Some applications of differential subordination for certain starlike functions

Lemma 1.1. (see [1], see also [16, Lemma 1.3, p. 28]) Let w be a nonconstant function meromorphic in D with w(0) = 0. If |w(z0)| = max{|w(z)| : |z| ≤|z0|} (z ∈D), then there exists a real number k (k ≥1) such that z0w′(z0) = kw(z0). In this paper, for analytic function p(z) in the unit disk D we find some conditions that imply p(z) ≺√1 + cz. Also, some interesting corollaries are obtained. 2. Main Results The first result is the following.

Theorem 2.1. Assume that p is an analytic function in D with p(0) = 1. Further assume that |A| ≤1, |B| < 1, 0 < c ≤1 and that γ satisfies the following inequality (2.1) γ ≥2(|A| + |B|) c(1 −|B|) (1 + c). If f satisfies the subordination 1 + γ zp′(z) p(z) ≺1 + Az 1 + Bz (z ∈D), then p(z) ≺ √ 1 + cz (z ∈D).

Corollary 2.1. Let |A| ≤1, |B| < 1, 0 < c ≤1 and let γ ≥2(|A| + |B|) c(1 −|B|) (1 + c). If f satisfies the following subordination 1 + γ  1 + zf ′′(z) f ′(z) −zf ′(z) f(z)  ≺1 + Az 1 + Bz (z ∈D), then f ∈S∗(qc). Putting c = 1 in Corollary 2.1, we have:

Corollary 2.2. Let |A| ≤1, B| < 1 and let γ ≥4(|A| + |B|) 1 −|B| . If f satisfies the following subordination 1 + γ  1 + zf ′′(z) f ′(z) −zf ′(z) f(z)  ≺1 + Az 1 + Bz (z ∈D), then f ∈SL∗.

Corollary 2.3. Let γ ≥4. If f satisfies the following inequality Re  1 + γ  1 + zf ′′(z) f ′(z) −zf ′(z) f(z)  > 0 for all z ∈D then f ∈SL∗. Putting p(z) = z p f ′(z)/f(z) in Theorem 2.1, we have:

Corollary 2.4. Let |A| ≤1, |B| < 1, 0 < c ≤1 and let γ ≥2(|A| + |B|) c(1 −|B|) (1 + c). If the function f satisfies the following subordination 1 + γ  1 + 1 2 zf ′′(z) f ′(z) −zf ′(z) f(z)  ≺1 + Az 1 + Bz (z ∈D),

Corollary 2.5. Assume that |A| ≤1, |B| < 1 and that γ ≥4(|A| + |B|) 1 −|B| . If 1 + γ  1 + 1 2 zf ′′(z) f ′(z) −zf ′(z) f(z)  ≺1 + Az 1 + Bz

Corollary 2.6. Assume that |A| ≤1, |B| < 1 and that γ ≥4(|A| + |B|) 1 −|B| . If 1 + γ 1 2 zf ′′(z) f ′(z)  ≺1 + Az 1 + Bz (z ∈D), then f is univalent in D by [13].

Corollary 2.7. Let |A| ≤1, |B| < 1, 0 < c ≤1 and let γ ≥2(|A| + |B|) c(1 −|B|) (1 + c). If f satisfies the following subordination 1 + γ zf ′(z) f(z) −1  ≺1 + Az 1 + Bz (z ∈D), then

Corollary 2.8. Let c ∈(0, 1] and let γ ≥2(1 + 1/c). If f satisfies the following inequality Re  1 + γ zf ′(z) f(z) −1  > 0 (z ∈D), then

Lemma 2.1. ([12]) Let q be univalent in the unit disk D and θ and φ be analytic in a domain U containing q(D) with φ(w) ̸= 0 when w ∈q(D). Set Q(z) = zq′(z)φ(q(z)), h(z) = θ(q(z)) + Q(z). Suppose that Q is starlike (univalent) in D, and Re zh′(z) Q(z)  = Re θ′(q(z)) φ(q(z)) + zQ′(z) Q(z)  > 0 (z ∈D). If p is analytic in D, with p(0) = q(0), p(D) ⊂U and

Lemma 2.2. (see [10], see also [11, p. 24]) Assume that Q is the set of analytic functions that are injective on D\E(f), where E(f) : {ω : ω ∈∂D and limz→ω f(z) = ∞}, and are such that f ′(ω) ̸= 0 for (ω ∈∂D\E(f). Let ψ ∈Q with ψ(0) = a and let ϕ(z) = a + amzm + · · · be analytic in D with ϕ(z) ̸≡a and m ∈N. If ϕ ̸≺ψ in D, then there exist points z0 = r0eiθ ∈D and ω0 ∈∂D\E(ψ), for which ϕ(|z| < r0) ⊂ψ(D), ϕ(z0) = ψ(ω0) and z0ϕ′(z0) = kω0ψ′(ω0), for some k ≥m.

Theorem 2.2. Let p be an analytic function on D and with p(0) = 1 and let c ∈(0, 1]. If the function p satisfies the subordination (2.9) 1 3p3(z) + zp′(z) ≺1 3 √ 1 + cz
3 + cz 2√1 + cz (z ∈D), then p also satisfies the subordination p(z) ≺ √

Corollary 2.9. Let c ∈(0, 1]. If a function f satisfies the subordination 1 3 zf ′(z) f(z) 3 +  1 + zf ′′(z) f ′(z) −zf ′(z) f(z)  zf ′(z) f(z)  ≺1

Theorem 2.3. Let k ≥1 and let 0 < c ≤1. If p is an analytic function in D with p(0) = 1 and it satisfies the condition (2.11) Re {p(z)(p(z) + zp′(z))} > 1 + c(1 + k/2), for all z ∈D then p(z) ≺ √ 1 + cz (z ∈D).

Corollary 2.10. Let 0 < c ≤1 and let k ≥1. If f satisfies the following inequality Re (zf ′(z) f(z) 2  2 + zf ′′(z) f ′(z) −zf ′(z) f(z) ) > 1 + c(1 + k/2), for all z ∈D, then f ∈S∗(qc). References [1] I.S. Jack, Functions starlike and convex of order α, J. London Math. Soc. 3 (1971), 469–474. [2] W. Janowski, Some extremal problems for certain families of analytic functions I, Ann. Polon. Math. 28 (1973), 297–326.