disuguaglianza giornalino 22

[bruno222]

$ \ \frac{1} { a^2+1} +\frac{1}{ b^2+1}+\frac{1}{c^2+1}+\frac{1}{d^2+1}\geq \frac{1}{ab+1}+\frac{1}{bc+1}+\frac{1}{cd+1}+\frac{1}{da+1} $


con $ \ a,b,c,d \geq 1 $

[Hammond]

$ \displaystyle \frac{1}{ a^2+1} + \frac{1}{ b^2+1} \geq \frac{2}{ab+1} $

$ (ab+1)(a^2+b^2+2)\ge2(a^2+1)(b^2+1) $

$ a^3b+b^3a+a^2+b^2+2ab+2\ge2a^2b^2+2a^2+2b^2+2 $

$ ab{(a-b)}^2\ge{(a-b)}^2 $