Estimate for initial MacLaurin coefficients of certain subclasses of bi-univalent functions, Miskolc Math. Notes, 17 (2016), 739–748. https://doi.org/10.18514/MMN.2017.1565 doi: 10.18514/MMN.2017.1565
![]() |
[10] | M. Liu, On certain subclass of p-valent functions, Soochow J. Math., 20 (2000), 163–171. |
[11] |
S. S. Miller, P. T. Mocanu, Second order differential inequalities in the complex plane, J. Math. Anal. Appl., 65 (1978), 289–305. https://doi.org/10.1016/0022-247X(78)90181-6 doi: 10.1016/0022-247X(78)90181-6
![]() |
[12] |
A. Murugan, S. M. E. Deeb, M. R. Almutiri, J. S. Ro, P. Sharma, S. Sivasubramanian, Certain new subclasses of bi-univalent function associated with bounded boundary rotation involving salagean derivative, AIMS Math., 9 (2024), 27577–27592. https://doi.org/10.3934/math.20241339 doi: 10.3934/math.20241339
![]() |
[13] |
K. I. Noor, On subclasses of close-to-convex functions of higher order, Int. J. Math. Math. Sci., 15 (1992), 279–290. https://doi.org/10.1155/S016117129200036X doi: 10.1155/S016117129200036X
![]() |
[14] |
M. Obradovic, A class of univalent functions, Hokkaido Math. J., 27 (1998), 329–335. https://doi.org/10.14492/hokmj/1351001289 doi: 10.14492/hokmj/1351001289
![]() |
[15] | S. Owa, M. Obradovic, Certain subclasses of Bazilevič functions of type α, Int. J. Math. Math. Sci., 9 (1986), 97–105. |
[16] |
S. Owa, On certain Bazilevič functions of order β, Int. J. Math. Math. Sci., 15 (1992), 613–616. https://doi.org/10.1155/S0161171292000796 doi: 10.1155/S0161171292000796
![]() |
[17] |
K. S. Padmanabhan, R. Parvatham, Properties of a class of functions with bounded boundary rotation, Ann. Pol. Math., 31 (1975), 311–323. https://doi.org/10.4064/ap-31-3-311-323 doi: 10.4064/ap-31-3-311-323
![]() |
[18] |
B. Pinchuk, Functions with bounded boundary rotation, Isr. J. Math., 10 (1971), 7–16. https://doi.org/10.1007/BF02771515 doi: 10.1007/BF02771515
![]() |
[19] | H. Saitoh, On certain class of mulivalent functions, Math. Japonica, 37 (1992), 871–875. |
[20] |
T. M. Seoudy, A. E. Shammaky, On certain class of Bazilevič functions associated with the Lemniscate of Bernoulli, J. Funct. Spaces, 2020 (2020), 6622230. https://doi.org/10.1155/2020/6622230 doi: 10.1155/2020/6622230
![]() |
[21] |
R. Singh, On Bazilevič functions, P. Am. Math. Sci., 38 (1973), 261–267. https://doi.org/10.1090/S0002-9939-1973-0311887-9 doi: 10.1090/S0002-9939-1973-0311887-9
![]() |
[22] |
N. L. Sharma, T. Bulboaca, Logarithmic coefficient bounds for the class of bazilevic functions, Anal. Math. Phys., 14 (2024), 52. https://doi.org/10.1007/s13324-024-00909-y doi: 10.1007/s13324-024-00909-y
![]() |
[23] |
P. Sharma, S. Sivasubramanian, N.vE. Cho, Initial coefficient bounds for certain new subclasses of bi-bazilevic functions and exponentially bi-convex functions with bounded boundary rotation, Axioms, 13 (2024), 25. https://doi.org/10.3390/axioms13010025 doi: 10.3390/axioms13010025
![]() |
[24] |
H. M. Srivastava, S. Khan, S. N. Malik, F. Tchier, A. Saliu, Q. Xin, Faber polynomial coefficient inequalities for bi-bazilevic functions associated with the Fibonacci number series and the square-root functions, J. Inequal. Appl., 16 (2024). https://doi.org/10.1186/s13660-024-03090-9 doi: 10.1186/s13660-024-03090-9
![]() |
[25] |
Y. Tian, Z. Wang, Finite-time extended dissipative filtering for singular T-S fuzzy systems with nonhomogeneous Markov jumps, IEEE T. Cybernetics, 52 (2022), 4574–4584. https://doi.org/10.1109/TCYB.2020.3030503 doi: 10.1109/TCYB.2020.3030503
![]() |
[26] | N. Tuneski, M. Darus, Fekete-Szegö functional for non-Bazilevič functions, Acta Math. Acad. Paed. Nyireyháziensis, 18 (2002), 63–65. |
[27] |
S. Umar, M. Arif, M. Raza, S. K. Lee, On a subclass related to Bazilevič functions, AIMS Math., 5 (2020), 2040–2056. https://doi.org/10.3934/math.2020135 doi: 10.3934/math.2020135
![]() |
[28] |
A. K. Wanas, S. A. Sehen, A. O. P. Szabo, Toeplitz matrices for a class of bazilevic functions and the λ-pseudo-starlike functions, Axioms, 13 (2024), 521. https://doi.org/10.3390/axioms13080521 doi: 10.3390/axioms13080521
![]() |
[29] | Z. Wang, C. Gao, M. Liao, On certain generalized class of non-Bazilevič functions, Acta Math. Acad. Paed. Nyireyháziensis, 21 (2005), 147–154. |
[30] | D. G. Yang, Properties of certain classes of multivalent functions, Bull. Inst. Math. Acad., 22 (1994), 361–367. |