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Die Ungleichungen von Vietoris. (English) Zbl 0542.33004

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 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 \(\mu\).

MSC:

33B15 Gamma, beta and polygamma functions
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References:

[1] Uhlmann, W.: Ranggrößen als Schätzfunktionen. Metrika7 (1963), 23-40. · Zbl 0243.62021 · doi:10.1007/BF02613959
[2] Vietoris, L.: Vergleich unbekannter Mittelwerte auf Grund von Versuchsreihen. I. Sitzungsber. d. Österr. Akad. d. Wiss., Abt. II,188, 329-341 (1979); II. Sitzungsber. d. Österr. Akad. d. Wiss., Abt. II,189, 95-100 (1980); III.Sitzungsber. d. Österr. Akad. d. Wiss., Abt. II.190, 469-473 (1982); IV.Sitzungsber. d. Österr. Akad. d. Wiss., Abt. II.191, 53-58 (1982).
[3] Vietoris, W.: Über gewisse die unvollständige Betafunktion betreffende Ungleichungen Sitzungsber. d. Österr. Akad. d. Wiss., Abt. II.191, 85-92 (1982). · Zbl 0502.33010
[4] Vietoris, W.: Eine Verallgemeinerung der Gleichung (n+1)!=n!(n+1) und zugehörige vermutete Ungleichungen. Mh. Math.97, 157-160 (1984). · Zbl 0532.33001 · doi:10.1007/BF01653245
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