a

b

c

a^{4}+b^{4}+c^{4}\lt2(a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2})

a^{3}+b^{3}+c^{3}+2abc\lt ab(a+b)+bc(b+c)+ca(c+a)

S

p

S=\sqrt{p(p-a)(p-b)(p-c)}=\frac{1}{4}\sqrt{(a+b+c)(a+b-c)(a+c-b)(b+c-a)}.

0\lt S^{2}=\frac{((a+b)^{2}-c^{2})(c^{2}-(a-b)^{2})}{16}=\frac{(a+b)^{2}c^{2}-(a^{2}-b^{2})^{2}-c^{4}+c^{2}(a-b)^{2}}{16}=

=\frac{(a^{2}c^{2}+2abc^{2}+b^{2}c^{2})-(a^{4}-2a^{2}b^{2}+b^{4})-c^{4}+(c^{2}a^{2}-2abc^{2}+b^{2}c^{2})}{16}=

=\frac{2(a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2})-a^{4}-b^{4}-c^{4}}{16}~\Rightarrow~a^{4}+b^{4}+c^{4}\lt2(a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2}).

0\lt\frac{(a+b-c)(a+c-b)(b+c-a)}{8}=\frac{((a+b)-c)(c^{2}-(a-b)^{2})}{8}=

=\frac{((a+b)c^{2}-(a+b)(a-b)^{2}-c^{2}+c(a-b)^{2})}{8}=

=\frac{ac^{2}+bc^{2}-(a^{3}-a^{2}b-ab^{2}+b^{3})-c^{2}+a^{2}c-2abc+b^{2}c}{8}=

=\frac{ab(a+b)+bc(b+c)+ca(c+a)-a^{3}-b^{3}-c^{3}-2abc}{8}~\Rightarrow

\Rightarrow~a^{3}+b^{3}+c^{3}+2abc\lt ab(a+b)+bc(b+c)+ca(c+a).