On the class of functions strongly starlike of order α with respect to a point

Lemma 2.4. If 0 ≤k1 ≤k2 ≤2, w ∈C and Ω∈Z(k2; w), then Ω∈Z(k1; w). Since Z(k; w) ⊂Z(w) for all k ∈(0, 2], every domain in Z(k; w) is simply connected.

Lemma 2.5. If Ω∈Z(k; w) for k ∈(0, 2] and w ∈Ω, then Ek(w, ω)\{ω} ⊂Ωfor every ω ∈∂Ω. P r o o f. Fix ω ∈∂Ω. By Lemma 2.4, Ωis starlike w.r.t. w so [w, ω) ⊂Ω. Take the sequence wn = w + (1 −1/n)(ω −w), n ≥2, in [w, ω). It is clear that limn→∞wn = ω. Since wn ∈Ωit follows that Ek(w, wn) ⊂Ωfor all n ≥2. Therefore (2.1) ∞ [ n=2 Ek(w, wn) ⊂Ω. Notice also that (2.2) Ek(w, wn) ⊂Ek(w, wn+1) for n ≥2.

Theorem 2.7. A regular and univalent function f is in Sg(k; ξ, w), where k ∈[0, 2], ξ ∈U and w ∈C, if and only if for every ̺ > 0 the domain f(B(ξ, ̺)) is in Z(k; w), where w = f(ξ). P r o o f. Suppose first that f ∈Sg(k; ξ, w), where k ∈[0, 2], ξ ∈U and w = f(ξ). Hence Ω= f(U) ∈Z(k; w). Fix ̺ > 0 and set Ω(ξ, ̺) = f(B(ξ, ̺)). We will show that Ek(w, ω) ⊂Ω(ξ, ̺) for all ω ∈Ω(ξ, ̺). Since Ωis k-starlike domain w.r.t. w, we see that w + (ω −w)v ∈Ωfor all ω ∈Ωand v ∈Ek. Thus the function (2.5) g(z)

Theorem 3.1. If f ∈Sg(k; ξ, w) for k ∈[0, 2), ξ ∈U and w ∈C, then (3.1) arg (1 −ξz)(z −ξ)f ′(z) f(z) −w  < απ 2 , z ∈U, where α = (2/π) arccos(k/2). Conversely, let α ∈(0, 1], ξ ∈U and w ∈C. If (3.1) is satisfied for a function f regular in U, then f ∈Sg(k; ξ, w) for k = 2 cos(απ/2). P r o o f. For f regular in U and ξ ∈U we set Ω= f(U), Ω(ξ, ̺) = f(B(ξ, ̺)) and C(̺) = ∂B(ξ, ̺) for ̺ > 0. 1. We first consider the case k = 0. (i) Assume that f ∈Sg(ξ, w), where ξ ∈U and w ∈C. Thus w = f(ξ)

Theorem 3.1 gives an equivalence between k-starlikeness with respect to a fixed point w ∈C, a property which defines the class Sg(k; ξ, w), and an analytic condition (4.1) which describes the class S∗(α; ξ), where α = (2/π) arccos(k/2). For ξ = 0 and w = f(ξ) = 0 we get the results of Ma and Minda [3]. Then the inequality (4.1) reduces to (1.1) and with the normalization f ′(0) = 1 defines the class S∗(α) of strongly starlike func- tions, which coincides with the subclass of Sg(k; 0, 0), k = 2 cos(

Definition 2.2. Fix k ∈[0, 2]. A domain Ωin the plane is called k- starlike with respect to the point w ∈Ωprovided that Ek(w, ω) ⊂Ωfor every ω ∈Ω. The set of all k-starlike domains w.r.t. w ∈C will be denoted by Z(k; w). For simplicity of notation we denote the set Z(0; w) by Z(w) and the set Z(0; 0) of all domains starlike w.r.t. the origin by Z.

Definition 2.6. Fix k∈[0, 2]. A function f ∈A(ξ, w), where ξ ∈U and w ∈C, univalent in U will be called k-starlike w.r.t. w if the domain f(U) is k-starlike w.r.t. w, i.e. f(U) ∈Z(k; w). The set of all functions f ∈A(ξ, w), w = f(ξ), which are k-starlike w.r.t. w will be denoted by Sg(k; ξ, w). We write Sg(ξ, w) for Sg(0; ξ, w). If ξ = 0 and w = f(ξ) = 0, then k-starlike functions w.r.t. the origin will be called k-starlike (see [3]). For k = 0, ξ = 0 and w = f(ξ) = 0 we obtain the well known cl

Definition 4.1. For each α ∈(0, 1] and ξ ∈U we denote by S∗(α; ξ) the class of all functions f regular in U satisfying the condition (4.1) arg (1 −ξz)(z −ξ)f ′(z) f(z) −f(ξ)  < απ 2 , z ∈U. From Theorem 3.1 it follows that every function in S∗(α; ξ) is univalent