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Let $A_0 , B_0 , A_1, B_1 \geqslant 0 $ and let $0 \leqslant a,b <1 $. Then prove that $$ A_0^a B_0^b + A_1 ^a B_1 ^b \leqslant (A_0 + A_1 )^a (B_0 + B_1 )^b$$

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    except the case of $0^0$.2012-06-17

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$A_0 = A_1 = B_0 = B_1 = x$ and $a+b < 1$. Then $$A_0^a B_0^b + A_1^a B_1^b = 2x^{a+b}$$ $$(A_0 + A_1)^a (B_0 + B_1)^b = (2x)^{a+b}$$ Since $a+b < 1$, we have $$(A_0 + A_1)^a (B_0 + B_1)^b = (2x)^{a+b} < 2x^{a+b} = A_0^a B_0^b + A_1^a B_1^b$$ contradicting your claim.