  We investigate the fundamental relation between entropy production rate and
the speed of energy exchange between a system and baths in classical Markov
processes. We establish the fact that quick energy exchange inevitably induces
large entropy production in a quantitative form. More specifically, we prove
two inequalities on instantaneous quantities: One is applicable to general
Markov processes induced by heat baths, and the other is applicable only to
systems with the local detailed-balance condition but is stronger than the
former one. We demonstrate the physical meaning of our result by applying to
some specific setups. In particular, we show that our inequalities are tight in
the linear response regime.
