On the second Hankel determinant of concave functions

Proposition 1.1. Ωp ⊂Ωq for 0 < q < p < 1 and [ 0<p<1 Ωp = D ∪{−1} and \ 0<p<1 Ωp = {−(1 + z)2/4 : |z| ≤1}. Note that the set {−(1 + z)2/4 : |z| ≤1} is a closed Jordan domain, bounded by a cardioid with an inward-pointing cusp at the origin. By the above observations, we have Ωp ⊂H(Cop). In view of the coefficient regions of an for Cop, one might suspect that H(Cop) = Ωp for 0 < p < 1 and, in particular, H(Cop) ⊂D. This is, however, not the case. To state our result, we set M(p) = sup{|H(f)| : f ∈

Theorem 1.2. Let 0 < p < 1. Then M(p) > 1. Moreover, 1 3p < M(p) < 1 3p + 2 3. In Section 3, we will prove the above proposition and the theorem. Indeed, we give a description of the variability region of H(f) for f ∈Cop in Proposition 3.1 below. As a preliminary, we give an explicit form of the coefficient body of order 2 for the class Bp in Section 2. Our basic idea is to employ an higher-order analogue of Dieusonn´e’s lemma. 2. Higher-order analogue of Dieudonn´e’s lemma and its application We

Theorem 2.1. Let 0 < p < 1. A triple (c0, c1, c2) of complex numbers is con- tained in the coefficient body X2(Bp) if and only if c0 = P −1(1 −σ0), and c1 = P −2 1 + (P 2 −2)σ0 + σ2 0  + P −1(1 −|σ0|2)σ1 and c2 = P −3(1 −σ0)  1 + (P 2 −2)σ0 + σ2 0 

Lemma 2.2 (Dieudonn´e’s lemma). Let z0, τ0 ∈D with |τ0| ≤|z0| ̸= 0. Then the variability region of τ1 = ψ′(z0) for ψ ∈End(D) with ψ(0) = 0, ψ(z0) = τ0 is the closed disk given by (2.3) τ1 −τ0 z0 ≤ |z0|2 −|τ0|2 |z0|(1 −|z0|2). We remark that equality holds in Dieudonn´e’s lemma if and only if ψ is a finite Blaschke product of degree at most 2 (cf. [7, Theorem 3.6]). The following result can be regarded as Dieudonn´e’s lemma of the second order (see [7, Theorem 3.7]).

Lemma 2.3. Let z0, τ0 ∈D with |τ0| < |z0| ̸= 0 and suppose that τ1 ∈C satisfies (2.3). Then the variability region of τ2 = ψ′′(z0)/2! for ψ ∈End(D) with ψ(0) = 0, ψ(z0) = τ0 and ψ′(z0) = τ1 is the closed disk described by τ2 −τ1 −τ0/z0 z0(1 −|z0|2) + τ0(τ1 −τ0/z0)2 |z0|2 −|τ0|2 + |z0||τ1 −τ0/z0|2 |z0|2 −|τ0|2 ≤|z0|(1 −|τ0/z0|2) (1 −|z0|2)2 . We remark that equality holds precisely when ψ is a finite Blaschke product of degree at most 3. In [7, Theorem 3.7], the above inequality is stated as a nece

Lemma 2.4. Let 0 < p < 1. A triple (c0, c1, c2) of complex numbers is contained in the coefficient body X2(Bp) if and only if c0 = p −pw0 1 −p2w0 and c1 = (1 −p2)2w0 + p(1 −p2)(1 −|w0|2)w1 (1 −p2w0)2 and c2 = (1 −p2) (1 −p2w0)3  p(1 −p2)(1 −w0)w0 −(1 −p2)(1 + p2w0)(1 −|w0|2)w1 +p(w0 −p2)(1 −|w0|2)w2 1 + p(1 −p2w0)(1 −|w0|2)(1 −|w1|2)w2

Proposition 3.1. Let 0 < p < 1. Then the variability region of the second Hankel determinant H(f) of order 2 for f ∈Cop is given by H(Cop) = {Φp(σ0, σ1, σ2)/18P 3 : σ0, σ1, σ2 ∈D}. We note that the function Fζ given in (1.3) corresponds to the parameters (σ0, σ1, σ2) = ([ζ, p2], 0, 0). Since Φp(σ, 0, 0) = −18P(1 + (P 2 −2)σ + σ2), as a by-product, we have the following description of the set Ωp defined by (1.4).

Lemma 3.2. Ωp = {−P −2 1 + (P 2 −2)σ + σ2 : σ ∈D}, where P = (1 + p2)/p, 0 < p < 1. The description of the Lemma can now be used to show Propositition 1.1.

Proposition 3.1 above, we obtain Φp(t, −1, 0) = −18P  1 + (P 2 −2)t + t2 −3  1 −7P 2 + 2P 4 + (3P 2 −2)t + t2 (1 −t2) + P  2(1 −t2) + 3t(P 2 −1 + t)  (1 −t2). Setting hp(t) := −Φp(t, −1, 0)/18P 3 gives 18P 3hp(t) = −(P + 3)t4 −(3P 3 + 9P 2 −3P −6)t3 −(6P 4 −21P 2 −17P)t2