Let Ω be an open connected subset of ℂ containing 𝑈 = { 𝑧 ∈ ℂ ∶ |𝑧| ≤ \(\frac{1}{2}\)}.

Let 𝔍 = { 𝑓 ∶ Ω → ℂ ∶ 𝑓 is analytic and \(\rm \displaystyle sup_{z,w \in U}\) |𝑓(𝑧) − 𝑓(𝑤)| = 1 }.

Consider the following statements:

𝑃: There exists 𝑓 ∈ ℑ such that |𝑓′ (0)| ≥ 2.

𝑄: |𝑓⁽³⁾ (0)| ≤ 48 for all 𝑓 ∈ ℑ, where 𝑓⁽³⁾ denotes the third derivative of 𝑓.

Then