Let-a-b-c-d-gt-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-

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

\frac{\sqrt{a} + \sqrt{b} + \sqrt{c}}{3} ⩽ \frac{1}{\sqrt{d}}

Prove if \forall n \in N^{+}, then

\frac{\sqrt[n]{a} + \sqrt[n]{b} + \sqrt[n]{c}}{3} ⩽ \frac{1}{\sqrt[n]{d}}