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Question Number 151215 by mathdanisur last updated on 19/Aug/21

$$\mathrm{if}\mathrm{x};\mathrm{y};\mathrm{z}>0;\mathrm{x}+\mathrm{y}+\mathrm{z}=1\mathrm{and}\lambda ⩾\frac{1}{6}\mathrm{then}:$$ $$\mathit{\lambda }\mathrm{\Sigma }\frac{\mathrm{y}+\mathrm{z}}{\mathrm{x}}+3\mathrm{\Sigma }\mathrm{yz}⩾6\mathit{\lambda }+1$$

Answered by dumitrel last updated on 19/Aug/21

$$p=1$$ $$p^2⩾3q\Rightarrow q⩽\frac{1}{3}⩽2\lambda$$ $$q^2⩾3pr\Rightarrow \frac{q}{r}⩾\frac{3}{q}$$ $$\lambda \mathrm{\Sigma }(\frac{y+z}{x}+1-1)+3q⩾6\lambda +1\iff \lambda \mathrm{\Sigma }\frac{x+y+z}{x}-3\lambda +3q⩾6\lambda +1\iff$$ $$\lambda \frac{q}{r}+3q⩾9\lambda +1$$ $$\lambda \frac{q}{r}+3q⩾\lambda \frac{3}{q}+3q$$ $$prove\frac{3\lambda }{q}+3q⩾9\lambda +1\iff (3q-1)(q-3\lambda )⩾0true$$ $$$$

Commented bymathdanisur last updated on 19/Aug/21

$$\mathrm{Nice}\mathbf{S}\mathrm{er},\mathrm{Thank}\mathrm{You}$$

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