Differential Subordinations Involving Generalized Bessel Functions

Lemma 1.1. [7, p. 28] Let ψ ∈Ψn [Ω, q] with q(0) = a. If the analytic function p ∈H[a, n] satisfies the following inclusion relationship ψ p(z), zp′(z), z2p′′(z); z
∈Ω, for all z ∈D, then p ≺q.

Lemma 1.2. [8, p. 818] Let ψ ∈Ψ′ n [Ω, q] with q(0) = a. If p ∈Q(a) and ψ p(z), zp′(z), z2p′′(z); z
is univalent in D, then Ω⊂  ψ p(z), zp′(z), z2p′′(z); z
: z ∈D

Theorem 2.1. Let φ ∈ΦH[Ω, q]. If f ∈A satisfies the following inclusion relationship (2.1)  φ Bc κ+1f(z), Bc κf(z), Bc κ−1f(z); z
: z ∈D

Corollary 2.2. Let φ ∈ΦH[h, q]. If f ∈A satisfies (2.7) φ Bc κ+1f(z), Bc κf(z), Bc κ−1f(z); z
≺h(z), then Bc κ+1f(z) ≺q(z) (z ∈D). Our next result is an extension of Theorem 2.1 to the case when the behavior of q on ∂D is not known.

Corollary 2.3. Let Ω⊂C and let q be univalent in D with q(0) = 0. Let φ ∈ΦH[Ω, qρ] for some ρ ∈(0, 1), where qρ(z) = q(ρz). If f ∈A satisfies the following inclusion relationship φ Bc κ+1f(z), Bc κf(z), Bc κ−1f(z); z
∈Ω, then Bc κ+1f(z) ≺q(z) (z ∈D).

Theorem 2.4. Let h and q be univalent in D with q(0) = 0 and set qρ(z) = q(ρz) and hρ(z) = h(ρz). Let φ : C3 × D →C satisfy one of the following conditions (1) φ ∈ΦH[Ω, qρ] for some ρ ∈(0, 1), or (2) there exists ρ0 ∈(0, 1) such that φ ∈ΦH[hρ, qρ] for all ρ ∈(ρ0, 1). If f ∈A satisfies (2.7), then Bc κ+1f(z) ≺q(z) (z ∈D).

Theorem 2.5. Let h be univalent in D and φ : C3 × D →C. Suppose that the differential equation (2.8) φ  q(z), zq′(z) + (κ −1)q(z) κ , z2q′′(z) + 2(κ −1)zq′(z) + (κ −1)(κ −2)q(z) κ(κ −1) ; z  = h(z) has a solution q with q(0) = 0 and satisfies one of the following conditions (1) q ∈Q0 and φ ∈ΦH[h, q] (2) q is univalent in D and φ ∈ΦH[h, qρ] for some ρ ∈(0, 1) (3) q is univalent in D and there exists ρ0 ∈(0, 1) such that φ ∈ΦH[hρ, qρ] for all ρ ∈(ρ0, 1).

Corollary 2.2 and Theorem 2.4. Since q satisfies (2.8), it is also a solution of (2.7) and therefore q will be dominated by all dominants. Hence q is the best dominant. □

Corollary 2.6. Let φ ∈ΦH[Ω, M]. If f ∈A satisfies the following inclusion relationship φ Bc κ+1f(z), Bc κf(z), Bc κ−1f(z); z
∈Ω, then Bc κ+1f(z) ≺Mz (z ∈D). In the special case when Ω= {w : |w| < M} = q(D), the class ΦH[Ω, M] is simply denoted by ΦH[M].

Corollary 2.6 can now be written in the following form.

Corollary 2.7. Let φ ∈ΦH[M]. If f ∈A satisfies the following inequality φ Bc κ+1f(z), Bc κf(z), Bc κ−1f(z); z
 < M, then Bc κ+1f(z) < M.

Corollary 2.8. If M > 0, Re(κ) ≥1−k 2 , where k ≥1, and f ∈A satisfies the following inequality |Bc κf(z)| < M, then Bc κ+1f(z) < M.

Corollary 2.8 we have (2.10) |ϕκ,c(z)| < M ⇒|ϕκ+1,c(z)| < M. This result is the generalization of a result given by Prajapat [14]. Also observe that ϕ 3 2 ,1,1(z) = 3 sin √z √z −3 cos √z, ϕ 1 2 ,1,1 = √z sin √z and ϕ−1 2 ,1,1 = z cos √z where ϕp,b,c(z) is given by (1.8). Thus we can obtain some trigonometric inequalities for special cases of parameters p, b and c. For example from (2.10) for all z ∈D and M > 0 we have z cos √z < M ⇒

Corollary 2.9. Let κ ∈C \ {0} and M > 0. If f ∈A satisfies the following inequality Bc κf(z) −Bc κ+1f(z) < M |κ|, then Bc κ+1f(z) < M.

Corollary 2.9, respectively, we have

Theorem 2.5 shows that the result is sharp. The differential equation zq′(z) = Mz has a univalent solution q(z) = Mz. It follows from Theorem 2.5 that q(z) = Mz is the best dominant. Definition 2.3. Let Ωbe a set in C and q ∈Q1 ∩H1 . The class of admissible functions ΦH,1[Ω, q] consists of those functions φ : C3 × D →C that satisfy the admissibility condition φ(u, v, w; z) /∈Ω whenever u = q(ζ), v = 1 κ −1 kζq′(ζ) q(ζ) + κq(ζ) −1  (κ ∈C, κ ̸= 0, 1, 2, q(ζ) ̸= 0),

Theorem 2.10. Let φ ∈ΦH,1[Ω, q]. If f ∈A satisfies the following inclusion relationship (2.11)  φ  Bc κf(z) Bc κ+1f(z), Bc κ−1f(z) Bcκf(z) , Bc κ−2f(z) Bc κ−1f(z); z  : z ∈D

Corollary 2.11. Let φ ∈ΦH,1[h, q]. If f ∈A satisfies (2.17) φ  Bc κf(z) Bc κ+1f(z), Bc κ−1f(z) Bcκf(z) , Bc κ−2f(z) Bc κ−1f(z); z  ≺h(z), then

Corollary 2.12. Let φ ∈ΦH,1[Ω, M]. If f ∈A satisfies the following inclusion relationship φ  Bc κf(z) Bc κ+1f(z), Bc κ−1f(z) Bcκf(z) , Bc κ−2f(z) Bc κ−1f(z); z  ∈Ω, then Bc

Corollary 2.13. Let φ ∈ΦH,1[M]. If f ∈A satisfies the following inequality φ  Bc κf(z) Bc κ+1f(z), Bc κ−1f(z) Bcκf(z) , Bc κ−2f(z) Bc κ−1f(z); z  −1 < M, then for all z ∈D we have

Corollary 2.14. Let κ ∈C (κ ̸= 1) and M > 0. If f ∈A satisfies the following inequality

Theorem 2.15. Let φ ∈ΦH,2[Ω, q]. If f ∈A satisfies the following inclusion relationship (2.19)  φ Bc κ+1f(z) z , Bc κf(z) z , Bc κ−1f(z) z ; z 

Corollary 2.16. Let φ ∈ΦH,2[h, q]. If f ∈A satisfies (2.26) φ Bc κ+1f(z) z , Bc κf(z) z , Bc κ−1f(z) z ; z  ≺h(z),

Corollary 2.17. Let φ ∈ΦH,2[Ω, M]. If f ∈A satisfies the following inclusion relationship φ Bc κ+1f(z) z , Bc κf(z) z , Bc κ−1f(z) z ; z  ∈Ω, then

Corollary 2.18. Let φ ∈ΦH,2[M]. If f ∈A satisfies the following inequality φ Bc κ+1f(z) z , Bc κf(z) z , Bc κ−1f(z) z ; z  −1 < M,

Corollary 2.19. Let κ ̸= 0 and M > 0. If f ∈A satisfies the following inequality

Corollary 2.20. Let Re(κ) ≥−k 2, κ ̸= 0, k ≥1 and M > 0. If f ∈A satisfy the following inequality

Theorem 3.1. Let φ ∈Φ′ H[Ω, q]. If f ∈A, Bc κ+1f ∈Q0 and φ Bc κ+1f(z), Bc κf(z), Bc κ−1f(z); z
is univalent in D, then (3.1) Ω⊂  φ Bc

Theorem 3.1.

Corollary 3.2. Let q ∈H0, h be analytic in D and φ ∈Φ′ H[h, q]. If f ∈A, Bc κ+1f ∈Q0 and φ Bc κ+1f(z), Bc κf(z), Bc κ−1f(z); z
is univalent in D, then (3.2) h(z) ≺φ Bc κ+1f(z), Bc κf(z), Bc

Theorem 3.1 and Corollary 3.2 can only be used to obtain subordinants of differential superordination of the form (3.1) or (3.2). The following theorem proves the existence of the best subordinant of (3.2) for an appropriate φ.

Theorem 3.3. Let h be univalent in D and φ : C3 × D →C. Suppose that the differential equation φ  q(z), zq′(z) + (κ −1)q(z) κ , z2q′′(z) + 2(κ −1)zq′(z) + (κ −1)(κ −2)q(z) κ(κ −1) ; z  = h(z) has a solution q ∈Q0. If φ ∈Φ′ H[h, q], f ∈A, Bc κ+1f ∈Q0 and φ Bc

Corollary 3.4. Let h1 and q1 be analytic functions in D, h2 be a univalent function in D, q2 ∈Q0 with q1(0) = q2(0) = 1 and φ ∈ΦH[h2, q2] ∩Φ′ H[h1, q1]. If f ∈A, Bc κ+1f ∈Q0 ∩H0 and φ Bc κ+1f(z), Bc κf(z), Bc κ−1f(z); z
is univalent in D, then h1(z) ≺φ Bc κ+1f(z), Bc κf(z), Bc

Theorem 3.5. Let φ ∈Φ′ H,1[Ω, q]. If f ∈A, Bc κf/Bc κ+1f ∈Q1 and φ  Bc κf(z) Bc κ+1f(z), Bc κ−1f(z) Bcκf(z) , Bc κ−2f(z) Bc κ−1f(z); z 

Theorem 3.6. Let q ∈H1, h be analytic in D and φ ∈Φ′ H,1[h, q]. If f ∈A, Bc κf/Bc κ+1f ∈Q1 and φ  Bc κf(z) Bc κ+1f(z), Bc κ−1f(z) Bcκf(z) , Bc κ−2f(z) Bc κ−1f(z); z 

Corollary 3.7. Let h1 and q1 be analytic functions in D, h2 be a univalent function in D, q2 ∈Q1 with q1(0) = q2(0) = 1 and φ ∈ΦH,1[h2, q2] ∩Φ′ H,1[h1, q1]. If f ∈A, Bc κf/Bc κ+1f ∈Q1 ∩H1 and φ  Bc κf(z) Bc κ+1f(z), Bc κ−1f(z) Bcκf(z) , Bc κ−2f(z) Bc κ−1f(z); z

Theorem 3.8. Let φ ∈Φ′ H,2[Ω, q]. If f ∈A, Bc κ+1f(z) z ∈Q1 and φ Bc κ+1f(z) z , Bc κf(z) z , Bc κ−1f(z)

Corollary 3.9. Let q ∈H1, h be analytic in D and φ ∈Φ′ H,2[h, q]. If f ∈A, Bc κ+1f(z) z ∈Q1 and φ Bc κ+1f(z) z , Bc κf(z) z , Bc κ−1f(z)

Corollary 3.10. Let h1 and q1 be analytic functions in D, h2 be a univalent function in D, q2 ∈Q1 with q1(0) = q2(0) = 1 and φ ∈ΦH,2[h2, q2] ∩Φ′ H,2[h1, q1]. If f ∈A, Bc κ+1f(z) z ∈Q1 ∩H1 and φ Bc κ+1f(z) z , Bc κf(z) z , Bc