1, the most easily overlooked problem before inequality problem is "when to take the equal sign". The equal sign of your question holds if and only if (a, b, c) is an arrangement of (s, s, 0), where s is an arbitrary positive number. Because the conditions in your question are positive, in a strict sense, this question is actually a strict proof of inequality.

2. Based on the above, if we relax the conditions, that is, allow non-negative numbers instead of just positive numbers, then the equal sign can be established. But if you think about Cauchy inequality, when the equal sign holds, a=b=c, and we can just arrange (A, B, C) into (S, S, 0), then Cauchy is difficult to hold, at least I didn't think of how to use it.

3. I only use a very useful theorem of IMO gold medal winner certificate (called EMV theorem) to prove this problem. Essentially, it uses an analytical method. But I think you may be interested in EMV theorem, but you may be more interested in how to prove it directly. That's why I'm looking for you at StackExchange. The link is here (/questions/864002/prove-sum-limits-mathrmyc-sqrta2bc-leq3-over2abc-with-a-b-c), and I post the answer below.

The EMV theorem I mentioned was mentioned in the link I gave you (it was a link). In addition, the person who helped to answer this question did not use Cauchy, but proved it by a method close to analysis. You can continue to pay attention to that post and see if there is a more beautiful answer.

If you don't understand English, please call me back, and I can help you translate when I have time.