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Question about an integral.

  1. Oct 2, 2009 #1
    given [x] the 'integer function' would be the following bound valid ??

    [tex] \int_{0}^{\infty}dt [g(x/t)]f(t) \le \int_{0}^{\infty}dt g(x/t)f(t) [/tex]

    for a given functions f(t) and g(u) u=x/t , here g is a non-decreasing positive function for positive arguments (and real) of parameter u=x/t
  2. jcsd
  3. Oct 2, 2009 #2

    if f(t) >= 0 for t >=0, then using (g(x/t) + 1) would be a better idea ?
    Now if
    [tex] f(t) = f^{+}(t) - f^{-}(t) [/tex],
    where [tex]f^{+}(t) and f^{-}(t)[/tex] denotes max(f(t),0) and max(-f(t),0) respectively
    then one can use g(x/t) + 1 for f+(t) and g(x/t) - 1 for f-(t) to obtain a bound.
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