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