  Thanks to a recent result by Jean-Marc Schlenker, we establish an explicit
linear inequality between the normalized entropies of pseudo-Anosov
automorphisms and the hyperbolic volumes of their mapping tori. As its
corollaries, we give an improved lower bound for values of entropies of
pseudo-Anosovs on a surface with fixed topology, and a proof of a slightly
weaker version of the result by Farb, Leininger and Margalit first, and by Agol
later, on finiteness of cusped manifolds generating surface automorphisms with
small normalized entropies. Also, we present an analogous linear inequality
between the Weil-Petersson translation distance of a pseudo-Anosov map
(normalized by multiplying the square root of the area of a surface) and the
volume of its mapping torus, which leads to a better bound.
