THE OSCILLATION OF p-FUNCTIONS WITH EXPONENTIAL START

Let \mathscrP be the set of all standard p-functions. For u>0 let \mathscrP(u)=\left\p \in \mathscrP: \exists q \geqslant 0, p(t)=e^-q t, 0 \leqslant t \leqslant u\right\ In 9. Kingman proved: if p \in \mathscrP(u) then p(t) \leqslant e^-1+e^-q u(t \geqslant u), and Griffeath ^4 further proved: p(t) \leqslant e^-\left(1-e^-q u)\right.(t \geqslant u). In this paper we first give Griffeath's result a new proof, and then prove the following rusult: if p \in \mathscrP(u), s \geqslant u, M=\max _u< t< s p(t), m=p(\mathbfs), then p(t) \leqslant \max \left(M, m+e^-1+m\right)(t \geqslant u). The second result of this paper is as follows. Let \begingathered m(M, p)=\inf \p(t): 0 \leqslant t \leqslant 1, p(1)=M\, \quad p \in \mathscrP \\ I(M, u)=\inf \m(M, p): p \in \mathscrP(u)\ \quad I(M)=\inf \m(M, p): p \in \mathscrP\; \\ I^*(M)=\lim _u \downarrow 0 I(M, u) ; v_0=\inf \M>0: I(M)>0\; \\ v^*=\inf \left\M>0: I^*(M)>0\right\ \endgathered then v_0=v^*.