On some $n$-starlike integral operators

Lemma 1 ([2]). Let u = u1 +u2i, v = v1 +v2i and ψ(u, v) a complex-valued function satisfying: (a) ψ(u, v) is continuous in a domain Ωof C2, (b) (1, 0) ∈Ωand Reψ(1, 0) > 0, (c) Reψ(λ + (1 −λ)u2i, v1) ≤λ when (λ + (1 −λ)u2i, v1) ∈Ωand 2v1 ≤ −(1 −λ)(1 + u2 2) for real number 0 ≤λ < 1. If p ∈P such that (p(z), zp′(z)) ∈Ωand Re ψ(p(z), zp′(z)) > λ for z ∈E, then Re p(z) > λ in E.

Lemma 2 ([6]). Let η and µ be complex constants and h(z) a convex uni- valent function in E satisfying h(0) = 1, and Re(ηh(z) + µ) > 0. Suppose p ∈P satisfies the differential subordination: (3) p(z) + zp′(z) ηp(z) + µ ≺h(z), z ∈E. If the differential equation: (4) q(z) + zq′(z) ηq(z) + µ = h(z), q(0) = 1 has univalent solution q(z) in E, then p(z) ≺q(z) ≺h(z) and q(z) is the

Lemma 3 ([3]). Let f ∈A and ζ > 0 be real. (i) If for z ∈E, Dn+1f(z)ζ/Dnf(z)ζ is independent of n, then (5) Dn+1f(z)ζ Dnf(z)ζ = ζ Dn+1f(z) Dnf(z) . (ii) The equality (5) also holds if Dn+1f(z)/Dnf(z) is independent of n, z ∈E.

Theorem 1. Let α ≥0. Suppose for α > 0, the real number λ is defined such that 0 ≤αλ < 1. If f ∈Sn(λ), then J j(f) ∈Sn(α β λ), j = 1, 2.

Theorem 2. Let α ≥0. Suppose for α > 0, the real number λ is defined such that 0 ≤αλ < 1. If Re Dn+1J j m−1(f)β DnJ j m−1(f)β > αλ, then Dn+1J j m(f)β DnJ j m(f)β ≺q(z) where (9) q(z) = z1+µ(1 −z)−2(1−αλ)

Corollary 1. The classes Sn is closed under J (f). This result is more genral than the result of Miller et-al [13] (Theorem 2, pg. 162) in which case δ = γ. A major breakthrough with our method is the fact that the integral (2) passes through, preserving all the goemetry (starlikeness and convexity for example) of f without having to drop any member of the sets on which the parameters α ≥0 and β > 0 were defined, which was not the case in many earlier works. This will become more evident in the f

Corollary 2. If f ∈Sn, then J (f) =  zβ−1 Z z 0 f(t) t α dt  1 β = z + · · · also belongs to Sn. (ii) If α + δ = 1, β = 1 and γ = 0, we have

Corollary 3. If f ∈Sn, then J (f) = Z z 0 f(t) t α dt = z + · · · also belongs to Sn. (iii) If α + δ = β + γ = α + η + γ, we have

Corollary 4. If f ∈Sn, then J (f) = α + γ + η zγ Z z 0 tγ+ηf(t)αdt  1 β = z + · · · also belongs to Sn. From the above corollary, we can obtain various sequences of starlike and convex functions (and more generally, of Sn functions): For example if γ + η = 1, α = 1 and η = k = 0, 1, 2 · · · ; and if γ = 0, α = 1 and η = k = 0, 1, 2 · · · we obtain, respectively, the following sequences of Sn

Corollary 5. Let f ∈Sn. If δ is a real number and γ = 0, then J (f) =  β Z z 0 tδ−1f(t)αdt  1 β = z + · · · belongs to Sn( 1 2β).

Corollary 6. Let f ∈Sn. If δ is a real number and γ = 1, then J (f) = β + 1 z Z z 0 tδ−1f(t)αdt  1 β = z + · · · belongs to Sn( 3−4 ln 2 2β(2 ln 2−1)). On the final note, if we take β = 1 in Corollaries 5 and 6 we then have the following special cases

Corollary 7. Let f ∈Sn. If δ is a real number and γ = 0, then J (f) = Z z 0 tδ−1f(t)αdt = z + · · · belongs to Sn(1 2).

Corollary 8. Let f ∈Sn. If δ is a real number and γ = 1, then J (f) = 2 z Z z 0 tδ−1f(t)αdt = z + · · · belongs to Sn( 3−4 ln 2 2(2 ln 2−1)). Miller et-al [13] proved that if f ∈S∗, then J (f) ∈S∗( √ 17−3 4 ) and also if f ∈K, then J (f) ∈K(