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Math Help - Concave function and little-oh

  1. #1
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    Concave function and little-oh

    The problem is:

    Suppose f(x) is a strictly concave function. Prove that x=o(f^(-1)(x)).

    Do you have any suggestions on how to prove this?
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  2. #2
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    Quote Originally Posted by batman View Post
    The problem is:

    Suppose f(x) is a strictly concave function. Prove that x=o(f^(-1)(x)).

    Do you have any suggestions on how to prove this?
    Do you mean x=o(f^{-1}(x)) as x\to\infty, or as x\to0, or something else?

    In any case, I don't think that this result can be true without some additional conditions. For example, the function f(x) = \sqrt{x^2-1} is strictly concave on the interval [1,\infty), but f^{-1}(x)\approx x as x\to\infty.
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    Sorry, I didn't copy the exercise properly from the blackboard. It should have been:

    Suppose f(x) is strongly concave. Then, x=o(f^{-1}(x)) as x\rightarrow \infty.

    The function you mention is strictly concave, but nog strongly concave. Thank you anyway.
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  4. #4
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    Quote Originally Posted by batman View Post
    Sorry, I didn't copy the exercise properly from the blackboard. It should have been:

    Suppose f(x) is strongly concave. Then, x=o(f^{-1}(x)) as x\rightarrow \infty.
    That certainly makes a difference, but I still don't understand what's going on here. As far as I can see, a strongly concave function has to be bounded above, so f^{-1}(x) will not exist when x is large.
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