If $B$ is worse than $A$ on some inputs, how do their worst-case time complexities compare?
You are essentially asking the following question:


  Suppose that $f(n) \geq g(n)$ for some $n$. Is there a universal constant $C$ such that $\min_n f(n) \geq C \min_n g(n)$?


Hopefully this formulation makes it easier to answer.