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|ca-ab| + ... <= |b^2-c^2| + ...

Source: 4th QEDMO, created by myself but probably known

March 9, 2007
inequalitiestriangle inequalityinequalities proposed

Problem Statement

For any three nonnegative reals aa, bb, cc, prove that
∣ca−ab∣+∣ab−bc∣+∣bc−ca∣≤∣b2−c2∣+∣c2−a2∣+∣a2−b2∣\left|ca-ab\right|+\left|ab-bc\right|+\left|bc-ca\right|\leq\left|b^{2}-c^{2}\right|+\left|c^{2}-a^{2}\right|+\left|a^{2}-b^{2}\right|.
Generalization. For any nn nonnegative reals a1a_{1}, a2a_{2}, ..., ana_{n}, prove that
∑i=1n∣ai−1ai−aiai+1∣≤∑i=1n∣ai2−ai+12∣\sum_{i=1}^{n}\left|a_{i-1}a_{i}-a_{i}a_{i+1}\right|\leq\sum_{i=1}^{n}\left|a_{i}^{2}-a_{i+1}^{2}\right|.
Here, the indices are cyclic modulo nn; this means that we set a0=ana_{0}=a_{n} and an+1=a1a_{n+1}=a_{1}.
darij
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