if-x-y-z-m-n-R-then-cyc-b-1-m-x-n-y-2-3-m-n-2-

Question Number 149870 by mathdanisur last updated on 07/Aug/21

if x; y; z; m; n \in R^{+} then :

\sum_{cyc} \frac{b^{- 1}}{{(m \sqrt{x} + n \sqrt{y})}^{2}} ⩾ \frac{3}{{(m + n)}^{2}}

Commented by mathdanisur last updated on 08/Aug/21

Sorry dear S er \dots

\sum_{cyc} \frac{y^{- 1}}{{(m \sqrt{x} + n \sqrt{y})}^{2}} ⩾ \frac{3}{{(m + n)}^{2}}

Commented by mathdanisur last updated on 08/Aug/21

\sum_{c y c} \frac{y^{- 1}}{{(m \sqrt{x} + n \sqrt{y})}^{2}} \overset{a m - g m}{⩾} \sum_{c y c} \frac{1}{{(\frac{m (x + y)}{2} + n y)}^{2}} = 4 \sum_{c y c} \frac{1}{{(m x + (m + 2 n) y)}^{2}}

⩾ 4 \cdot \frac{{(1 + 1 + 1)}^{3}}{(\sum_{c y c} {(m x + (m + 2 n) y)}^{2}} = \frac{27}{{(m + n)}^{2} {(x + y + z)}^{2}} (1)

3 = x^{2} + y^{2} + z^{2} ⩾ \frac{{(x + y + z)}^{2}}{3} \Leftrightarrow 9 ⩾ {(x + y + z)}^{2} (2)

(1) a n d (2) w e o b t a i n \sum_{c y c} \frac{y^{- 1}}{{(m \sqrt{x} + n \sqrt{y})}^{2}} ⩾ \frac{27}{{(n + m)}^{2} \cdot 9} = \frac{3}{{(m + n)}^{2}}

Dear Ser, wrong or true . ?

Commented by mathdanisur last updated on 08/Aug/21

Thank you very much dear \boldsymbol Ser \dots

Thank you very much dear Ser \dots