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Concavity of two functions

  1. Oct 24, 2013 #1
    Given two functions and their respective concavity, is there any method to determine the concavity of the product of these two functions? Also, can you determine concavity if these two functions are composite?
     
  2. jcsd
  3. Oct 25, 2013 #2

    mfb

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    Not without additional information. If the functions are differentiable (twice), you can find a nice condition for that.
    There is nothing special about composite functions, they just might need more work to analyze.
     
  4. Oct 25, 2013 #3
    Q: Is the product of two concave functions concave? i.e. If [itex]f[/itex] and [itex]g[/itex] are concave, and [itex]h(x)=f(x)g(x)[/itex], then is [itex]h[/itex] concave?
    A: Not necessarily. Consider the case of [itex]f(x)=-1[/itex], the constant function.

    Q: Is the composition of two concave functions concave? i.e. If [itex]f[/itex] and [itex]g[/itex] are concave, and [itex]h(x)=f(g(x))[/itex], then is [itex]h[/itex] concave?
    A: Not necessarily. Consider the case of [itex]f(y)=-y[/itex].

    In both settings, [itex]f[/itex] (weakly concave) was chosen so that [itex]h=-g[/itex]. Then the only way [itex]h[/itex] can be concave is if [itex]g[/itex] was both concave and convex, i.e. a straight line.

    A nice exercise: If [itex]f,g[/itex] are both (weakly) concave and f is (weakly) increasing, show that the composition [itex]h(x)=f(g(x))[/itex] is also concave.
     
  5. Oct 25, 2013 #4
    An example of this that shows up in probability sometimes: If [itex]g[/itex] is concave, then so is [itex]\log(g)[/itex]. Some probabilistic theorems involve an assumption that some function has a concave logarithm, which is (according to the exercise I gave you, since the [itex]\log[/itex] function is increasing and concave) a less stringent requirement than the function itself being concave.
     
  6. Oct 25, 2013 #5
    Ok, so you say not without additional information. What is the minimal amount of information that must be provided to assess such a question? Also, what if the function could not have a degree of -1, 0 or 1, how would this change everything?

    Thanks!
     
    Last edited: Oct 25, 2013
  7. Oct 27, 2013 #6

    mfb

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    If you know f and g are both differentiable, just look at the second derivatives of f(x)*g(x) or f(g(x)).
    If you don't know that, I don't think there is a general answer. There are some special cases, economicsnerd mentioned some of them.
     
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