If-a-b-c-gt-0-and-n-N-then-a-2n-b-2n-c-2n-a-n-b-n-b-n-c-n-c-n-a-n-3-a-2-b-2-c-2-a-b-c-

Tinku Tara June 4, 2023 Algebra 0 Comments

Question Number 154493 by mathdanisur last updated on 18/Sep/21

If a;b;c>0 and n∈N^+ then: ((a^(2n) + b^(2n) + c^(2n) )/(a^n b^n + b^n c^n + c^n a^n )) ≥ ((√(3∙(a^2 + b^2 + c^2 )))/(a + b + c))

$$\mathrm{If}\:\:\mathrm{a};\mathrm{b};\mathrm{c}>\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{n}\in\mathbb{N}^{+} \:\:\mathrm{then}: \\ $$$$\frac{\mathrm{a}^{\mathrm{2}\boldsymbol{\mathrm{n}}} \:+\:\mathrm{b}^{\mathrm{2}\boldsymbol{\mathrm{n}}} \:+\:\mathrm{c}^{\mathrm{2}\boldsymbol{\mathrm{n}}} }{\mathrm{a}^{\boldsymbol{\mathrm{n}}} \mathrm{b}^{\boldsymbol{\mathrm{n}}} \:+\:\mathrm{b}^{\boldsymbol{\mathrm{n}}} \mathrm{c}^{\boldsymbol{\mathrm{n}}} \:+\:\mathrm{c}^{\boldsymbol{\mathrm{n}}} \mathrm{a}^{\boldsymbol{\mathrm{n}}} }\:\geqslant\:\frac{\sqrt{\mathrm{3}\centerdot\left(\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{b}^{\mathrm{2}} \:+\:\mathrm{c}^{\mathrm{2}} \right)}}{\mathrm{a}\:+\:\mathrm{b}\:+\:\mathrm{c}} \\ $$

Facebook Tweet Pin

If-a-b-c-gt-0-and-n-N-then-a-2n-b-2n-c-2n-a-n-b-n-b-n-c-n-c-n-a-n-3-a-2-b-2-c-2-a-b-c-

Leave a Reply Cancel reply