For complex-valued functions $f\in L^1(\mathbb R^2_+)$,  where $\mathbb R_+:= [0,\infty)$ we give sufficient conditions under which the double cosine or cosine-sine Fourier transform of $f$ belongs to a generalized Lipschitz class defined by the mixed modulus of smoothness of orders $m,n\in\mathbb N=\{1,2,\cdots\}$ in uniform metric. The sharpness of these conditions is established  under some restriction for non-negative functions.