Let $f:\mathbb{R}_{+}^{*}$$\rightarrow$$\mathbb{R}_{+}^{*}$ such that $f'''(x)>0$ for all $x$ $\in$ $\mathbb{R}_{+}^{*}$. Prove that: $f(a^{2}+b^{2}+c^{2})+2f(ab+bc+ac)$ $\geq$ $f(a^{2}+2bc)+f(b^{2}+2ca)+f(c^{2}+2ab)$, for all $a,b,c$ $\in$ $\mathbb{R}_{+}^{*}$.
Problem
Source: 2017 Olympic Revenge, Problem 4
Tags: algebra, inequalities
rkm0959
20.02.2017 17:00
Apparently done by Vasc? http://math.stanford.edu/~notzeb/higher.pdf
Arefe
31.05.2020 18:19
What does $\mathbb{R}_{+}^{*}$ mean ? and if $f$ is a function then what does $f'''$ mean ?
TuZo
31.05.2020 19:14
rkm0959 wrote: Apparently done by Vasc? http://math.stanford.edu/~notzeb/higher.pdf Quote: Forbidden You don't have permission to access /~notzeb/higher.pdf on this server.
Mr.C
15.08.2020 05:57
TuZo wrote: rkm0959 wrote: Apparently done by Vasc? http://math.stanford.edu/~notzeb/higher.pdf Quote: Forbidden You don't have permission to access /~notzeb/higher.pdf on this server. the link that the user gave doesnt work ... would mind posting a solution , i really want to see one since this problem is quite beautiful and verry hard
phoenixfire
15.08.2020 11:51
Arefe wrote: What does $\mathbb{R}_{+}^{*}$ mean ? and if $f$ is a function then what does $f'''$ mean ? I have the same question.
Mr.C
15.08.2020 18:12
$f'''$ means taking the derivative of $f$ 3 times for example in using jensen you have to first prove $f''(x)>0$ or $f''(x)<0$ for every $x$ here it sayd take derivative 3 times which i dknt know even what it is!!!
phoenixfire
15.08.2020 18:33
Arefe wrote: What does $\mathbb{R}_{+}^{*}$ mean ? And this?
Mr.C
15.08.2020 19:04
usually means $x\ge0$
KaiRain
16.08.2020 14:50
Mr.C wrote: the link that the user gave doesnt work ... would mind posting a solution , i really want to see one since this problem is quite beautiful and verry hard Maybe it helps you, https://artofproblemsolving.com/community/q7h549887p3203652.
Mr.C
16.08.2020 16:08
i guess we can say that KaiRain wrote: Mr.C wrote: the link that the user gave doesnt work ... would mind posting a solution , i really want to see one since this problem is quite beautiful and verry hard Maybe it helps you, https://artofproblemsolving.com/community/q7h549887p3203652. this right here , absolutly $KILLS$ the problem ...
Mikeglicker
16.08.2020 22:07
Wait, are questions with derivatives a part of olimpiad mathematics?
csav10
17.08.2020 08:43
We can prove this inequality with Vasc's EV-Theorem. Denoting $$x=a^2+2bc,\ \ \ y=b^2+2ca, \ \ \ z=c^2+2ab,$$the inequality becomes $$f(x)+f(y)+f(z)\le f(a^2+b^2+c^2)+2f(ab+bc+ca).$$Assume that $$a^2+b^2+c^2=constant,\ \ \ \ \ ab+bc+ca=constant,,$$which involve $$(a+b+c)^2=(a^2+b^2+c^2)+2(ab+bc+ca)=constant,$$therefore $$x+y+z=(a+b+c)^2=constant,$$$$x^2+y^2+z^2=(a^2+b^2+c^2)^2+2(ab+bc+ca)^2=constant.$$According to EV-Theorem, since $f'''(x)\ge 0$ for $x\in \mathbb{R}_{+}^{*}$, the sum $f(x)+f(y)+f(z)$ is maximum for $x=y\le z$, that is $$a^2+2bc=b^2+2ca\le c^2+2ab.$$From a^2+2bc=b^2+2ca, we get $a=b$ or $a+b=2c$. If $a+b=2c$, the condition $ b^2+2ca\le c^2+2ab$ is equivalent to $(b-c)^2\le 0$, which involves $b=c$. Thus it suffices to prove the requested inequality for two equal variable. The equality occurs for $a=b$ or $b=c$ or $c=a$.