Well obviously if n = 1, you have an equality.
Furthermore, if n > 1, you can say without loss of generality that
[math]1 \le i < n \implies 0 < c_i \le c_{i+1}.[/math]
[math]\text {Assume } n = 2.[/math]
[math]\text {Let } p = c_2 - c_1 \implies p \ge 0 \text { and } c_2 = c_1 + p.[/math]
[math]\text {Let } q = (c_1 + c_2)^3 = (2c_1 + p)^3 = 8c_1^3 + 12c_1^2p + 6c_1p^2 + p^3.[/math]
[math]\text {Let } r = n^2(c_1^3 + c_2^3) = 4(c_1^3 + c_1^3 + 3c_1^2p + 3c_1p^2 + p^3) =\\ 8c_1^3 + 12c_1^2p + 12c_1p^2 + 4p^2.[/math]
[math]\therefore r - q = 6c_1p^2 + 3p^3 \ge 0 \ \because \ c_1 > 0 \land p \ge 0.[/math]
[math]\therefore r \ge q \implies (c_1 + c_2)^3 \le n^2(c_1^3 + c_2^3) \text { if } n = 2.[/math]
That answers your question about when can we have equality. An example is n = 2 and [imath]c_1 = c_2[/imath].
More importantly, I suspect that you can use what was just shown to prove by induction the general case and find other examples of equality.
Give it a go.
