Using Cauchy-Schwarz Inequality twice:

\(\displaystyle{a}^{{4}}+{b}^{{4}}+{c}^{{4}}\geq{a}^{{2}}{b}^{{2}}+{b}^{{2}}{c}^{{2}}+{c}^{{2}}{a}^{{2}}\geq{a}{b}^{{2}}{c}+{b}{a}^{{2}}{c}+{a}{c}^{{2}}{b}={a}{b}{c}{\left({a}+{b}+{c}\right)}\)

\(\displaystyle{a}^{{4}}+{b}^{{4}}+{c}^{{4}}\geq{a}^{{2}}{b}^{{2}}+{b}^{{2}}{c}^{{2}}+{c}^{{2}}{a}^{{2}}\geq{a}{b}^{{2}}{c}+{b}{a}^{{2}}{c}+{a}{c}^{{2}}{b}={a}{b}{c}{\left({a}+{b}+{c}\right)}\)