Final answer:
Mr. Brocklehurst punishes perceived vanity by imposing harsher punishments on the student. He treats weaker students with leniency while enforcing stricter discipline on those he deems arrogant.
Step-by-step explanation:
Mr. Brocklehurst punishes the seeming "vanity" of one of his students by inflicting harsher punishments on them. He administers justice with discrimination, giving more leniency to weaker students but imposing a double portion of punishment on those he deems arrogant or stubborn. Brocklehurst justifies this as "doing his duty by their parents" and makes sure to remind the punished student that they should be thankful for the discipline.