  We extend a result by Huneke and Watanabe bounding the multiplicity of
$F$-pure local rings of prime characteristic in terms of their dimension and
embedding dimensions to the case of $F$-injective, generalized Cohen-Macaulay
rings. We then produce an upper bound for the multiplicity of any local
Cohen-Macaulay ring of prime characteritic in terms of their dimensions,
embedding dimensions and HSL numbers. Finally, we extend the upper bounds for
the multiplicity of generalized Cohen-Macaulay rings in characteristic zero
which have dense $F$-injective type.
