a, b, c > 0 に対して、

(1) 3 + √{(a^2 + b^2 + c^2)(1/a^2 + 1/b^2 + 1/c^2)} ≧ (2/3)(a+b+c)(1/a + 1/b + 1/c)

(2) √{(a^4 + b^4 + c^4)(1/a^4 + 1/b^4 + 1/c^4)} ≧ 1 + √[1 + √{(a^5 + b^5 + c^5)(1/a^5 + 1/b^5 + 1/c^5)}]

(3) a^4/(a^3 + b^3) + b^4/(b^3 + c^3) + c^4/(c^3 + a^3) ≧ (a+b+c)/3

(4) {(a-b)/c}^2 + {(b-c)/a}^2 + {(c-a)/b}^2 ≧ (2√2)*{(a-b)/c + (b-c)/a + (c-a)/b}

(5) a/{√(2b^2+2c^2)} + b/(c+a) + c/(a+b) ≧ 3/2

(6) a+b+c=3 のとき、44 ≧ (a^2+2)(b^2+2)(c^2+2) ≧ 27



参考 (2) https://artofproblemsolving.com/community/q1h1328831p7152622