Stud. Univ. Babe¸s-Bolyai Math. 62(2017), No. 3, 377–394

Theorem 1.3. Let F, G ∈A be any convex univalent functions in ∆. If f ≺F and g ≺G, then f ∗g ≺F ∗G in ∆. Observe that, in Theorem 1.3, nothing is said about the normalization of F and G.

Theorem 2.1. If f ∈P(φ) ∩S, n ∈N then (Kδ c )nf(z) ≺(Kδ c )nφ(z), where Kδ c is Komatu integral operator.

Corollary 2.2. Let g′ ∈P(φ), α < 1. If we take φ(z) = 1−z(2α−1) 1−z , n = 1 and h1(z) = ∞ X n=1 2 n + 1zn, (z ∈∆). Then 1 z Z z 0 g(t)

Theorem 3.1. Let f ∈Rτ γ(A, B) [3]. Then f is in the class Rτ δ,γ,ρ,c if and only if ∞ X k=2 (1 + B)k{ρ + γ(k −1)} c + 1 c + k δ ak ≤|τ(A −B)|. (3.1)

Corollary 3.2. Let f(z) ∈Rτ δ,γ,ρ,c(A, B), then ak ≤ |τ(A −B)| (1 + B)k{ρ + γ(k −1)}  c+1 c+k δ ; k ≥2. 4. Fekete-Szeg¨o inequality We recall the following lemma to prove our results:

Lemma 4.1. [6] If p1(z) = 1+c1z +c2z2 +c3z3 +...(z ∈∆) is a function with positive real part, then for any complex number ε, |c3 −εc2 2| ≤2 max{1, |2ε −1|} and the result is sharp for the functions given by p1(z) = 1 + z2 1 −z2 or p1(z) = 1 + z 1 −z .

Theorem 4.2. Let φ(z) = 1 + B1z + B2z2 + B3z3 + ... (4.1) where φ(z) ∈A with φ′(0) > 0. If f(z) given by (1.1) belongs to Rτ δ,γ,ρ,c(φ) (γ, ρ ∈[0, 1); τ ∈C \ {0}; δ > 0; c > −1), z ∈∆, then for any complex number ν |a3 −νa2 2| ≤ |τ| B1 3(ρ + 2γ) c + 3 c + 1 δ max

Corollary 4.3. If f(z) given by (1.1) belongs to Rτ δ,γ,ρ,c(A, B), then |a3 −νa2 2| ≤|τ|(A −B) 3(ρ + 2γ) c + 3 c + 1 δ max  1, B −3ν(A −B)τ(ρ + 2γ)(c + 2)2δ (ρ + γ)2(c + 3)δ(c + 1)δ

Lemma 5.2. Let α > 0, β and η be real. Then, for k > max{0, β −η} −1, Iα,β,η 0,z zk = Γ(k + 1)Γ(k −β + η + 1) Γ(k −β + 1)Γ(k + α + η + 1)zk−β. (5.1)

Theorem 5.3. Let f ∈Rτ δ,γ,ρ,c(A, B), then |Iα,β,η 0,z f(z)| ≤ Γ(2 −β + η)|z|1−β Γ(2 −β)Γ(2 + α + η)  1 + (2 −β + η)|τ(A −B)||z| (2 −β)(2 + α + η)(1 + B)(ρ + γ)  c+1 c+2 δ

Theorem 6.1. Let f1(z) = z and fk(z) = z + |τ(A −B)| k(1 + B){ρ + γ(k −1)}  c+1 c+k δ zk. Then f ∈Rτ δ,γ,ρ,c(A, B) if and only if f(z) can be expressed in the form f(z) = λ1f1(z) + ∞ X k=2 λkfk(z)

Corollary 6.2. The extreme points of the class Rτ δ,γ,ρ,c(A, B) are the functions f1(z) and fk(z), (k ≥2). 7. Radii of starlikeness and convexity

Theorem 7.1. Let f ∈Rτ δ,γ,ρ,c(A, B). Then f(z) is starlike of order α (0 ≤α < 1) in |z| < r1 where r1 = inf k " (1 −α)k(1 + B){ρ + γ(k −1)}( c+1 c+k)δ (k −α)|τ(A −B)| # 1 k−1 .

Theorem 7.2. Let f ∈Rτ δ,γ,ρ,c(A, B). Then f is convex of order α (0 ≤α < 1) in |z| < r2 where r2 = inf k " (1 −α)(1 + B){ρ + γ(k −1)}( c+1 c+k)δ (k −α)|τ(A −B)| # 1 k−1 . 8. Neighborhood results Definition 8.1. For f ∈A of the form (1.1) and µ ≥0. We define a (n, µ)−neigh-

Theorem 8.2. If µ = |τ(A −B)| (1 + B)(ρ + nγ)  c+1 c+n+1 δ then, Rτ δ,γ,ρ,c(A, B) ⊂Nn,µ(e)

Theorem 8.3. If g ∈Rτ δ,γ,ρ,c(A, B) and α = 1 − µ(1 + B)(ρ + nγ)  c+1 c+n+1 δ n(1 + B)(ρ + nγ)  c+1 c+n+1 δ −|τ(A −B)| .

Lemma 9.1. [7] If f(z) and g(z) are analytic in ∆with f(z) ≺g(z), then Z 2π 0 |f(reiθ)|ηdθ ≤ Z 2π 0 |g(reiθ)|ηdθ, (9.2) η ≥0, z = reiθ and 0 < r < 1. Application of Lemma (9.1) to function of f in the class Rτ δ,γ,ρ,c(A, B), gives the following result.

Theorem 9.2. Let η > 0. If f ∈Rτ δ,γ,ρ,c(A, B) is given by (1.1) and f2(z) is defined by f2(z) = z + |τ(A −B)| 2(1 + B)(ρ + γ)  c+1 c+2 δ z2 (9.3) = z + 1 φA