For feasible mass functions $Q_1,Q_2$ at distortion levels $d_1,d_2$, the mixture $Q=\lambda Q_1+(1-\lambda)Q_2$ has nonzero-symbol mass at most $\lambda d_1+(1-\lambda)d_2$. Concavity of <information entropy> gives
$$
H(Q)\geq\lambda H(Q_1)+(1-\lambda)H(Q_2).
$$
Taking maximizing sequences proves
$$
\phi(\lambda d_1+(1-\lambda)d_2)
\geq\lambda\phi(d_1)+(1-\lambda)\phi(d_2).
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 224 / 3 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-224/3/a/solution