The maximal oscillation problem for p-functions (Q908575)

\textit{J. F. C. Kingman} [see J. London Math. Soc., II. Ser. 6, 747-752 (1973; Zbl 0283.60096)] introduced the p-function in the study of regenerative phenomena. The p-function is standard if \(\lim_{t\downarrow 0} p(t)=1\). Denote by \({\mathcal P}\) the class of all standard p-functions. For \(0<M\leq 1\), let \[ m(M,p)=Inf\{p(t):\quad 0\leq t\leq 1,p(1)=M\}, \] \[ I(M)=Inf\quad \{m(M,p):\quad p\in {\mathcal P}\},\quad v_ 0=Inf\{M:\quad I(M)>0\}. \] D. G. Kendall conjectured that \(v_ 0\leq 1/2\). The present paper proves the conjecture.