I know some of it, but I would like to learn other ones. I think this question is useful, as it will be helpful for other beginners. Such as: uvw, AM GM, Cauchy–Schwarz inequality, Jensen inequality. Could you please add something? Maybe books containing it?

I know some of it, but I would like to learn other ones. I think this question is useful, as it will be helpful for other beginners. Such as: uvw, AM GM, Cauchy–Schwarz inequality, Jensen inequality. Could you please add something? Maybe books containing it?

let $a,b,c$ are real numbers,and such $a+b+c=3$,show that $$\dfrac{1}{2a^2-6a+9}+\dfrac{1}{2b^2-6b+9}+\dfrac{1}{2c^2-6c+9}\le\dfrac{3} {5}\cdots (1)$$ I find sometimes,and I find this same problem: let $a,b,c$ are real numbers,and such $a+b+c=3$,show that $$ \frac{1}{5a^2-4a+11}+\frac{1}{5b^2-4b...

I'm looking for some ideas to deal with my inequality. P/s: I'd like to create simple look inequalities so we can call them as artificial inequalities. I agree that my inequality may be wrong but I checked it carefully before posting here. Problem. Let $a,b,c\ge 0: a+b+c=4.$ Prove that$$\color{b...

The Problem. Let $a,b,c$ be non negative real numbers such that $a+b+c=1$, show that the follwoing inequality holds true $$ \frac{2a}{-6a^2+5a+3}+\frac{2b}{-6b^2+5b+3}+\frac{2c}{-6c^2+5c+3}\geq \frac{1}{2}. $$ My Attempt(s) I first approached the problem simply by using Cauchy-Schwartz inequalit...