  Convex functions of quantum states play a key role in quantum physics, with
examples ranging from Bell inequalities to von Neumann entropy. However, in
experimental scenarios, direct measurements of these functions are often
impractical. We address this issue by introducing two methods for determining
rigorous confidence bounds for convex functions based on informationally
incomplete measurements. Our approach outperforms existing protocols by
providing tighter bounds for a fixed confidence level and number of
measurements. We evaluate the performance of our methods using both numerical
and experimental data. Our findings demonstrate the efficacy of our approach,
paving the way for improved quantum state certification in real-world
applications.
