January 13th, 2019, 10:52 AM  #1 
Senior Member Joined: Oct 2016 From: Arizona Posts: 170 Thanks: 31 Math Focus: Frame Theory is pretty awesome, and it's ripe for undergraduate research!  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, 11:44 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,262 Thanks: 1198 
$\lim \limits_{x\to 0} \dfrac{d^2}{dx^2}\left(\dfrac 1 x \ln(1+x)\right) = \dfrac 2 3 > 0$ It's convex at 0. 
January 13th, 2019, 12:17 PM  #3  
Senior Member Joined: Oct 2016 From: Arizona Posts: 170 Thanks: 31 Math Focus: Frame Theory is pretty awesome, and it's ripe for undergraduate research!  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 12:35 PM.  
January 13th, 2019, 12:49 PM  #4  
Senior Member Joined: Sep 2015 From: USA Posts: 2,262 Thanks: 1198  Quote:
 
January 13th, 2019, 01:09 PM  #5 
Senior Member Joined: Oct 2016 From: Arizona Posts: 170 Thanks: 31 Math Focus: Frame Theory is pretty awesome, and it's ripe for undergraduate research!  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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