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Question Number 142914 by loveineq last updated on 07/Jun/21
Let a≥b≥c≥d>0 and a+b+c+d = 4.  Prove that                           (((√a)+(√b)+(√c))/3) ≤ (1/( (√d)))  Prove if ∀n∈N^+ , then                           (((a)^(1/n) +(b)^(1/n) +(c)^(1/n) )/3) ≤ (1/( (d)^(1/n) ))
$$\mathrm{Let}\:{a}\geqslant{b}\geqslant{c}\geqslant{d}>\mathrm{0}\:\mathrm{and}\:{a}+{b}+{c}+{d}\:=\:\mathrm{4}. \\ $$$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\sqrt{{a}}+\sqrt{{b}}+\sqrt{{c}}}{\mathrm{3}}\:\leqslant\:\frac{\mathrm{1}}{\:\sqrt{{d}}} \\ $$$$\mathrm{Prove}\:\mathrm{if}\:\forall{n}\in\mathbb{N}^{+} ,\:\mathrm{then} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\sqrt[{{n}}]{{a}}+\sqrt[{{n}}]{{b}}+\sqrt[{{n}}]{{c}}}{\mathrm{3}}\:\leqslant\:\frac{\mathrm{1}}{\:\sqrt[{{n}}]{{d}}} \\ $$$$ \\ $$

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