January 13th, 2019, 09:52 AM  #1 
Senior Member Joined: Oct 2016 From: Arizona Posts: 209 Thanks: 37 Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems.  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).

January 13th, 2019, 10:44 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,552 Thanks: 1402 
$\lim \limits_{x\to 0} \dfrac{d^2}{dx^2}\left(\dfrac 1 x \ln(1+x)\right) = \dfrac 2 3 > 0$ It's convex at 0. 
January 13th, 2019, 11:17 AM  #3  
Senior Member Joined: Oct 2016 From: Arizona Posts: 209 Thanks: 37 Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems.  Quote:
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. Last edited by ProofOfALifetime; January 13th, 2019 at 11:35 AM.  
January 13th, 2019, 11:49 AM  #4  
Senior Member Joined: Sep 2015 From: USA Posts: 2,552 Thanks: 1402  Quote:
 
January 13th, 2019, 12:09 PM  #5 
Senior Member Joined: Oct 2016 From: Arizona Posts: 209 Thanks: 37 Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems.  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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