I have an inequality frobenius matrix norm as follow:
$f(Y) = \|-2YA + BY + CYD\|_F$
$A,B,C,D,Y$ are all matrices, and assume all $R^{n \times n}$.
My goal is to define an upperbound for $f(Y)$, basically, I want to pull out of $Y$
I have an solution as follow $f(Y) \le \|-2YA\|_F + \|BY\|_F + \|CYD\|_F = \|2Y\|_F\|A\|_F + \|B\|_F\|Y\|_F + \|C\|_F\|Y\|_F\|D\|_F$
As we can get the following: $f(Y) \le (\|2A\|_F + \|B\|_F + \|C\|_F\|D\|_F) \times \|Y\|_F$
Am I right for all proof? if not, tell me what's wrong on the inequality chain. Thanks very much.
$f(Y) \le \|-2YA\|_F + \|BY\|_F + \|CYD\|_F \color{red}{\le} \|2Y\|_F\|A\|_F + \|B\|_F\|Y\|_F + \|C\|_F\|Y\|_F\|D\|_F$
By submultiplicative propery of frobenius norm.
In general, $\|AB\|_F \ne \|A\|_F\|B\|_F$