Sheza M. El-Deeb; Georgia I. Oros

Proposition 1.1 ([14]). (i) If (B, FB) = (U, FU), then we have B = U, where B = sup(B, FB) and U = sup(U, FU). (ii) If (B, FB) ⊆(U, FU), then we have B ⊆U, where B = sup(B, FB) and U = sup(U, FU). Let f, g ∈H(D). We denote (1.2) f(D) = {f(z): 0 < Ff(D)f(z) ⩽1, z ∈D} = sup(f(D), Ff(D)) and (1.3) g(D) = {g(z): 0 < Fg(D)g(z) ⩽1, z ∈D} = sup(g(D), Fg(D)). Definition 1.3 ([14]). Let z0 ∈D and f, g ∈H(D). The function f is said to be fuzzy subordinate to g, written f ≺F g or f(z) ≺F g(z), when followin

Proposition 1.2 ([14]). Assume that z0 ∈D and f, g ∈H(D). If f(z) ≺F g(z), z ∈D, then (i) f(z0) = g(z0), (ii) f(D) ⊆g(D), Ff(D)f(z) ⩽Fg(D)g(z), z ∈D, where f(D) and g(D) are defined by (1.2) and (1.3), respectively. 398

Lemma 2.1 ([13]). Let ψ ∈A and G(z) = z−1 R z 0 ψ(t) dt, z ∈U. If ℜ(1 + zψ′′(z)/ψ′(z)) > −1 2, z ∈U, then G ∈K.

Lemma 2.2 ([15], Theorem 2.6). Assume that h is a convex function with h(0) = a and ν ∈C∗= C \ {0} with ℜ(ν) ⩾0. If p ∈H[a, n] with p(0) = a, Φ: C2 × U →C, Φ(p(z), zp′(z); z) = p(z) + ν−1zp′(z) is analytic function in U and FΦ(C2×U)  p(z) + 1 ν zp′(z)  ⩽Fh(U)h(z), i.e. p(z) + 1 ν zp′(z) ≺F h(z), z ∈U, then Fp(U)p(z) ⩽Fq(U)q(z) ⩽Fh(U)h(z), i.e. p(z) ≺F q(z), z ∈U, where

Lemma 2.3 ([15], Theorem 2.7). Let g be a convex function in U and let ψ(z) = g(z) + nγzg′(z), where z ∈U, n ∈N and γ > 0. If p(z) = g(0) + pnzn + pn+1zn+1 + . . . is holomorphic in U and Fp(U)(p(z) + γzp′(z)) ⩽Fψ(U)ψ(z), i.e. p(z) + γzp′(z) ≺F ψ(z), z ∈U, then Fp(U)(p(z)) ⩽Fg(U)g(z), i.e. p(z) ≺F g(z), z ∈U; this result is sharp. For the general theory of fuzzy differential subordination and its applications, we refer the reader to [2], [12]–[14], [16]. The objective of the present section is to

Theorem 3.1. Let k be a convex function in U and suppose that h(z) = k(z) + zk′(z)/(λ + 2). If f ∈MF m,γ(n, α, η) and (3.1) G(z) = Iλf(z) = λ + 2 zλ+1 Z z 0 tλf(t) dt, then (3.2) F(In,α m,γf)′(U)(In,α m,γf(z))′ ⩽Fh(U)h(z), i.e. (In,α

Theorem 3.2. Consider h(z) = (1 + (2η −1)z)/(1 + z), η ∈[0, 1), λ > 0 and Iλ is given by (3.1). Then (3.8) Iλ(MF m,γ(n, α, η)) ⊂MF m,γ(n, α, ζ), where (3.9) ζ = 2η −1 + (λ + 2)(2 −2η) Z 1 0 tλ+2 t + 1 dt. P r o o f. The function h is convex and using the same technique as in the proof

Theorem 3.3. Assume that k is a convex function in U, k(0) = 1, and let h(z) = k(z) + zk′(z). If f ∈A and satisfies the fuzzy differential subordination (3.11) F(In,α m,γf)′(U)(In,α m,γf(z))′ ⩽Fh(U)h(z), i.e. (In,α m,γf(z))′ ≺F h(z), then (3.12) FIn,α m,γf(U) In,α m,γf(z) z

Theorem 3.4. For h ∈H(U), h(0) = 1, which satisfies ℜ(1+zh′′(z)/h′(z)) > −1 2, if f ∈A and verifies the fuzzy differential subordination (3.13) F(In,α m,γf)′(U)(In,α m,γf(z))′ ⩽Fh(U)h(z), i.e. (In,α m,γf(z))′ ≺F h(z), then (3.14) FIn,α m,γf(U) In,α m,γf(z)

Corollary 3.1. Let h = (1 + (2β −1)z)/(1 + z) be a convex function in U with h(0) = 1, 0 ⩽β < 1. If f ∈A and verifies the fuzzy differential subordination F(In,α m,γf)′(U)(In,α m,γf(z))′ ⩽Fh(U)h(z), i.e. (In,α m,γf(z))′ ≺F h(z), then k(z) = 2β −1 + 2(1 −β) z ln(1 + z), and the function k is convex and it is the fuzzy best dominant. 4. Conclusion All the above results give us information about fuzzy differential subordinations for a linear operator In,α