Research article

Certain subfamily of multivalently Bazilevič and non-Bazilevič functions involving the bounded boundary rotation

  • Received: 06 February 2025 Revised: 13 May 2025 Accepted: 27 May 2025 Published: 30 May 2025
  • MSC : 30C45

  • 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

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  • 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),     (pN={1,2,3,...}), (1.1)

    which are p-valent (multivalent of order p) in Δ={xC:|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: {φ(x)}>0 and {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:

    {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={φH:{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     (xΔ), (1.2)

    satisfying the following inequality

    2π0|{g(x)}σ1σ|dθkπ, (1.3)

    where k2, 0σ<1, and x=reiθΔ. 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)=12π2π01+(12σ)xeis1xeisdμ(s)     (xΔ), (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)=k24μ2(s), then (1.4) becomes

    g(x)=(k+24)12π2π01+(12σ)xeis1xeisdμ1(s)(k24)12π2π01+(12σ)xeis1xeisdμ2(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)(k24)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 {g(x)}>0.

    Remark 2. It is well-known from [13] that the family Qk(σ) is a convex set.

    Remark 3. For 0σ1<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 φHp is said to be in the subfamily BNα,βp,k(λ,σ) if it satisfies the following condition:

    (1αβα+βλ)[φ(x)xp]αβ+αβα+βλxφ(x)pφ(x)[φ(x)xp]αβQk(σ),

    which is equivalent to

    2π0|{(1αβα+βλ)[φ(x)xp]αβ+αβα+βλxφ(x)pφ(x)[φ(x)xp]αβ}σ1σ|dθkπ,

    where α,β0; α+β0; λ>0; k2; 0σ<1; xΔ; and all powers are principal ones.

    Example 1. Let φ(x):ΔC be an analytic function given by

    φ(x)=xp(1+p(α+β)(1σ)k[p(α+β)+λ]x)1αβHp   (αβ).

    Clearly φ(x)Hp (with all powers being principal ones). After some calculations, we find that

    (1αβα+βλ)[φ(x)xp]αβ+αβα+βλ[φ(x)xp]αβxφ(x)pφ(x)=1+(1σ)kx.

    Now, if x=reiθ(0r<1), then

    {(1αβα+βλ)[φ(x)xp]αβ+αβα+βλxφ(x)pφ(x)[φ(x)xp]αβ}σ1σ=1+krcosθ,

    and

    2π0|{(1αβα+βλ)[φ(x)xp]αβ+αβα+βλxφ(x)pφ(x)[φ(x)xp]αβ}σ1σ|dθ=2π0(1+krcosθ)dθ=2πkπ(k2).

    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=1L1M(1M<L1), we have BNα,02,p(λ,1L1M)=Bαp(λ,L,M)

    ={φHp:(1λ)[φ(x)xp]α+λxφ(x)pφ(x)[φ(x)xp]α1+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)

    ={φHp:xφ(x)pφ(x)[φ(x)xp]α1+Lx1+Mx},

    where the subfamily Bαp(L,M) was introduced by Yang [30];

    (ⅱ) BNα,02,1(1,1L1M)=Bα(L,M)

    ={φH:xφ(x)φ(x)[φ(x)x]α1+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)

    ={φH:{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=1L1M(1M<L1), we have BN0,β2,p(λ,1L1M)=Nβp(λ,L,M)

    ={φHp:(1+λ)[xpφ(x)]βλxφ(x)pφ(x)[xpφ(x)]β1+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)

    ={φH:(1+λ)[xφ(x)]βλxφ(x)φ(x)[xφ(x)]β1+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)

    ={φH:{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

    ={φH:{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)

    ={φHp:(1λ)[φ(x)xp]α+λxφ(x)pφ(x)[φ(x)xp]αQk(σ)},

    and Bαk,1(λ,σ)=Bαk(λ,σ)

    ={φH:(1λ)[φ(x)x]α+λxφ(x)φ(x)[φ(x)x]αQk(σ)};

    (ⅱ) BN0,βk,p(λ,σ)=Nβk,p(λ,σ)

    ={φHp:(1+λ)[xpφ(x)]βλxφ(x)pφ(x)[xpφ(x)]βQk(σ)},

    and Nβk,1(λ,σ)=Nβk(λ,σ)

    ={φH:(1+λ)[xφ(x)]βλxφ(x)φ(x)[xφ(x)]βQk(σ)};

    (ⅲ) BN1,0k,p(λ,σ)=Bk,p(λ,σ)

    ={φHp:(1λ)φ(x)xp+λφ(x)pxp1Qk(σ)},

    and Bk,1(λ,σ)=Bk(λ,σ)

    ={φH:(1λ)φ(x)x+λφ(x)Qk(σ)};

    (ⅳ) BN0,1k,p(λ,σ)=Nk,p(λ,σ)

    ={φHp:(1+λ)xpφ(x)λxp+1φ(x)pφ2(x)Qk(σ)},

    and Nk,1(λ,σ)=Nk(λ,σ)

    ={φH:(1+λ)xφ(x)λx2φ(x)φ2(x)Qk(σ)};

    (ⅴ) BNα,0k,p(1,σ)=Bαk,p(σ)

    ={φHp:xφ(x)pφ(x)[φ(x)xp]αQk(σ)},

    and Bαk,1(σ)=Bαk(σ)

    ={φH:xφ(x)φ(x)[φ(x)x]αQk(σ)};

    (ⅵ) BN0,βk,p(1,σ)=Nβk,p(σ)

    ={φHp:xφ(x)pφ(x)[xpφ(x)]βQk(σ)},

    and Nβk,1(σ)=Nβk(σ)

    ={φH:xφ(x)φ(x)[xφ(x)]βQk(σ)};

    (ⅶ) B0k,p(σ)=Sk,p(σ)

    ={φHp:xφ(x)pφ(x)Qk(σ)},

    and Sk,1(σ)=Sk(σ)

    ={φH: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 DC2.

    (ⅱ) (0,1)D and Θ(1,0)>0.

    (ⅲ) {Θ(iγ2,δ1)}>0 whenever (iγ2,δ1)D and δ11+γ222.

    If g(x) given by (1.2) is analytic in Δ such that (g(x),xg(x))D and {Θ(g(x),xg(x))}>0 for xΔ, then {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|(1σ)k     (jN). (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+αβ12b2p+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 φBNα,βk,p(λ,σ), then

    [φ(x)xp]αβQk(σ1), (2.1)

    where σ1 is given by

    σ1=2p(α+β)σ+λ2p(α+β)+λ. (2.2)

    Proof. Let φBNα,βk,p(λ,σ) and set

    [φ(x)xp]αβ=(1σ1)g(x)+σ1     (xΔ)=(k+24){(1σ1)g1(x)+σ1}(k24){(1σ1)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

    (1αβα+βλ)[φ(x)xp]αβ+αβα+βλxφ(x)pφ(x)[φ(x)xp]αβ=(1σ1)g(x)+σ1+λ(1σ1)xg(x)p(α+β)Qk(σ),

    this implies that

    11σ{(1σ1)gi(x)+σ1σ+λ(1σ1)xgi(x)p(α+β)}Q    (xΔ;i=1,2).

    We form the functional Θ(γ,δ) by choosing γ=gi(x), δ=xgi(x),

    Θ(γ,δ)=(1σ1)γ+σ1σ+λ(1σ1)δp(α+β).

    Clearly, the first two conditions of Lemma 1 are satisfied. Now, we verify the condition (ⅲ) of Lemma 1 as follows:

    {Θ(iγ2,δ1)}=σ1σ+{λ(1σ1)δ1p(α+β)}σ1σλ(1σ1)(1+γ22)2p(α+β)=2p(α+β)(σ1σ)λ(1σ1)λ(1σ1)γ222p(α+β)=A+Bγ222C,

    where

    A=2p(α+β)(σ1σ)λ(1σ1),B=λ(1σ1)<0,C=2p(α+β)>0.

    We note that {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(1σ1)<0 gives us 0σ1<1. Therefore, applying Lemma 1, gi(x)Q(i=1,2) and consequently g(x)Qk(σ1) for xΔ. This completes the proof of Theorem 1.

    Putting β=0 in Theorem 1, we obtain the next result.

    Corollary 1. If φBαk,p(λ,σ), then

    [φ(x)xp]αQk(σ2),

    where σ2 is given by

    σ2=2pαρ+λ2pα+λ.

    Putting α=0 in Theorem 1, we get the following result.

    Corollary 2. If φNβk,p(λ,σ), then

    [xpφ(x)]βQk(σ3),

    where σ3 is given by

    σ3=2pβρ+λ2pβ+λ.

    Theorem 2. If φBNα,βk,p(λ,σ), then

    [φ(x)xp]αβ2Qk(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 φBNα,βk,p(λ,σ) and let

    [φ(x)xp]αβ=(k4+12)[(1σ4)g1(x)+σ4]2(k4+12)[(1σ4)g1(x)+σ4]2=[(1σ4)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

    (1αβα+βλ)[φ(x)xp]αβ+αβα+βλxφ(x)pφ(x)[φ(x)xp]αβ={[(1σ4)g(x)+σ4]2+[(1σ4)g(x)+σ4]2λ(1σ4)xg(x)p(α+β)}Qk(σ),

    this implies that

    11σ{[(1σ4)gi(x)+σ4]2+[(1σ4)gi(x)+σ4]2λ(1σ4)xgi(x)p(α+β)σ}Q  (i=1,2).

    We form the functional Θ(γ,δ) by choosing γ=gi(x), δ=xgi(x),

    Θ(γ,δ)=[(1σ4)γ+σ4]2+[(1σ4)γ+σ4]2λ(1σ4)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:

    {Θ(iγ2,δ1)}=σ24(1σ4)2γ22+2λσ4(1σ4)δ1p(α+β)σσ24σ(1σ4)2γ22λσ4(1σ4)(1+γ22)p(α+β)=A+Bγ222C,

    where

    A=p(α+β)(σ24σ)λσ4(1σ4),B=(1σ4)(1σ4+λσ4)<0,C=p(α+β)2>0.

    We note that {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 0σ4<1. Therefore, applying Lemma 1, gi(x)Q (i=1,2), and consequently g(x)Qk(σ4) for xΔ. This completes the proof of Theorem 2.

    Putting β=0 in Theorem 2, we obtain the following.

    Corollary 3. If φBαk,p(λ,σ), then

    [φ(x)xp]α2Qk(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 φNβk,p(λ,σ), then

    [xpφ(x)]β2Qk(σ6),

    where σ6 is given by

    σ6=λ+λ2+4(pβ+λ)ρpβ2(pβ+λ).

    For a function φHp, the generalized Bernardi-Libera-Livingston integral operator Φp,μ:HpHp, with μ>p, is given by (see [19])

    Φp,μ(φ(x))=μ+pxμx0ωμ1φ(ω)dω   (μ>p). (2.7)

    It is easy to verify that for all φHp 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 φHp satisfies the next condition

    (1αβα+βλ)[Φp,μ(φ(x))xp]αβ +αβα+βλφ(x)Φp,μ(φ(x))[Φp,μ(φ(x))xp]αβQk(σ), (2.9)

    with Φp,μ is the integral operator defined by (2.7), then

    [Φp,μ(φ(x))xp]αβQk(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]αβ=(k4+12){(1σ7)g1(x)+σ7}(k4+12){(1σ7)g2(x)+σ7}=(1σ7)g(x)+σ7   (xΔ), (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

    (1αβα+βλ)[Φp,μ(φ(x))xp]αβ +αβα+βλφ(x)Φp,μ(φ(x))[Φp,μ(φ(x))xp]αβ=(1σ7)g(x)+λ(1σ7)xg(x)(p+μ)(α+β)Qk(σ)  (xΔ).

    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 φHp satisfies the next condition

    (1λ)[Φp,μ(φ(x))xp]α +λφ(x)Φp,μ(φ(x))[Φp,μ(φ(x))xp]αQk(σ),

    with Φp,μ is defined by (2.7), then

    [Φp,μ(φ(x))xp]αQk(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 φHp satisfies the next condition

    (1+λ)[xpΦp,μ(φ(x))]β λφ(x)Φp,μ(φ(x))[xpΦp,μ(φ(x))]βQk(σ),

    with Φp,μ is defined by (2.7), then

    [xpΦp,μ(φ(x))]βQk(σ9),

    where σ9 is given by

    σ9=2(p+μ)(α+β)σ+λ2(p+μ)(α+β)+λ.

    Theorem 4. If 0λ1<λ2, then

    BNα,βk,p(λ2,σ)BNα,βk,p(λ1,σ).

    Proof. If we consider an arbitrary function φBNα,βk,p(λ2,σ), then

    φ2(x)=(1αβα+βλ2)[φ(x)xp]αβ+αβα+βλ2xφ(x)pφ(x)[φ(x)xp]αβQk(σ).

    According to Theorem 1, we have

    φ1(x)=[φ(x)xp]αβQk(σ1),

    where σ1 is given by (2.2). From (2.2), it follows that σ1σ, and from Remark 3, we conclude that Qk(σ1)Qk(σ); hence, φ1(x)Qk(σ).

    A simple computation shows that

    (1αβα+βλ1)[φ(x)xp]αβ+αβα+βλ1xφ(x)pφ(x)[φ(x)xp]αβ=(1λ1λ2)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 0λ1<λ2, which implies that φBNα,βk,p(λ1,σ).

    Putting β=0 in Theorem 4, we obtain the following.

    Corollary 7. If 0λ1<λ2, then

    Bαk,p(λ2,σ)Bαk,p(λ1,σ).

    Putting α=0 in Theorem 4, we get the following.

    Corollary 8. If 0λ1<λ2, then

    Nβk,p(λ2,σ)Nβk,p(λ1,σ).

    Theorem 5. If φBNα,βk,p(λ,σ) given by (1.1) with αβ, p(α+β)+λ0 and p(α+β)+2λ0, then

    |bp+1|p|α+β|(1σ)k|αβ||p(α+β)+λ|, (2.13)

    and

    |bp+2+αβ12b2p+1|p|α+β|(1σ)k|αβ||p(α+β)+2λ|. (2.14)

    Proof. If φBNα,βk,p(λ,σ), from Definition 1, we have

    (1αβα+βλ)[φ(x)xp]αβ+αβα+βλxφ(x)pφ(x)[φ(x)xp]αβ=G(x), (2.15)

    where G(x)Qk(σ) is given by

    G(x)=1+d1x+d2x2+d3x3+... . (2.16)

    Since

    (1αβα+βλ)[φ(x)xp]αβ+αβα+βλxφ(x)pφ(x)[φ(x)xp]αβ=1+(αβ)[p(α+β)+λ]p(α+β)bp+1x+(αβ)[p(α+β)+2λ]p(α+β)(bp+2+αβ12b2p+1)x2+..., (2.17)

    Comparing the coefficients in (2.15) by using (2.16) and (2.17), we obtain

    (αβ)[p(α+β)+λ]p(α+β)bp+1=d1, (2.18)
    (αβ)[p(α+β)+2λ]p(α+β)(bp+2+αβ12b2p+1)=d2. (2.19)

    Therefore,

    bp+1=p(α+β)(αβ)[p(α+β)+λ]d1,

    and

    bp+2+αβ12b2p+1=p(α+β)(αβ)[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 φBαk,p(λ,σ) is given by (1.1) with pα+λ0 and pα+2λ0, then

    |bp+1|p(1σ)k|pα+λ|,

    and

    |bp+2+α12b2p+1|p(1σ)k|pα+2λ|.

    Putting k=2 and σ=1L1M(1M<L1) in Corollary 9, we obtain the following corollary, which improves the result of Liu [10, Theorem 4 with n=1].

    Corollary 10. If φBαp(λ,L,M) is given by (1.1) with pα+λ0 and pα+2λ0, then

    |bp+1|2p(LM)|pα+λ|(1M),

    and

    |bp+2+α12b2p+1|2p(LM)|pα+2λ|(1M).

    Putting α=0 in Theorem 5, we get the following.

    Corollary 11. If φNβk(λ,σ) given by (1.1) with pβ+λ0 and pβ+2λ0, then

    |bp+1|p(1σ)k|pβ+λ|,

    and

    |bp+2β+12b2p+1|p(1σ)k|pβ+2λ|.

    Putting k=2 and σ=1L1M(1M<L1) 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 φNβk(λ,L,M) given by (1.1) with pβ+λ0 and pβ+2λ0, then

    |bp+1|2p(LM)|pβ+λ|(1M),

    and

    |bp+2β+12b2p+1|2p(LM)|pβ+2λ|(1M).

    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.



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