  We prove a conjecture of Teissier asserting that if $f$ has an isolated
singularity at $P$ and $H$ is a smooth hypersurface through $P$, then
$\widetilde{\alpha}_P(f)\geq
\widetilde{\alpha}_P(f\vert_H)+\frac{1}{\theta_P(f)+1}$, where
$\widetilde{\alpha}_P(f)$ and $\widetilde{\alpha}_P(f\vert_H)$ are the minimal
exponents at $P$ of $f$ and $f\vert_H$, respectively, and $\theta_P(f)$ is an
invariant obtained by comparing the integral closures of the powers of the
Jacobian ideal of $f$ and of the ideal defining $P$. The proof builds on the
approaches of Loeser and Elduque-Mustata. The new ingredients are a result
concerning the behavior of Hodge ideals with respect to finite maps and a
result about the behavior of certain Hodge ideals for families of isolated
singularities with constant Milnor number. In the opposite direction, we show
that for every $f$, if $H$ is a general hypersurface through $P$, then
$\widetilde{\alpha}_P(f)\leq \widetilde{\alpha}_P(f\vert_H)+\frac{1}{{\rm
mult}_P(f)}$, extending a result of Loeser from the case of isolated
singularities.
