  Linear multistep methods (LMMs) applied to approximate the solution of
initial value problems---typically arising from method-of-lines
semidiscretizations of partial differential equations---are often required to
have certain monotonicity or boundedness properties (e.g.
strong-stability-preserving, total-variation-diminishing or
total-variation-boundedness properties). These properties can be guaranteed by
imposing step-size restrictions on the methods. To qualitatively describe the
step-size restrictions, one introduces the concept of step-size coefficient for
monotonicity (SCM, also referred to as the strong-stability-preserving (SSP)
coefficient) or its generalization, the step-size coefficient for boundedness
(SCB). A LMM with larger SCM or SCB is more efficient, and the computation of
the maximum SCM for a particular LMM is now straightforward. However, it is
more challenging to decide whether a positive SCB exists, or determine if a
given positive number is a SCB. Theorems involving sign conditions on certain
linear recursions associated to the LMM have been proposed in the literature
that allow us to answer the above questions: the difficulty with these theorems
is that there are in general infinitely many sign conditions to be verified. In
this work we present methods to rigorously check the sign conditions. As an
illustration, we confirm some recent numerical investigations concerning the
existence of SCBs in the BDF and in the extrapolated BDF (EBDF) families. As a
stronger result, we determine the optimal values of the SCBs as exact algebraic
numbers in the BDF family (with $1\le k\le 6$ steps) and in the
Adams--Bashforth family (with $1\le k\le 3$ steps).
