Taylor's theorem and $g(0)=g'(0)=0$ give
$$
|g(s)|\leq\frac12|s|^2,
\qquad
|g'(s)|\leq|s|
\quad(|s|\leq1).
$$
Combining these with the product estimate in a <Hölder space> yields
$$
\boxed{|g(u)|_{0,\alpha;B}\leq|u|_{0,\alpha;B}^2.}
$$
Writing
$$
g(u_1)-g(u_2)=(u_1-u_2)
\int_0^1g'(u_2+t(u_1-u_2))\,dt
$$
and using the same product estimate gives
$$
\boxed{|g(u_1)-g(u_2)|_{0,\alpha;B}
\leq(|u_1|_{0,\alpha;B}+|u_2|_{0,\alpha;B})
|u_1-u_2|_{0,\alpha;B}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 107 / 3 / c / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-107/3/c/i/solution