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

if x;y;z;m;n∈R^+ then: Σ_(cyc) (b^(−1) /((m(√x) + n(√y))^2 )) ≥ (3/((m + n)^2 ))

Commented bymathdanisur last updated on 08/Aug/21

Sorry dear Ser... Σ_(cyc) (y^(−1) /((m(√x) + n(√y))^2 )) ≥ (3/((m + n)^2 ))

Commented bymathdanisur last updated on 08/Aug/21

Σ_(cyc) (y^(−1) /((m(√x)+n(√y))^2 ))≥^(am-gm) Σ_(cyc) (1/((((m(x+y))/2)+ny)^2 )) = 4Σ_(cyc) (1/((mx+(m+2n)y)^2 )) ≥ 4∙(((1+1+1)^3 )/((Σ_(cyc) (mx+(m+2n)y)^2 )) = ((27)/((m+n)^2 (x+y+z)^2 )) (1) 3=x^2 +y^2 +z^2 ≥(((x+y+z)^2 )/3) ⇔ 9≥(x+y+z)^2 (2) (1) and (2) we obtain Σ_(cyc) (y^(−1) /((m(√x)+n(√y))^2 ))≥((27)/((n+m)^2 ∙9))=(3/((m+n)^2 )) Dear Ser, wrong or true.?

Commented bymathdanisur last updated on 08/Aug/21

Thank you very much dear Ser...