if-x-y-z-gt-0-such-that-x-y-z-3-and-0-then-prove-that-x-y-3-y-2-y-z-3-z-2-z-x-3-x-2-3-1-

Question Number 161433 by HongKing last updated on 17/Dec/21

if x;y;z>0 such that x+y+z=3 and 𝛌≥0 then prove that: (x/(y^3 +λy^2 )) + (y/(z^3 +λz^2 )) + (z/(x^3 +λx^2 )) ≥ (3/(λ+1))

$$\mathrm{if}\:\:\mathrm{x};\mathrm{y};\mathrm{z}>\mathrm{0}\:\:\mathrm{such}\:\mathrm{that}\:\:\mathrm{x}+\mathrm{y}+\mathrm{z}=\mathrm{3} \\ $$$$\mathrm{and}\:\:\boldsymbol{\lambda}\geqslant\mathrm{0}\:\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{x}}{\mathrm{y}^{\mathrm{3}} +\lambda\mathrm{y}^{\mathrm{2}} }\:+\:\frac{\mathrm{y}}{\mathrm{z}^{\mathrm{3}} +\lambda\mathrm{z}^{\mathrm{2}} }\:+\:\frac{\mathrm{z}}{\mathrm{x}^{\mathrm{3}} +\lambda\mathrm{x}^{\mathrm{2}} }\:\geqslant\:\frac{\mathrm{3}}{\lambda+\mathrm{1}} \\ $$

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