January 13th, 2019, 09:52 AM  #1 
Senior Member Joined: Oct 2016 From: Arizona Posts: 193 Thanks: 34 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: 2,382 Thanks: 1281 
$\lim \limits_{x\to 0} \dfrac{d^2}{dx^2}\left(\dfrac 1 x \ln(1+x)\right) = \dfrac 2 3 > 0$ It's convex at 0. 
January 13th, 2019, 11:17 AM  #3  
Senior Member Joined: Oct 2016 From: Arizona Posts: 193 Thanks: 34 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: 2,382 Thanks: 1281  Quote:
 
January 13th, 2019, 12:09 PM  #5 
Senior Member Joined: Oct 2016 From: Arizona Posts: 193 Thanks: 34 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! 

Tags 
convex 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Convex, concave or neither is convex/concave?  Brazen  Calculus  0  January 28th, 2013 12:38 PM 
Quadrilateral Convex  galc127  Algebra  0  August 31st, 2012 10:23 PM 
Convex set, convex function  Vasily  Applied Math  1  June 30th, 2012 02:57 PM 
Convex hull  magyaddello  Algebra  0  November 26th, 2010 08:42 AM 
convex combination of closed and convex sets  frederico  Real Analysis  0  April 6th, 2009 11:31 AM 