Calculate, or estimate, $\max_{t\in (a,b]\\ x,y\in\mathbb R} \left|F(x,y,t)+k\sqrt{t} \, e^{ixy}\, \operatorname{sinc}(xt) \right|$. Let $F$ be a complex-valued function. Let also $0&lt;a&lt;b$ and $k&gt;0$.

Given the 
$$\max_{t\in (a,b]\\ x,y\in\mathbb R} \left|F(x,y,t)\right|\, ,$$ 
I would like to calculate (or estimate) the following
$$\max_{t\in (a,b]\\ x,y\in\mathbb R} \left|F(x,y,t)+k\sqrt{t} \, e^{ixy}\, \operatorname{sinc}(xt) \right|\, ,$$
where, for $x\neq 0$
$$\operatorname{sinc} x = \frac{\sin x}{x}$$
and $\operatorname{sinc}(0)=1$.

My attempt is a simple estimation, where I used the well-known inequality
$$\left||z|-|w|\right|\leq |z+w|, \quad z,w\in\mathbb C\, .$$
Any suggestions, please? Are there better ways?

Thanks.
This's my best shot (but i'm not sure about the first 2 row) and in advance sorry for the bad language :
http://www.mediafire.com/view/1jiff26xniifl1t/1.jpg
This's my best shot (but i'm not sure about the first 2 row) and in advance sorry for the bad language :
http://www.mediafire.com/view/1jiff26xniifl1t/1.jpg