Proof of inequality 1. If a > 0, b > 0 and a + B = 1, verify 1 / (a + 1) + 1 / (B + 1) 0, b > 0, verify: (a ^ 2 + B ^ 2) / √ ab ≥ a + B

1) a>0,b>0 ab>0 1/(a +1) +1/(b+ 1)=(a+b+2)/(ab+a+b+1)=3/(2+ab)0,b>0 a+b>2√ab, (a^2+b^2)/(a+b)=a+b-2ab/a+b≥2√ab-2ab/(2√ab)=√ab (a^2+b^2)/√ab≥a+b

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