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Question Number 144901 by loveineq last updated on 30/Jun/21
Let a,b > 0 and a+b+1 = 3ab. Prove that              (a/(a^2 +1))+(b/(b^2 +1)) ≤ 1 ≤ (a^3 /(a^2 +1))+(b^3 /(b^2 +1))    Let a,b > 0, n ∈ Z^+  and a+b+1 = 3ab. Prove or disprove              (a^(n−1) /(a^n +1))+(b^(n−1) /(b^n +1)) ≤ 1 ≤ (a^(n+1) /(a^n +1))+(b^(n+1) /(b^n +1))
$$\mathrm{Let}\:{a},{b}\:>\:\mathrm{0}\:\mathrm{and}\:{a}+{b}+\mathrm{1}\:=\:\mathrm{3}{ab}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\frac{{a}}{{a}^{\mathrm{2}} +\mathrm{1}}+\frac{{b}}{{b}^{\mathrm{2}} +\mathrm{1}}\:\leqslant\:\mathrm{1}\:\leqslant\:\frac{{a}^{\mathrm{3}} }{{a}^{\mathrm{2}} +\mathrm{1}}+\frac{{b}^{\mathrm{3}} }{{b}^{\mathrm{2}} +\mathrm{1}} \\ $$$$ \\ $$$$\mathrm{Let}\:{a},{b}\:>\:\mathrm{0},\:{n}\:\in\:\mathbb{Z}^{+} \:\mathrm{and}\:{a}+{b}+\mathrm{1}\:=\:\mathrm{3}{ab}.\:\mathrm{Prove}\:\mathrm{or}\:\mathrm{disprove} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\frac{{a}^{{n}−\mathrm{1}} }{{a}^{{n}} +\mathrm{1}}+\frac{{b}^{{n}−\mathrm{1}} }{{b}^{{n}} +\mathrm{1}}\:\leqslant\:\mathrm{1}\:\leqslant\:\frac{{a}^{{n}+\mathrm{1}} }{{a}^{{n}} +\mathrm{1}}+\frac{{b}^{{n}+\mathrm{1}} }{{b}^{{n}} +\mathrm{1}} \\ $$$$ \\ $$

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