Assume $ g:[0,\infty)\to [0,\infty) $ be strictly convex and increasing monotonic function and $B_1:[0,\infty)\to [0,\infty) $ be convex and increasing monotonic function and $B_2:[0,\infty)\to [0,\infty) $ is increasing monotonic function and $B_1(t)>B_2(t)$ for every $ a<t<b $ and $B_1(t)=B_2(t)$ for $t=a$ , $t=b$ then I guess that we can show $$\int_a^b g(B'_1(t)) dt \le \int_a^b g(B'_2(t)) dt $$
I have tried to use Jensen's inequality or other similar inequalities and also I have tried to use calculus of variation methods but I have not reached to any good place.
Can somebody give a counterexample or give hint to proof this even with some modification on hypothesizes