Question

已知abcd均为正实数,且a+b+c+d=1,求证：a^/(1+a)+b^2/(1+b)+c^2/(1+c)+d^2/(

Answer 1

证明:

a、b、c、d是正数,且a+b+c+d=1,

故依Cauchy不等式得

[(a+1)+(b+1)+(c+1)+(d+1)][a^2/(a+1)+b^2/(b+1)+c^2/(c+1)+d^2/(d+1)]≥(a+b+c+d)^2

即5[a^2/(a+1)+b^2/(b+1)+c^2/(c+1)+d^2/(d+1)]≥1^2

∴a^2/(a+1)+b^2/(b+1)+c^2/(c+1)+d^2/(d+1)≥1/5.

证毕.