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The title gives away pretty much the entire question except for the fact that the class of functions I am interested in is a subset of twice continuously differentiable functions.

I think that if an odd function (defined on the whole real line) is concave on the positive real line, then it is convex on the negative real line. If it is supposed to be convex everywhere, then it is linear on the negative real line. The argument goes also the other way. So it is linear everywhere. So maybe I don't even need the monotonicity property.

This is the first question I am posting here that has no equations. So I hope that is allowed. Is my reasoning correct?

Calculon
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  • Hint: Can you show that if $g$ is concave on $(-\infty,0]$ then $h:x\mapsto-g(-x)$ is convex on $[0,+\infty)$? Thus... – Did May 16 '16 at 19:57
  • @Did I did that as a rough sketch. I just wanted to make sure that I didn't overlook something. I didn't post my proof here as for this particular problem I only need to know whether the claim is true or not. – Calculon May 16 '16 at 20:15
  • ?? Compare "Is my reasoning correct?" with "I only need to know whether the claim is true or not". – Did May 16 '16 at 20:18
  • @Did Well we are supposed to show our effort towards the solution. I thought the question would get closed if I just asked "Is the claim true or not?" Also, I should have said "I mainly care about the truth value of the statement". That was a mistake on my part. – Calculon May 16 '16 at 20:22

1 Answers1

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Yes: $f$ concave implies $-f$ convex, so concave+odd implies concave+convex, and concave+convex implies linear. Your reasoning is fine.

Gareth McCaughan
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