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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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