Citation: Ekram E. Ali, Georgia Irina Oros, Rabha M. El-Ashwah, Abeer M. Albalahi. Applications of fuzzy differential subordination theory on analytic p -valent functions connected with q-calculus operator[J]. AIMS Mathematics, 2024, 9(8): 21239-21254. doi: 10.3934/math.20241031
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[10] | Ekram E. Ali, Georgia Irina Oros, Rabha M. El-Ashwah, Abeer M. Albalahi . Applications of fuzzy differential subordination theory on analytic p -valent functions connected with q-calculus operator. AIMS Mathematics, 2024, 9(8): 21239-21254. doi: 10.3934/math.20241031 |
Let A stand for the collection of functions G of the type
G(ξ)=ξ+∞∑j=2ajξj, | (1.1) |
that are holomorphic in the open unit disk Λ:={ξ∈C:|ξ|<1} of the complex plane, and let S indicate the subclass of functions of A which are univalent in Λ. For functions G∈A given by (1.1) and H∈A given by H(ζ)=ζ+∞∑j=2bjζj, we define the convolution product (or Hadamard) of G and H by
(G∗H)(ζ)=(H∗G)(ζ)=ζ+∞∑j=2ajbjζj,ζ∈Λ. | (1.2) |
Let G and F be two holomorphic functions in Λ. The function G is said to be subordinated to F if there are Schwarz function w(ξ), that is, holomorphic in Λ with w(0)=0 and |w(ξ)|<1, ξ∈Λ, such as G(ξ)=F(w(ξ)) for all ξ∈Λ. This subordination notion is indicated by
G≺ForG(ξ)≺F(ξ). |
If the function F is univalent in Λ, then we have the inclusion equivalence
G(ξ)≺F(ξ)⇔G(0)=F(0)andG(Λ)⊂F(Λ). |
The subfamilies of S which are the starlike and the convex function in Λ defined by
S∗:={G∈A:ReξG′(ξ)G(ξ)>0,ξ∈Λ} | (1.3) |
and
C:={G∈A:Re(ξG′(ξ))′G′(ξ)>0,ξ∈Λ}, | (1.4) |
respectively. Equivalently, we have
S∗(φ)={G∈A:ξG′(ξ)G(ξ)≺φ(ξ)},C(φ)={G∈A:(ξG′(ξ))′G′(ξ)≺φ(ξ)}, |
where
φ(ξ)=1+ξ1−ξ. | (1.5) |
Janowski defined in [4] the extended function family S∗[A,B] of starlike functions called the Janwoski class of functions as follows: A function G∈A is in the family S∗[A,B] if
ξG′(ξ)G(ξ)≺1+Aξ1+Bξ(−1≤B<A≤1). |
The above subordination could be written as
ξG′(ξ)G(ξ)=1+Ap(ξ)1+Bp(ξ)(−1≤B<A≤1), | (1.6) |
where an analytical function with a real positive part in Λ is denoted by p(ξ).
The Janowski convex and Janowski starlike functions are obtained by reducing the above-described classes to the requirement −1≤B<A≤1. For the special cases A:=1−2α and B:=−1, where 0≤α<1, we obtain the families, namely the family of starlike and convex functions of order α (0≤α<1) previously defined by Robertson in [6], and considered respectively by
S∗(α):={G∈A:ReξG′(ξ)G(ξ)>α,ξ∈Λ},C(α):={G∈A:Re(ξG′(ξ))′G′(ξ)>α,ξ∈Λ}. |
Babalola defined the operator Iρυ:A→Aas
IσυG(ζ)=(ρσ∗ρ−1σ,υ∗G)(ζ), | (1.7) |
where
ρσ,υ(ζ)=ζ(1−ζ)σ−υ+1, σ−υ+1>0, ρσ=ρσ,0, |
and ρ−1σ,υ is
(ρσ,υ∗ρ−1σ,υ)(ζ)=ζ1−ζ (σ,υ∈N={1,2,3,...}). |
For G∈A, then (1.7) is equivalent to
IσυG(ζ)=ζ+∞∑j=2[Γ(σ+j)Γ(σ+1)⋅(σ−υ)!(σ+j−υ−1)!]ajζj. |
Making use the binomial series
(1−δ)t=t∑i=0(ti)(−1)i δi (t∈N), |
for G∈A, El-Deeb [3] introduced the linear differential operator as follows:
Dσ,0m,δ,υG(ζ)=G(ζ), |
Dσ,1t,δ,υG(ζ)=Dσt,δ,υG(ζ)=(1−δ)tIσυG(ζ)+[1−(1−δ)t]ζ(IσυG)′(ζ)=ζ+∞∑j=2[1+(j−1)ct(δ)][Γ(σ+j)Γ(σ+1)⋅(σ−υ)!(σ+j−υ−1)!]ajζj...Dσ,nt,δ,υG(ζ)=Dσt,δ,υ(Dσ,n−1t,δ,υG(ζ))=(1−δ)tDσ,n−1t,δ,υG(ζ)+[1−(1−δ)t]ζ(Dσ,n−1t,δ,υG(ζ))′=ζ+∞∑j=2[1+(j−1)ct(δ)]n[Γ(σ+j)Γ(σ+1)⋅(σ−υ)!(σ+j−υ−1)!]ajζj=ζ+∞∑j=2ψnj[Γ(σ+j)Γ(σ+1)⋅(σ−υ)!(σ+j−υ−1)!]ajζj, (δ>0; t,σ,υ∈N; n∈N0=N∪{0}), | (1.8) |
where
ψnj=[1+(j−1)ct(δ)]n, | (1.9) |
and
ct(δ)=t∑i=1(ti)(−1)i+1 δi (t∈N). |
From (1.8), we obtain that
ct(δ) ζ (Dσ,nt,δ,υG(ζ))′=Dσ,n+1t,δ,υG(ζ)−[1−ct(δ)]Dσ,nt,δ,υG(ζ). | (1.10) |
In this article using the El-Deeb operator defined in (1.8), we define a new sub-family of A:
Rm,n,σt,δ,υ(A,B)={G∈A:Dσ,mt,δ,υG(ζ)Dσ,nt,δ,υG(ζ)≺1+Aξ1+Bξ}, | (1.11) |
where −1≤A<B≤1; δ>0; t,σ,υ∈N and n,m∈N0, that will lead us to the study of Fekete-Szegö problem. Further, coefficient estimates, characteristic properties and partial sums results will be established.
Specializing the values of A and B one can obtain the particular cases
(i) Rm,n,σt,δ,υ(1−2α,−1)=:Wm,n,σt,δ,υ(α)={G∈A:Re(Dσ,mt,δ,υG(ζ)Dσ,nt,δ,υG(ζ))>α, (0≤α<1)}; |
and
(ii) Rm,n,σt,δ,υ(1,−1)=:Fm,n,σt,δ,υ={G∈A:Re(Dσ,mt,δ,υG(ζ)Dσ,nt,δ,υG(ζ))>0}. |
To solve the Fekete-Szegö type inequality for G∈Rm,n,σt,δ,υ(A,B) we will use the next results (the first part is due to Carathéodory [1]):
Lemma 1. [1,5] If P(ξ)=1+p1ξ+p2ξ2+⋯∈P where P the class of holomorphic functions with positive real part in Λ, with P(0)=1, then
|pn|≤2,n≥1, | (2.1) |
and for the complex number μ∈C we have
|p2−μp21|≤2max{1;|1−2μ|}. | (2.2) |
If μ is a real parameter, then
|p2−μp21|≤{−4μ+2,ifμ≤0,2if0≤μ≤1,4μ−2ifμ≥1. | (2.3) |
When μ>1 or μ<0, equality (2.3) holds true if and only if P1(ξ)=1+ξ1−ξ or one of its rotations. When 0<μ<1, the equality (2.3) holds if and only if P2(ξ)=1+ξ21−ξ2 or one of its rotations. When μ=0, equality (2.3) holds if and only if
P3(ξ)=(1+c2)1+ξ−ξ+1+(1−c2)−ξ+11+ξ(0≤c≤1) |
or one of its rotations. When μ=1, the equality (2.3) holds true if P(ξ) is a reciprocal of one of the functions such that the equality holds true in the case when μ=0.
Theorem 1. If G∈A defined as (1.1), belongs to Rm,n,σt,δ,υ(A,B), then
|a2|≤(σ−υ+1)(A−B)(σ+1)|ψm2−ψn2|, | (2.4) |
|a3|≤(σ−υ+1)(σ−υ+2)(A−B)(σ+1)(σ+2)|ψm2−ψn2|×max{1;|−B+(A−B)(ψn+m2−ψ2n2)(ψm2−ψn2)2|}, | (2.5) |
and for a complex number τ, we have
|a3−τa22|≤(σ−υ+1)(σ−υ+2)(A−B)(σ+1)(σ+2)|ψm3−ψn3|max{1;|Ω(τ,σ,υ,A,B)|}, | (2.6) |
where
Ω(τ,σ,υ,A,B)=1−2Θ(τ,σ,υ,A,B), |
Θ(τ,σ,υ,A,B)=12(1+B−(A−B)(ψn+m2−ψ2n2)(ψm2−ψn2)2 |
+τ(A−B)(σ+2)(σ−υ+1)(ψm3−ψn3)(σ+1)(σ−υ+2)(ψm2−ψn2)2), | (2.7) |
and ψnj is given by (1.9).
Proof. We will show that the relations (2.4)–(2.6) and (2.16) hold true for G∈Rm,n,σt,δ,υ(A,B). If G∈Rm,n,σt,δ,υ(A,B), then
Dσ,mt,δ,υG(ζ)Dσ,nt,δ,υG(ζ)≺1+Aξ1+Bξ |
which yields
Dσ,mt,δ,υG(ζ)Dσ,nt,δ,υG(ζ)≺1+Aw(ξ)1+Bw(ξ)=G(w(ξ)),(−1≤B<A≤1). | (2.8) |
Since we can write w(ξ) as
w(ξ)=1−h(ξ)1+h(ξ)=p1ξ+p2ξ2+p3ξ3+…2+p1ξ+p2ξ2+p3ξ3+…, |
where h(ξ)∈P and have the form h(ξ)=1+p1ξ+p2ξ2+p3ξ3+…, so
G(w(ξ))=1+12(A−B)p1ξ+(A−B)4[2p2−(1+B)p21]ξ2+…, | (2.9) |
and therefore
Dσ,mt,δ,υG(ζ)Dσ,nt,δ,υG(ζ)=1+(σ+1)(σ−υ+1)(ψm2−ψn2)a2ζ+((σ+1)(σ+2)(σ−υ+1)(σ−υ+2)(ψm3−ψn3)a3−(σ+1)2(σ−υ+1)2((ψn+m2−ψ2n2))a22)ζ2+…. | (2.10) |
If we compare the first coefficients of (2.9) and (2.10), we get
a2=(σ−υ+1)(A−B)2(σ+1)(ψm2−ψn2)p1, | (2.11) |
a3=(σ−υ+1)(σ−υ+2)(A−B)2(σ+1)(σ+2)(ψm3−ψn3)×(p2−p212[1+BA−B−((A−B)(ψn+m2−ψ2n2)(ψm2−ψn2)2)]) | (2.12) |
and by using (2.1) in (2.11) and (2.2) in (2.12), we get
|a2|≤(σ−υ+1)(A−B)(σ+1)|ψm2−ψn2|, | (2.13) |
|a3|≤(σ−υ+1)(σ−υ+2)(A−B)(σ+1)(σ+2)|ψm3−ψn3|×max{1;|−B+(A−B)(ψn+m2−ψ2n2)(ψm2−ψn2)2|}. | (2.14) |
For a complex nubmer τ, and from (2.11) together with (2.12), we have
|a3−τa22|=(σ−υ+1)(σ−υ+2)(A−B)2(σ+1)(σ+2)(ψm3−ψn3)|p2−Θ(τ,σ,υ,A,B)p21|, | (2.15) |
where Θ(τ,σ,υ,A,B) is denoted by (2.7). Now, we apply Lemma 1 to (2.15) and obtain the required results.
Theorem 2. If the function G∈A defined as (1.1) belongs to Rm,n,σt,δ,υ(A,B), then for any real parameter τ we obtain
|a3−τa22|≤(σ−υ+1)(σ−υ+2)(A−B)2(σ+1)(σ+2)|ψm3−ψn3|{1−2Θ(τ,σ,υ,A,B),ifτ<φ1,1,ifφ1≤τ≤φ2,2Θ(τ,σ,υ,A,B)−1,ifτ>φ2, | (2.16) |
where Θ(τ,σ,υ,A,B) is given by (2.7),
φ1=(σ+1)(σ−υ+2)(ψm2−ψn2)2(A−B)(σ+2)(σ−υ+1)(ψm3−ψn3)×(−1−B+(A−B)(ψn+m2−ψ2n2)(ψm2−ψn2)2), |
and
φ2=(σ+1)(σ−υ+2)(ψm2−ψn2)2(A−B)(σ+2)(σ−υ+1)(ψm3−ψn3)×(1−B+(A−B)(ψn+m2−ψ2n2)(ψm2−ψn2)2). |
Proof. The proof can be produced directly by making use of Lemma 1 in (2.15), so we choose to omit it.
The "Koebe one quarter theorem" [2] ensures that the image of Λ under each univalent function G∈A consists a disk of radius 14. Thus each univalent function G has an inverse G−1 satisfying
G−1(G(ξ))=ξ,(ξ∈Λ)andG(G−1(w))=w,(|w|<r0(G),r0(G)≥14). |
A function G∈A is called bi-univalent in Λ if both G and G−1 are univalent in Λ. We mention that the collection of bi-univalent functions defined in the unit disk Λ is not empty. For example, the functions ξ, ξ1−ξ, −log(1−ξ) and 12log1+ξ1−ξ are members of bi-univalent function family, however the Koebe function is not a member.
Theorem 3. If G∈Rm,n,σt,δ,υ(A,B) and the inverse function of G is G−1(w)=w+∞∑j=2djwj, then
|d2|≤(σ−υ+1)(A−B)(σ+1)|ψm2−ψn2| | (3.1) |
|d3|≤(σ−υ+1)(σ−υ+2)(A−B)(σ+1)(σ+2)|ψm3−ψn3|max{1;|2Θ(2,σ,υ,A,B)−1|}, | (3.2) |
and for any μ∈C, we have
|d3−μd22|≤(σ−υ+1)(σ−υ+2)(A−B)(σ+1)(σ+2)|ψm3−ψn3|×max{1;|2Θ(2,σ,υ,A,B)+μ(A−B)(σ+2)(σ−υ+1)(ψm3−ψn3)(σ+1)(σ−υ+2)(ψm2−ψn2)2−1|}, |
where Θ(2,σ,υ,A,B) given by (2.7).
Proof. Since
G−1(w)=w+∞∑n=2dnwn | (3.3) |
is the inverse of the function G, it can be seen that
ξ=G−1(G(ξ))=G(G−1(ξ)),|ξ|<r0(G). | (3.4) |
From (1.1) and (3.4), we obtain that
ξ=G−1(ξ+∞∑n=2anξn),|ξ|<r0(G). | (3.5) |
Therefore from (3.4) and (3.5) we get
ξ+(a2+d2)ξ2+(a3+2a2d2+d3)ξ3+⋯=ξ,|ξ|<r0(G). | (3.6) |
Equating the corresponding coefficients of the relation (3.6), we conclude that
d2=−a2, | (3.7) |
d3=2a22−a3. | (3.8) |
First, from the relations (2.11) and (3.7) we have
d2=−(σ−υ+1)(A−B)2(σ+1)(ψm2−ψn2)p1. | (3.9) |
To find |d3|, from (3.8) we have
|d3|=|a3−2a22|. |
Hence, by using (2.15) for real τ=2 we deduce that
|d3|=|a3−2a22|=(σ−υ+1)(σ−υ+2)(A−B)2(σ+1)(σ+2)|ψm3−ψn3||p2−Θ(2,σ,υ,A,B)p21|=(σ−υ+1)(σ−υ+2)(A−B)(σ+1)(σ+2)|ψm3−ψn3|max{1;|2Θ(2,σ,υ,A,B)|−1}, | (3.10) |
where Θ(2,σ,υ,A,B) given by (2.7). For any complex number μ, a simple computation gives us that
d3−μd22=(σ−υ+1)(σ−υ+2)(A−B)2(σ+1)(σ+2)(ψm3−ψn3)(p2−Θ(2,σ,υ,A,B)p21)−μ[(σ−υ+1)(A−B)]2[2(σ+1)(ψm2−ψn2)]2p21.=(σ−υ+1)(σ−υ+2)(A−B)2(σ+1)(σ+2)(ψm3−ψn3)×(p2−p212[2Θ(2,σ,υ,A,B)+μ(A−B)(σ+2)(σ−υ+1)(ψm3−ψn3)(σ+1)(σ−υ+2)(ψm2−ψn2)2]). | (3.11) |
By taking modulus on both sides of (3.11) and applying Lemma 1 and (2.1), we find that
|d3−μd22|≤(σ−υ+1)(σ−υ+2)(A−B)(σ+1)(σ+2)|ψm3−ψn3|×max{1;|2Θ(2,σ,υ,A,B)+μ(A−B)(σ+2)(σ−υ+1)(ψm3−ψn3)(σ+1)(σ−υ+2)(ψm2−ψn2)2−1|}, |
and this completes our proof.
By applying the techniques introduced by Silverman in [7], we will introduce some characteristic properties of the functions G∈Rm,n,σt,δ,υ(A,B) such as partial sums results, necessary and sufficient conditions, radii of close-to-convexity, distortion bounds, radii of starlikeness and convexity.
Theorem 4. If G∈A and be defined as (1.1) belongs to Rm,n,σt,δ,υ(A,B), then
∞∑j=2((1−B)ψmj+(A−1)ψnj)(σ+1(σ−υ+1))|aj|≤(A−B), | (4.1) |
where ψnj given by (1.9).
Proof. Letting G∈Rm,n,σt,δ,υ(A,B), by (1.11) we deduce that
Dσ,mt,δ,υG(ζ)Dσ,nt,δ,υG(ζ)=1+Aw(ξ)1+Bw(ξ),ξ∈Λ, | (4.2) |
where w(ξ) is a Schwarz function, or equivalently
|Dσ,mt,δ,υG(ζ)−Dσ,nt,δ,υG(ζ)ADσ,nt,δ,υG(ζ)−BDσ,mt,δ,υG(ζ)|<1,ξ∈Λ. |
Thus, the above relation leads us to
|Dσ,mt,δ,υG(ζ)−Dσ,nt,δ,υG(ζ)ADσ,nt,δ,υG(ζ)−BDσ,mt,δ,υG(ζ)|=|∞∑j=2(ψmj−ψnj)(σ+1(σ−υ+1))ajξj(A−B)ξ+∞∑j=2(Aψnj−Bψmj)(σ+1(σ−υ+1))ajξj|≤∞∑j=2(ψmj−ψnj)(σ+1(σ−υ+1))|aj|rj−1(A−B)−∞∑j=2(Aψnj−Bψmj)(σ+1(σ−υ+1))|aj|rj−1<1, |
and taking |ξ|=r→1− simple computation yields (4.1).
Example 1. For
G(ξ)=ξ+∞∑j=2(A−B)(1−B)ψmj+(A−1)ψnj(σ−υ+1σ+1)ℓjξj,ξ∈Λ, |
such that ∞∑j=2ℓj=1, we get
∞∑j=2((1−B)ψmj+(A−1)ψnj)(σ+1σ−υ+1)|aj|=∞∑j=2((1−B)ψmj+(A−1)ψnj)(σ+1σ−υ+1)×(A−B)(1−B)ψmj+(A−1)ψnj(σ−υ+1σ+1)ℓj=(A−B)∞∑j=2ℓj=(A−B). |
Then G∈Rm,n,σt,δ,υ(A,B), and we note that the inequality (4.1) is sharp.
Corollary 1. Let G∈Rm,n,σt,δ,υ(A,B) given by (1.1). Then
|aj|≤(A−B)(1−B)ψmj+(A−1)ψnj(σ−υ+1σ+1),forj≥2, | (4.3) |
where ψnj is defined by (1.9). The approximation is sharp for the function
G∗(ξ):=ξ−(A−B)(1−B)ψmj+(A−1)ψnj(σ−υ+1σ+1)ξj,j≥2. | (4.4) |
Theorem 5. If G∈Rm,n,σt,δ,υ(A,B), then
r−(A−B)(1−B)ψmj+(A−1)ψnj(σ−υ+1σ+1)r2≤|G(η)|≤r+(A−B)(1−B)ψmj+(A−1)ψnj(σ−υ+1σ+1)r2. | (4.5) |
For the function defined by
ˆG(ξ):=ξ−(A−B)(1−B)ψmj+(A−1)ψnj(σ−υ+1σ+1)ξ2,|ξ|=r<1, | (4.6) |
the approximation is sharp.
Proof. For |ξ|=r<1 we have
|G(ξ)|=|ξ+∞∑j=2ajξj|≤|ξ|+∞∑j=2aj|ξ|j=r+∞∑j=2aj|r|j. |
Moreover, since for |ξ|=r<1 we get rj<r2 for all j≥2, the above relation implies that
|G(ξ)|≤r+r2∞∑j=2|aj|. | (4.7) |
Similarly, we get
|G(ξ)|≥r−r2∞∑j=2|aj|. | (4.8) |
From the relation (4.1) we have
∞∑j=2((1−B)ψmj+(A−1)ψnj)(σ+1σ−υ+1)|aj|≤(A−B), |
but
((1−B)ψm2+(A−1)ψn2)(σ+1σ−υ+1)∞∑j=2|aj|≤∞∑j=2((1−B)ψmj+(A−1)ψnj)(σ+1σ−υ+1)|aj|≤(A−B). |
Therefore,
∞∑j=2aj≤(σ−υ+1σ+1)(A−B)(1−B)ψm2+(A−1)ψn2, | (4.9) |
and by using (4.3) in (4.7) and (4.8) we get the desired result.
The next distortion theorem for the family Rm,n,σt,δ,υ(A,B) could be similarly obtained:
Theorem 6. If G∈Rm,n,σt,δ,υ(A,B), then
1−2(σ−υ+1)(A−B)(σ+1)((1−B)ψm2+(A−1)ψn2)r≤|G′(ξ)|≤1+2(σ−υ+1)(A−B)(σ+1)((1−B)ψm2+(A−1)ψn2)r. |
The equality holds if the function is ˆG given by (4.6).
Proof. Since the proof is quite analogous with those of Theorem 5, so it will be omitted.
The next result deals with the fact that a convex combination of functions of the class Rm,n,σt,δ,υ(A,B) belongs to the same class, as follows:
Theorem 7. Let Gi∈Rm,n,σt,δ,υ(A,B) given by
Gi(ξ)=ξ+∞∑j=2ai,jξj,i=1,2,3,…,m. | (4.10) |
Then H∈Rm,n,σt,δ,υ(A,B), where
H(ξ):=m∑i=1ciGi(ξ),andm∑i=1ci=1. | (4.11) |
Proof. By Theorem 4 we have
∞∑j=2((1−B)ψmj+(A−1)ψnj)(σ+1σ−υ+1)|aj|≤(A−B), |
and,
H(ξ)=m∑i=1ci(ξ+∞∑j=2ai,jξj)=ξ+∞∑j=2(m∑i=1ciai,j)ξj. |
Therefore
∞∑j=2((1−B)ψmj+(A−1)ψnj)(σ+1σ−υ+1)|m∑i=1ciai,j|≤m∑i=1[∞∑j=2((1−B)ψmj+(A−1)ψnj)(σ+1σ−υ+1)|ai,j|]ci=m∑i=1(A−B)ci=(A−B)m∑i=1ci=(A−B), |
thus H(ξ)∈Rm,n,σt,δ,υ(A,B).
Regarding the arithmetic means of the functions of the family Rm,n,σt,δ,υ(A,B) the next result holds:
Theorem 8. If Gi∈Rm,n,σt,δ,υ(A,B) are given by (4.10), then
G(ξ):=ξ+1k∞∑j=2(k∑i=1ai,jξj)∈Rm,n,σt,δ,υ(A,B). | (4.12) |
Where G is the arithmetic mean of Gi, i=1,2,3,…,k.
Proof. From the definition relation (4.12) we get
G(ξ)=1kk∑i=1fi(ξ)=1kk∑i=1(ξ+∞∑j=2ai,jξj)=ξ+∞∑j=2(1kk∑i=1ai,j)ξj, |
and to prove that G(ξ)∈Rm,n,σt,δ,υ(A,B), according to the Theorem 4.1 it is sufficient to prove that
∞∑j=2((1−B)ψmj+(A−1)ψnj)(σ+1σ−υ+1)(1kk∑i=1|ai,j|)≤(A−B). |
A simple computation shows that
∞∑j=2((1−B)ψmj+(A−1)ψnj)(σ+1σ−υ+1)(1kk∑i=1|ai,j|)=1kk∑i=1(∞∑j=2((1−B)ψmj+(A−1)ψnj)(σ+1σ−υ+1)|ai,j|)≤1kk∑i=1(A−B)=(A−B). |
Therefore G∈Rm,n,σt,δ,υ(A,B).
Theorem 9. If G∈Rm,n,σt,δ,υ(A,B), then G is a starlike functions of order ϑ (0≤ϑ<1), |ξ|<r∗1,
r∗1=infj≥2((1−ϑ)((1−B)ψmj+(A−1)ψnj)(σ+1)(j−ϑ)(σ−υ+1)(A−B))1j−1. |
The equality holds for G given in (4.4).
Proof. Let G∈Rm,n,σt,δ,υ(A,B). We see that G is a starlike functions of order ϑ, if
|ξG′(ξ)G(ξ)−1|<1−ϑ. |
By simple calculation, we deduce
∞∑j=2(j−ϑ1−ϑ)|aj||ξ|j−1<1. | (4.13) |
Since G∈Rm,n,σt,δ,υ(A,B), from (4.1) we get
∞∑j=2((1−B)ψmj+(A−1)ψnj)(σ+1)(σ−υ+1)(A−B)|aj|<1. | (4.14) |
The relation (4.13) will holds true if
∞∑j=2(j−ϑ1−ϑ)|aj||ξ|j−1<∞∑j=2((1−B)ψmj+(A−1)ψnj)(σ+1)(σ−υ+1)(A−B)|aj|, |
which implies that
|ξ|j−1<((1−ϑ)((1−B)ψmj+(A−1)ψnj)(σ+1)(j−ϑ)(σ−υ+1)(A−B)), |
or, equivalently
|ξ|<((1−ϑ)((1−B)ψmj+(A−1)ψnj)(σ+1)(j−ϑ)(σ−υ+1)(A−B))1j−1, |
which yields the starlikeness of the family.
Theorem 10. If G∈Rm,n,σt,δ,υ(A,B), then G is a close-to-convex function of order ϑ (0≤ϑ<1), |ξ|<r∗2,
r∗2=infj≥2((1−ϑ)(σ+1)((1−B)ψmj+(A−1)ψnj)j(σ−υ+1)(A−B))1j−1. |
Proof. Let G∈Rm,n,σt,δ,υ(A,B). If G is close-to-convex function of order ϑ, then we find that
|G′(ξ)−1|<1−ϑ, |
that is
∞∑j=2j1−ϑ|aj||ξ|j−1<1. | (4.15) |
Since G∈Rm,n,σt,δ,υ(A,B), by (4.1) we have
∞∑j=2(σ+1)((1−B)ψmj+(A−1)ψnj)(σ−υ+1)(A−B)|aj|<1. | (4.16) |
The relation (4.13) will holds true if
∞∑j=2j1−ϑ|aj||ξ|j−1<∞∑j=2(σ+1)((1−B)ψmj+(A−1)ψnj)(σ−υ+1)(A−B)|aj|, |
which implies that
|ξ|j−1<((1−ϑ)(σ+1)((1−B)ψmj+(A−1)ψnj)j(σ−υ+1)(A−B)), |
or, equivalently
|ξ|<((1−ϑ)(σ+1)((1−B)ψmj+(A−1)ψnj)j(σ−υ+1)(A−B))1j−1, |
which yields the desired result.
In this paper, we introduced a new class Rm,n,σt,δ,υ(A,B) of holomorphic functions defined in the open unit disk, which is connected to the combination of the Binomial series and the Babalola operator. We employed differential subordination involving Janowski-type functions to investigate these properties. Utilizing well-established results, such as Carathéodory's inequality for functions with real positive parts, as well as the Keogh-Merkes and Ma-Minda inequalities, we established upper bounds for the first two initial coefficients of the Taylor-Maclaurin power series expansion. Additionally, we derived an upper bound for the Fekete-Szegő functional for functions within this family.
We also extended our findings to include similar results for the first two coefficients and for the Fekete-Szegő inequality for functions G−1 when G∈Rm,n,σt,δ,υ(A,B). Furthermore, we determined coefficient estimates, distortion bounds, radius problems, and the radius of starlikeness and close-to-convexity for these newly defined functions.
Kholood M. Alsager: Conceptualization, validation, formal analysis, investigation, supervision; Sheza M. El-Deeb: Methodology, formal analysis, investigation; Ala Amourah: Methodology, validation, writing-original draft; Jongsuk Ro: Writing-original draft, writing-review. All authors have read and agreed to the published version of the manuscript.
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government(MSIT) (No. NRF-2022R1A2C2004874). This work was supported by the Korea Institute of Energy Technology Evaluation and Planning(KETEP) and the Ministry of Trade, Industry & Energy(MOTIE) of the Republic of Korea (No. 20214000000280).
The authors declare no conflict of interest.
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