lean_workbook_plus_41244

Wlog assume $a \ge b \ge c$ . Consider sum $f(a,b,c)={1 \over a}+{1 \over b}+{1 \over c}+{3 \over a+b+c}\ge{8 \over 3}\left({1 \over a+b}+{1 \over b+c}+{1 \over c+a}\right)$ , $f(a,b,c)-f\left(a,{b+c \over 2},{b+c \over 2}\right)=(b-c)^{2}\left[{1 \over bc(b+c)}-{8 \over 3}\cdot{1 \over (a+b)(a+c)(2a+b+c)}\right]$ Clearing denominators, $S_{0}=6a^{3}+12abc+9a^{2}b+3ab^{2}+9a^{2}c+3ac^{2}-5bc(b+c)$ .