Stud. Univ. Babe¸s-Bolyai Math. 67(2022), No. 2, 329–344

Proposition 2.5.4 in [9] implies that h is holomorphically accretive if and only if ℜℓx(h(x)) ≥0 for all x ∈B \ {0}. The main feature of the class of holomorphically accretive mappings is that they generate semigroups of holomorphic self-mappings on B, so they are of most impor- tance in dynamical systems [24, 9]. A very fruitful characterization of holomorphically accretive mappings is:

Proposition 2.4 (Theorem 7.3 in [24], see also [9]). A mapping h ∈Hol(B, X) is holomorphically accretive if and only if it satisfies the so-called range condition (RC), that is, (Id +rh)(B) ⊇B for each r > 0, and the inverse mapping Jr := (Id +rh)−1 is a well-defined holomorphic self-mapping of B. The mapping Jr that occurs in this proposition is called the non-linear resolvent of h. In other words, the non-linear resolvent is the unique solution w = Jr(x) ∈B of the functional equation w + rh(w) =

Proposition 2.7. Let A ∈L(X) be strongly accretive, and let f ∈Hol(B, X) be a normalized and locally biholomorphic mapping. Then f ∈bSA(B) if and only if the mapping h := (Df)−1Af belongs to NA. This proposition inter alia implies that a spirallike mapping f relative to A linearizes the semigroup u(t, x) generated by h = (Df)−1Af in the sense that f ◦u(t, f −1(x)) = e−tAx on f(B). In the one-dimensional case, any linear oper- ator is scalar, hence can be chosen to be A = eiβ Id. In this case the

Lemma 3.1. Let p(z) = a + p1z + p2z2 + o(z2) and φ(z) = a + b1z + b2z2 + o(z2) be holomorphic functions on D such that φ ≺p. Then for every µ ∈C the following sharp inequality holds: |b2 −µb2 1| ≤max |p1|, |p2 −µp2 1|
.

Lemma 3.2. Let h ∈Hol(B, X) with h(0) = 0 and B ∈L(X) with ρ := ∥B∥≤1. For any x ∈∂B and ℓ∈X∗denote ϕ(t) := ℓ(h(tBx)) t , t ∈D \ {0}. (i) The function ϕ can be analytically extended to the disk 1 ρD with the Taylor expansion ϕ(t) = b0 + b1t + b2t2 + o(t2), where b0 = ℓ(Dh(0)Bx), b1 = 1 2! ℓ D2h(0)[(Bx)2]
and

Lemma 3.3. Let f ∈Hol(B, X) be a mapping of one-dimensional type. Then for every n ∈N the entire mapping x 7→Dnf(0)[xn] is also of one-dimensional type. Therefore for any x ∈∂B, ℓx ∈T(x) and constants µj ∈C, j = 1, 2, . . . , we have

Theorem 4.1. Let x ∈∂B, ℓx ∈T(x) and τ = g−1(ℓx(Ax)). Assume that g  τ −z 1 −zτ  = q0 + q1z + q2z2 + o(z2). Given f ∈Hol(B, X) denote ea2 2 = 1 2ℓx  D2f(0) 

Corollary 4.3. Every f ∈bSgα 1 (B) satisfies a3 −(ν −1)a2 2 −ea2 2 ≤α|ℓ| cos arg ℓ 1 α · max {1, Q1,α} , where Q1,α = ℜℓ 1 α 4α(ν −1)ℓ α−1 α +

Corollary 4.4. Every f ∈bSgα 2 (B) satisfies a3 −(ν −1)a2 2 −ea2 2 ≤ℜℓ· max {1, Q2,α} , where Q2,α = |1 + 4(ν −1)(1 −α)ℜℓ| . In particular, taking α = 0, we return to inequality (4.5) for all spirallike map- pings relative to the linear operator A. Another interesting (and, as we mentioned, dual) case occurs when ℓx(h(x)) ∥x∥ lies in some circle tangent to the imaginary axis. We can then set g = gα 3 .

Corollary 4.5. Every f ∈bSgα 3 (B) satisfies a3 −(ν −1)a2 2 −ea2 2 ≤(ℜℓ−|ℓ|2α) · max {1, Q3,α} , where Q3,α = 1 −2ℓα + 4(ν −1)(ℜℓ−|ℓ|2α) . Recall that for A = eiβ Id, the class bSgα 3 (B) consists of so-called spirallike map- pings of type β of order α.

Corollary 4.7. If f ∈bSA(B) satisfies Assumption 2, then for any ν ∈C, a3 −  ν −1 + 1 ℓx (Ax)  a2 2 ≤|q1| 2 max  1,

Corollary 4.8. If f ∈Hol(B, X) is a spirallike mapping of type β, that satisfies As- sumption 2. Then for any µ ∈C we have a3 −µa2 2 ≤|q1| 2 max  1,

Lemma 5.1. (a) For any r > 0, the operator Br := DJr(0) = (Id +rA)−1 is strongly contractive, that is, ρr := ∥Br∥< 1. (b) If h is of one-dimensional type, then A is a scalar operator and Jr, r > 0, is of one-dimensional type too.

Theorem 5.2. Let h ∈NA(g) and Jr be the nonlinear resolvent of h for some r > 0. For x ∈∂B and ℓr := ℓBrx ∈T(Brx), let ea2 2 := ℓr  (Id +rA) 1 2!D2Jr(0)  x, (Id +rA) 1 2!D2Jr(0)[x2]  , a2

Corollary 5.3. Assume that A = λ Id, ℜλ > 0 and g = g0. Then for any ν ∈C we have a3 −2ea2 2 −(ν −2)a2 2 ≤|1 + λ2|r |1 + λr|3 max  1, λ −(2 −ν)r 1 + λ2 |1 + λr|

Corollary 5.4. If h ∈NA(g) satisfies Assumption 2, then a3 −(ν −2 + 2δ)a2 2 ≤r|q1|∥xr∥ρ2 r max (1, Qr(x)) , (5.10) where Qr(x) is defined by (5.4) and δ = ℓr(Brxr) ∥xr∥2 .

Corollary 5.5. If h ∈NA(g) is one-dimensional type with A = λ Id, then for any ν ∈C we have (Id +rA) 1 3!D3Jr(0)[x3] −µ(Id +rA) 1 2!D2Jr(0)  x, (Id +rA) 1 2!D2Jr(0)[x2]  = a3 −µa2 2 ≤ r|q1| |1 + λr|3 · max

Lemma 3.3 states that this is equal to ℓx 1 + rλ 3! D3Jr(0)[x3] −µ1 + rλ 2! κ(x)D2Jr(0)[x2]  = |a3 −µa2κ(x)| = a3 −µa2 2 . Set µ = ν −2 + 2δ. Then we proceed by Corollary 5.4: ≤