lean_workbook_plus_32437

Prove or disprove the following inequalities for positive numbers a, b, and c:\n\ni) $\dfrac{{{{\left( {a - b} \right)}^4}}}{{{a^2}{b^2}}} + \dfrac{{{{\left( {b - c} \right)}^4}}}{{{b^2}{c^2}}} + \dfrac{{{{\left( {c - a} \right)}^4}}}{{{c^2}{a^2}}} \ge \dfrac{1}{2}\cdot\left[\dfrac{{{{\left( {a - b} \right)}^2}}}{{ab}} + \dfrac{{{{\left( {b - c} \right)}^2}}}{{bc}} + \dfrac{{{{\left( {c - a} \right)}^2}}}{{ca}} \right]^2.$\nii) $\dfrac{{{{\left( {a - b} \right)}^4}}}{{{a^2}{b^2}}} + \dfrac{{{{\left( {b - c} \right)}^4}}}{{{b^2}{c^2}}} + \dfrac{{{{\left( {c - a} \right)}^4}}}{{{c^2}{a^2}}} \ge \dfrac{1}{2}\cdot\left[\dfrac{{{{\left( {a - b} \right)}^2}}}{{ab}} + \dfrac{{{{\left( {b - c} \right)}^2}}}{{bc}} + \dfrac{{{{\left( {c - a} \right)}^2}}}{{ca}} \right].$