June 17th, 2017, 03:45 PM  #1 
Member Joined: May 2017 From: France Posts: 57 Thanks: 1  A strange Inequality
Hi, Let $\displaystyle g\in C^1([0,1],\mathbb R)$ Is it true that : $\displaystyle 4\times (\max(g)\min(g))\geq \min(g')$ ? Cordially. 
June 27th, 2017, 05:52 AM  #2 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,390 Thanks: 100 
It seems to work if you sketch out a few representative curves. From there you could probably develop a proof if you're interested beyond the "gotcha" phase.

June 27th, 2017, 05:57 AM  #3 
Member Joined: May 2017 From: France Posts: 57 Thanks: 1 
Gotcha

June 27th, 2017, 06:48 AM  #4 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,390 Thanks: 100  
June 27th, 2017, 06:51 AM  #5 
Member Joined: May 2017 From: France Posts: 57 Thanks: 1 
Gotcha

June 27th, 2017, 07:00 AM  #6 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,327 Thanks: 2451 Math Focus: Mainly analysis and algebra 
I may have misunderstood, but $$g:[0,1]\mapsto \mathbb R \quad \text{$g,g'$ continuous}$$ Has a maximum value for $\min{(g')}$ of $\max{(g)}  \min{(g)}$. This when $\min{(g)}=g(0)$, $\max{(g)}=g(1)$ and $g$ is a straight line. If I haven't understood, the question seems trivial. What have I got wrong? Last edited by v8archie; June 27th, 2017 at 07:02 AM. 
June 27th, 2017, 07:19 AM  #7 
Member Joined: May 2017 From: France Posts: 57 Thanks: 1 
Are you say : $\displaystyle \max(g)\min(g) \geq \min(g')$ ? What about $\displaystyle g(x)=\sinh(x)$ ? 
June 27th, 2017, 08:33 AM  #8 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,327 Thanks: 2451 Math Focus: Mainly analysis and algebra  I believe so. $\sinh 1  \sinh 0 \approx 1.18  0 = 1.18$ $\min{(\cosh x)}=1$ Last edited by v8archie; June 27th, 2017 at 08:36 AM. 
June 27th, 2017, 09:05 AM  #9 
Member Joined: May 2017 From: France Posts: 57 Thanks: 1 
edit : I must think What about this inequality $\displaystyle 4(\max(g)\min(g))\geq \min(g'')$, when $\displaystyle g \in C^2([0,1])$ Last edited by Dattier; June 27th, 2017 at 09:16 AM. 
June 27th, 2017, 09:27 AM  #10 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,327 Thanks: 2451 Math Focus: Mainly analysis and algebra 
That sounds plausible. Again consider the case where $g''$ is constant (giving the maximum value for $\min{(g'')}$.


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