Inequalities #3 with the lengths of triangle sides

[Румен Симеонов]

Find [tex]m[/tex] and [tex]M[/tex], with minimal difference [tex]M-m[/tex] amongst all [tex]M-m[/tex] where
[tex]m\leqq\frac{a^3+b^3+c^3}{(a^2+b^2+c^2)(a+b+c)}\leqq M[/tex], for all [tex](a,b,c)[/tex] being ordered triples of the lengths of the ordered different sides of a nondegenerate triangle. For which such [tex](a,b,c)[/tex], the left/right inequality becomes equality?
Hint. Make use of Р. Симеонов's lemma, found in the bg section of this forum.