Inegalitate conditionata cu produsul 1(OWN)

[Claudiu Mindrila]

Demonstrati ca pentru orice \( a,\ b,\ c\in\left(\frac{1}{2},\ +\infty\right) \) astfel incat \( abc=1 \) are loc inegalitatea \( \frac{a^{4}}{b+c-1}+\frac{b^{4}}{c+a-1}+\frac{c^{4}}{a+b-1}\ge3. \)

Claudiu Mindrila, R. M. T. 3/2009

[opincariumihai]

\( \frac{a^4}{b+c-1}\geq 2a^2-2b-2c+2 \) si analoagele care insumate duc la
\( \frac{a^{4}}{b+c-1}+\frac{b^{4}}{c+a-1}+\frac{c^{4}}{a+b-1}
\geq 2\sum{a^2}-4\sum{a}+6 \geq\frac{2}{3} (\sum{a}-3)^2 \).

[Marius Mainea]

\( LHS\ge \frac{(a^2+b^2+c^2)^2}{2\sum ab-6}\ge \frac{3\sqrt[3]{a^2b^2c^2}(a^2+b^2+c^2)}{2\sum ab-6}\ge RHS
\)