How to Determine Concavity of the Product of Two Functions?

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Discussion Overview

The discussion revolves around determining the concavity of the product and composition of two functions, particularly focusing on the conditions required for such determinations. It includes theoretical considerations and examples related to concavity in mathematical functions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants inquire about methods to determine the concavity of the product of two functions, suggesting that additional information is necessary.
  • It is proposed that if the functions are differentiable twice, there exists a condition to assess the concavity of their product.
  • Questions are raised about whether the product of two concave functions is necessarily concave, with examples provided to illustrate that this is not always the case.
  • Concerns are expressed regarding the composition of two concave functions, with similar examples indicating that the composition may not be concave either.
  • Participants discuss the implications of specific conditions, such as weak concavity and increasing functions, on the concavity of compositions.
  • There is a request for clarification on the minimal information needed to assess concavity, with suggestions that knowledge of the second derivatives is crucial.
  • Some participants mention that without knowing differentiability, a general answer may not be possible, although special cases exist.

Areas of Agreement / Disagreement

Participants generally agree that additional information is required to determine concavity, particularly regarding differentiability. However, there are competing views on the implications of concavity in products and compositions, and the discussion remains unresolved regarding the general conditions for concavity.

Contextual Notes

Limitations include the dependence on the differentiability of the functions and the specific forms of the functions involved. The discussion highlights the complexity of assessing concavity without clear conditions.

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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?
 
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Given two functions and their respective concavity, is there any method to determine the concavity of the product of these two functions?
Not without additional information. If the functions are differentiable (twice), you can find a nice condition for that.
Also, can you determine concavity if these two functions are composite?
There is nothing special about composite functions, they just might need more work to analyze.
 
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.
 
economicsnerd said:
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.

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.
 
mfb said:
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.

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:
MathewsMD said:
Ok, so you say not without additional information. What is the minimal amount of information that must be provided to assess such a question?
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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