[Anonymous314]

ผมได้ว่า
$$\max(k)=\log_23-1$$
Hint:
\[{\left( {\frac{a}{{b + c}}} \right)^k} + {\left( {\frac{b}{{c + a}}} \right)^k} + {\left( {\frac{c}{{a + b}}} \right)^k} \ge \min \left( {2,\frac{3}{{{2^k}}}} \right)\]
for $k \in (0,1)$