A New Bound for Standard $p$-Functions

Abstract

Let $p(t), t \in (0, \infty)$ be the standard $p$-function of a regenerative phenomenon as defined in Kingman's theory. Let $p(1) = M$ and $\min \{p(t), 0 \leqq t \leqq 1\} = m$. Griffeath (1973) has derived a new upper bound for $M$ for given $m$ by using the Kingman inequalities of order $\leqq 3$. Here Griffeath's result is generalized by using the Kingman equalities of order $\leqq n$. Further taking limits as $n \rightarrow \infty$ a new upper bound is obtained which is uniformly strictly superior to the present known upper bound. Thus a part of the uncharted region in the $M - m$ diagram becomes charted by being shown inaccessible. This gives also an improved upper bound for the constant $\nu_0$.