I have created a new one this is the following :
Let $a,b,c>0$ such that $a^a+b^b+c^c=3$ then we have : $$a+b+c\leq 3$$
I have tried to use Jensen's inequality like this :
$$(a+b+c)\ln(\frac{a+b+c}{3})\leq a\ln(a)+b\ln(b)+c\ln(c)\leq 3\ln(\frac{a^a+b^b+c^c}{3})$$
My question is how to prove it without using Jensen's inequality ?
Thanks a lot !