Sharp norm estimate of Schwarzian derivative for a class of convex functions

Theorem 1.1 (Nehari [N1], K¨uhnau [K], Ahlfors-Weill [AW]). Let f ∈ A . If f is univalent, then ∥Sf∥≤6. Conversely, if ∥Sf∥≤2, then f is univalent. Moreover, let 0 ≤k < 1. If f extends to a k-quasiconformal mapping of the Riemann sphere bC then ∥Sf∥≤6k. Conversely, if ∥Sf∥≤ 2k, then f extends to a k-quasiconformal mapping of bC. Here, a mapping f : bC →bC of the Riemann sphere bC = C ∪{∞} is called k-quasiconformal if f is a sense-preserving homeomorphism of bC and has locally integrable partial

Theorem 1.2 (Mocanu). Kγ(β) ⊂S ∗ β for 0 < β < 1. In other words, Kα ⊂S ∗ γ−1(α) for 0 < α < 1, where γ−1 denotes the inverse function of γ : [0, 1] →[0, 1]. For sharp or improved relations of this kind, see a paper [KS2] of the authors. We summarize important properties of strongly starlike functions as fol- lows.

Theorem 1.3. A strongly starlike function f of order α ∈(0, 1) extends to a sin(πα/2)-quasiconformal mapping of bC and therefore ∥Sf∥≤6 sin(πα/2). The first part is due to Fait, Krzy˙z and Zygmunt [FKZ] and the second one is obtained from the first in combination with Theorem 1.1 (as was pointed out by Chiang [Ch]). By Theorems 1.2 and 1.3, we see that a function f ∈Kα extends to a sin(πγ−1(α)/2)-quasiconformal mapping of bC and satisfies ∥Sf∥≤

Theorem 1.4. A convex function f satisfies ∥Sf∥≤2. The bound is sharp. This result was repeatedly proved in the literature (see [Rob], [N2], [L1]), and was refined by Suita [NS] in the following form: the sharp ienequality ∥Sf∥≤8α(1−α) holds for a function f ∈A with Re [1+zf ′′(z)/f ′(z)] > α and 1/2 ≤α < 1. Obviously, the estimate ∥Sf∥≤6 sin(πγ−1(α)/2) for f ∈Kα is not better than Theorem 1.4 when α is close to 1. We will give a sharp norm estimate for f ∈Kα. Main Theorem 1.5. Let f be a strongly

Corollary 1.6. A function f ∈Kα extends to an α-quasiconformal map- ping of bC for 0 < α < 1. By using Mathematica Ver. 7, we found that sin(πγ−1(α)/2) < α when 0 < α < 0.3354 (see Figure 1). Therefore, the corollary gives a better bound only when α > 0.3355, though it has obvious merit of simplicity. For some reason, the second author was even led in [Su1] to the ex- pectation that each function in S ∗ α might extend to an α-quasiconformal mapping of bC. This was recently disproved by Yuliang S

Lemma 3.1. The functions Pα(z) and Qα(z) = 2zP ′ α(z) + 1 −Pα(z)2 have non-negative Taylor coefficients about z = 0 for 0 < α ≤1.

Lemma 4.1. For every non-negative integer n, the following inequality holds: X 0≤k,l,m k+l+m=n 1 (2k + 1)(2l + 1)(2m + 1) ≤1.

Lemma 4.2. The functions P given in (2.1) and Q(z) = 2zP ′(z)+1−P(z)2 have non-negative Taylor coefficients about z = 0.