Correspondence between spirallike functions and starlike functions

Theorem 1.1 (Representation Theorem). Let f ∈Fλ for a λ ∈(−π/2, π/2). Then the limits β(t) = lim r→1−argλ f(reit) and f(eit) = lim r→1−f(reit) ∈bC = C ∪{∞} exist for every t ∈R in such a way that β(t) is non-decreasing in t and β(t + 2π) = β(t) + 2π. Moreover, f is represented by (1.2) f(z) = z exp  −eiλ cos λ π Z 2π

Theorem 1.2. Let f ∈Fλ. Then lim r→1− log M(r, f) log 1 1−r = A(f) cos2 λ π . He suspected in [6] that M(r, f) = O[(1 −r)−q0] if A(f) ̸= 0. We will show that this is not true in general.

Theorem 1.3. Let λ ∈(−π/2, π/2) and 0 < A < 2π. Then there is a λ-spirallike function f with A(f) = A so that M(r, f) = O[(1 −r)−A(f) cos2 λ/π] does not hold. 2. Preliminaries We first summarize basic properties of the λ-argument. The following elementary lemma is convenient in various computations.

Lemma 2.1. For λ ∈(−π/2, π/2), θ ∈R and w ∈C \ {0}, argλ w = arg w −(tan λ) log |w| (mod 2π).

Lemma 2.2. For nonzero complex numbers w1, w2 and λ ∈(−π/2, π/2), argλ (w1w2) = argλ w1 + argλ w2 (mod 2π). Also, by Lemma 2.1, we have a canonical way to take a harmonic branch of argλ h for a non-vanishing analytic function h.

Lemma 2.3. Let λ ∈(−π/2, π/2) and let h be a non-vanishing analytic function in the unit disk D with h(0) = 1. Then there is a unique harmonic function u on D with u(0) = 0 such that argλ h(z) = u(z) (mod 2π) for z ∈D.

Lemma 3.1. Let λ ∈(−π/2, π/2). There corresponds to f ∈Fλ a unique starlike func- tion g ∈S ∗in such a way that (3.1) f(z) z = g(z) z eiλ cos λ , z ∈D. This relation serves as a key to reduce a problem concerned with spirallike functions to one with starlike functions. Note here, however, that the relation (3.1) does not give a transformation of the image domain f(D) onto g(D) because the term z is involved. To realize the connection, we extend notions for starlike functions to spirallike ones

Theorem 3.2. Let f ∈Fλ for a λ with −π/2 < λ < π/2. Then the limits Uλ(t) = lim r→1−argλ f(reit) reit and f(eit) = lim r→1−f(reit) ∈bC exist for every t ∈R, and βλ(t, f) = Uλ(t) + t is a non-decreasing function in t with βλ(t + 2π, f) = βλ(t, f) + 2π. Moreover, the left and right limits of βλ(t, f) satisfy the following relation: (3.2) βλ(t, f) = 1 2 βλ(t+, f) + βλ(t−, f)

Lemma 3.3. Suppose that f ∈Fλ and g ∈S ∗are related by (3.1). Then A(f) = Bλ(f) = B(g) = A(g). We are ready to prove Theorem 1.2.

Lemma 4.1. Let g0(z) = 1 1 −z log 1 1 −z = ∞ X n=1  1 + 1 2 + · · · + 1 n  zn,

Lemma 4.2. There exists a number C0 > 2 such that the following inequalities hold for z ∈D whenever C ≥C0 : Re " 1 (1 −z) log C 1−z # > 1 2 log C 2 , Re

Lemma 4.3. Let 0 < α < 2. Choose positive numbers β and c so that c ≤ 1 log C0 and α + cβ 1 −c log 2 < 2, where C0 is the number appearing in Lemma 4.2. Then the function g(z) = z (1 −z)α  1 + c log 1

Lemma 4.3 for α = A/π. We now define a function f by the relation (3.1). Then f ∈Fλ and, by (3.5), M(r, f) ≍M(r, g)cos2 λ ≍(1 −r)−α cos2 λlog 1 1−r
β cos2 λ as r →1 −. Therefore, M(r, f) = O[(1 −r)−A cos2 λ/π] does not hold. □ References [1] T. Ba¸sg¨oze and F. R. Keogh, The Hardy class of a spiral-like function and its derivative, Proc. Amer. Math. Soc. 26 (1970), 266–269. [2] P. L. Duren, Theory of Hp Spaces, Academic Press, New York and London, 1970. [3]