On subclasses of analytic functions based on a quantum symmetric conformable differential operator with application

Lemma 3.1 If K is smooth (analytic) in ∪, ⋎∈C( 1+ξ 1–ξ ) is convex and g ∈S∗( 1+ξ 1–ξ ) is starlike then ⋎∗(Kg) ⋎∗g (∪) ⊆co  K(∪)  , (3.8) where co(K(∪)) is the closed convex hull of K(∪).

Lemma 3.2 For analytic functions h,ℏ∈∪, the subordination h ≺ℏimplies that  2π 0 h(ξ) p dθ ≤  2π 0 ℏ(ξ) p dθ, (3.9) where ξ = reiθ, 0 < r < 1, and p is a positive number. Some of the few studies in q-calculus are realized by comparison between two differ- ent values of calculus. Class Ξ ν,k q1,q2(σ) shows the relation between the q1- and q2-calculus depending on the operator (2.7).

Theorem 4.1 Let ⋎∈ and let the function g := Ψ k q2 ∗⋎∈S∗( 1+ξ 1–ξ ), ξ ∈∪. If ⋎∈Ξ 0,k q1,q2(σ), q1 ̸= q2 and the function Φk(ξ) ∈C( 1+ξ 1–ξ ) then ⋎∈Ξ ν,k q1,q2(σ), σ(0) = 1.

Corollary 4.2 Let ⋎be a function from  and σ(ξ) be a convex univalent function in ∪ such that σ(0) = 1. Then Ξ 0,k q1,q2(σ) ⊂Ξ ν,k q1,q2(σ). In general, we have the following result:

Theorem 4.3 Let ⋎∈ and let the function G := Ψ k q2 ∗Φk ν ∗⋎∈S∗( 1+ξ 1–ξ ), ξ ∈∪. If ρ1 := Ψq1 ∗Φν ≺r ρ2 := Ψq2 ∗Φν for some r < 1 and the function ρ2 ∈C( 1+ξ 1–ξ ) then Ξ ν,k q1,q2(σ) ⊂Ξ ν,k+1 q1,q2 (σ). (4.6)

Theorem 4.4 Let ⋎∈ and let the function H := Ψ k q2 ∗Φk ν1 ∗⋎∈S∗( 1+ξ 1–ξ ), ξ ∈∪. If Φk ν1 ≺r Φk ν2 for some r < 1 then Ξ ν1,k q1,q2(σ) ⊂Ξ ν2,k q1,q2(σ). (4.12)

Theorem 5.1 Consider the operator [Sk ν ]q ⋎(ξ), ⋎∈. If the coefficients of ⋎satisfy the inequality | ⋎n | ≤( 1 nν )k, ν ∈(0,1) then  2π 0  [Sk ν ]q ⋎(ξ) ξ  p dθ ≤  2π 0

Theorem 5.2 Consider the operator [Sk ν ]q ⋎(ξ), ⋎∈. If the coefficients of ⋎satisfy the inequality | ⋎n | ≤( 1 nν )k, ν ∈(0,1) then  2π 0 Sk ν

Theorem 5.3 Consider the operator Sκ,k q ψ(z), ψ ∈Λ. If the coefficients of ψ satisfy the inequality |ϑn| ≤( 1 nκ )k, κ ∈(0,∞) then there is a probability measure μ on (∂U)2, for all δ > 1.

Theorem 6.1 If the function ⋎∈Vq(ψ) is given by (2.1), then | ⋎2 | ≤ 1 [Q2]kq , | ⋎3 | ≤ 1 [Q3]kq . (6.6)

Lemma 6.2 Consider functions f1,f2,f3 : ∪→C such that ℜ(f1) ≥a ≥0. If f ∈H[1,n] (the set of analytic functions having the expansion f (ξ) = 1 + ϕ1ξ + ···) and ℜ  aξ 2f ′′(ξ) + f1(ξ)ξf ′(ξ) + f2(ξ)f (ξ) + f3(ξ)  > 0, a ≥0,ξ ∈∪, then ℜ(f (ξ)) > 0.

Lemma 6.3 Let ♭be convex in ∪and suppose f1,f2,f3 : ∪→C are analytic functions such that ℜ(f1) ≥a ≥0. If g ∈H[0,m] (the set of analytic functions with the expansion g(ξ) = g1ξ m + ···), m ≥1 and aξ 2g′′(ξ) + f1(ξ)ξg′(ξ) + f2(ξ)g(ξ) + f3(ξ) ≺♭(ξ), a ≥0,ξ ∈∪, then g(ξ) ≺♭(ξ).

Lemma 6.4 Let a,b,c ∈R be such that a ≥0, b ≥–a, c ≥–b. If q ∈H[0,1], where q(ξ) = q1ξ + ··· and aξ 2q′′(ξ) + bξq′(ξ) + cq(ξ) ≺ξ, a ≥0,ξ ∈∪, then q(ξ) ≺ ξ b+c, which is the best dominant.

Theorem 6.5 Let ⋎∈Vq(ξ) and F(ξ) = ξ([Skν ]q⋎(ξ))′ ([Skν ]q⋎(ξ)) . If ℜ(ξF(ξ)) > –1, ξ ∈∪, then [Sk ν ]q⋎∈ S∗(starlike with respect to the origin).

Theorem 6.6 Let ⋎∈ and F(ξ) = ξ([Skν ]q⋎(ξ))′ [Skν ]q⋎(ξ) . If ξF′(ξ) F(ξ)

Lemma 6.3, we have P(ξ) = 1 + ([Sk ν ]q ⋎(ξ))′′ ([Skν ]q ⋎(ξ))′ – ξ([Sk ν ]q ⋎(ξ))′ [Skν ]q ⋎(ξ) ≺ψ(ξ). Consequently, we get ⋎∈Vq(ψ). □

Theorem 6.7 Let ⋎∈ and F(ξ) = ξ([Skν ]q⋎(ξ))′ [Skν ]q⋎(ξ) . If ξF′(ξ) F(ξ)

Lemma 6.4, we have P(ξ) = 1 + ([Sk ν ]q ⋎(ξ))′′ ([Skν ]q ⋎(ξ))′ – ξ([Sk ν ]q ⋎(ξ))′ [Skν ]q ⋎(ξ) ≺ξ. Consequently, we obtain ⋎∈Vq(ξ). □ 7 Conclusion In this paper, we presented different types of integral inequalities based on q-calculus and conformable differential operator. These inequalities described the relations between the quantum conformable differential operators for different orders. Acknowledgements The authors would like to thank the Associate Editor for her/his advice in preparing the a