Die Ungleichungen von Vietoris (Q795203)

The inequalities \[ P_{k,l}=(1/B(l+1,k))\int^{l/(k+l)}_{0}x^ l(1- x)^{k-1}dx=I_{l/(k+l)}(l+1,k)<{1\over2} \] and \[ \Phi_{k,l,\mu}=(1/B(k,k\mu +1))\int^{1}_{0}x^{k-1}(1- x)^{k\mu}I_ x(l+1,l\mu)dx<{1\over2} \] are shown to be valid for any positive real numbers k, l, \(\mu\). These inequalities seem to be of some interest in the theory of probability, since \textit{L. Vietoris} [in a series of papers published in Sitzungsber., Abt. II, Österr. Akad. Wiss., Math.-Naturwiss. Kl. namely: ibid. 188, 329-341 (1979; Zbl 0466.62025), ibid. 189, 95-100 (1980; Zbl 0466.62026), ibid. 190, 469-473 (1981; Zbl 0496.62023), ibid. 191, 85-92 (1982; Zbl 0502.33010)] has attributed to them the following meaning: If k, l, m, n are natural numbers satisfying the condition: \(1<m/k=n/l=1+\mu,\) then the value of \(\Phi_{k,l,\mu}\) may be considered as a lower bound for the measure of confidence in the assumption that an event which was observed l times within a sequence of n experiments is not more probable than another one which was observed k times within a sequence of m experiments. \(P_{k,l}\) is an approximation to \(\Phi_{k,l,\mu}\) for large numbers \(\mu\).