MATH.20:4.5 - Tighten the comparison or identify what prevents it

Inspect where the proof introduces slack. It may enlarge the feasible set, forget a dependency, replace a quantity by a worst-case value, or use one norm for effects that could be bounded separately.

Improve the part that can change the receiving result. Construct a better feasible candidate, strengthen the inequality, partition the domain, retain a lost relation, or add an available premise. A bound on a requested component can be enough when a bound on the whole object is costly or impossible.

Check attainability and equality conditions. When a feasible value meets a proved bound in the opposite direction, the optimum is established. When equality conditions conflict, the current bound can remain valid while failing to identify an attainable result. A sequence approaching a bound can establish an infimum or supremum without an attaining object.

If the gap reflects missing information, expose it. Two admissible constructions with different answers can show why the current assumptions do not determine a narrower result. That identifies a useful next assumption, observation or mathematical question.