Generalization of Cauchy Inequality

By applying Chebeshev's inequality $3(\frac{a}{b+c}a^{2016}+\frac{b}{c+a}b^{2016}+\frac{c}{a+b}c^{2016})\geq (\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b})(a^{2016}+b^{2016}+c^{2016})$ But by our classic AM-HM problem ,we know that, $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\geq \frac{3}{2}$ Thus we get, $3(\frac{a}{b+c}a^{2016}+\frac{b}{c+a}b^{2016}+\frac{c}{a+b}c^{2016})\geq\frac{3}{2} (a^{2016}+b^{2016}+c^{2016})$ Finally we get, $(a^{2017}+b^{2017}+c^{2017})\geq \frac{1}{2} (a^{2016}+b^{2016}+c^{2016})$ Therefore K=2016