moldovan wrote: If $ a,b,c$ are positive real numbers, prove the inequality: $ \frac {a^2}{b} + \frac {b^3}{c^2} + \frac {c^4}{a^3} \ge - a + 2b + 3c.$ Oh, if $ a=b=c=1$, this inequality is wrong ?

moldovan wrote: If $ a,b,c$ are positive real numbers, prove the inequality: $ \frac {a^2}{b} + \frac {b^3}{c^2} + \frac {c^4}{a^3} \ge - a + 2b + 2c.$ P.S. Thank you for the observation. I've already corrected it. i think we can add 3 inequalities : $ \frac {a^2}{b} + b \ge 2a$ $ \frac {b^3}{c^2} + c + c \ge 3b$ $ \frac {c^4}{a^3} + a + a + a \ge 4c$