Starting with what we know:
[tex]a + b + c = 1[/tex] is the same as:
[tex](a + b + c)^{2} = 1^{2}[/tex]
[tex]a^{2} + b^{2} + c^{2} + 2(ab + bc + ac) = 1[/tex]
Now, let's use our inequality and bring it into our equation.
Since a > b and b >0, then a > 0
Thus, we can multiply both sides by a, b, or c since they are all positive values.
We can say: [tex]ab > b^{2}[/tex], since b > 0
Similarly, [tex]ac > c^{2}[/tex] and [tex]bc > c^{2}[/tex]
Now, we've got values for ab, ac, and bc.
Using this information back into our original equation:
[tex]a^{2} + b^{2} + c^{2} + 2(ab + bc + ac) = 1[/tex]
Since [tex]ab > c^{2}[/tex], then we can say:
[tex]a^{2} + b^{2} + c^{2} + 2(ab + bc + ac) > a^{2} + b^{2} + c^{2} + 2(b^{2} + 2c^{2})[/tex] and
[tex]a^{2} + 3b^{2} + 5c^{2} < 1[/tex] as required.
