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 ProofOfALifetime January 13th, 2019 09:52 AM

Is this convex?

I'm trying to use Jensen's to prove an inequality, but my solution depends on $$\frac{1}{x} \ln(1+x)$$ being convex when $x>0$. I'm not completely sure if this is true. The second derivative is inconclusive (at least it seems like that).

 romsek January 13th, 2019 10:44 AM

$\lim \limits_{x\to 0} \dfrac{d^2}{dx^2}\left(\dfrac 1 x \ln(1+x)\right) = \dfrac 2 3 > 0$

It's convex at 0.

 ProofOfALifetime January 13th, 2019 11:17 AM

Quote:
 Originally Posted by romsek (Post 604413) $\lim \limits_{x\to 0} \dfrac{d^2}{dx^2}\left(\dfrac 1 x \ln(1+x)\right) = \dfrac 2 3 > 0$ It's convex at 0.
That doesn’t make sense to me. Convex at 0? What I mean is isn't it supposed to be convex on an interval?

Sorry, but I was hoping that it would be convex when $x>0$, so on $(0,\infty)$. By the way, this is my first proof using Jensen's so I'm still learning.

 romsek January 13th, 2019 11:49 AM

Quote:
 Originally Posted by ProofOfALifetime (Post 604415) That doesn’t make sense to me. Convex at 0? What I mean is isn't it supposed to be convex on an interval? Sorry, but I was hoping that it would be convex when $x>0$, so on $(0,\infty)$. By the way, this is my first proof using Jensen's so I'm still learning.
Ok, my bad, This refers to the function being convex in an infinitesimal interval $(0,\delta)$, but at any rate the 2nd derivative of your function is positive in the interval $(0,\infty)$ so your function is convex for all non-negative reals.

 ProofOfALifetime January 13th, 2019 12:09 PM

Quote:
 Originally Posted by romsek (Post 604416) Ok, my bad, This refers to the function being convex in an infinitesimal interval $(0,\delta)$, but at any rate the 2nd derivative of your function is positive in the interval $(0,\infty)$ so your function is convex for all non-negative reals.
Thank you thank you thank you! This is all I needed to complete the proof I was doing! I appreciate it! :)

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