On Fourier transforms of functions of the R. Nevanlinna class in the half-plane

Let $f$ be a function holomorphic in the upper half-plane and belonging to the Nevanlinna class $N(\mathbb{C}_+)$. Assume that

$$\limsup\limits_{y \to +\infty} \frac{\ln\vert f(iy)\vert}{y} \le 0$$

and that the boundary values of $f$ on the real axis lie in $L^1(\mathbb{R})$. It is shown that if $\vert\widehat{f}(x)\vert\le\frac{1}{\lambda(\vert x\vert)}$, $x\in{\mathbb{R}_-}$, where $\widehat{f}$ is the Fourier transform of $f$ and $\lambda$ is a logarithmically convex positive function on ${\mathbb{R}_+}$, then the condition $\int_{1}^{+\infty}\frac{\ln \lambda(x)}{x^{3/2}} dx=+\infty$ implies that $\widehat{f}(x)=0$ for all $x\in{\mathbb{R}_-}$. Conversely, if one of the conditions listed above fails, then there exists $f\in N(\mathbb{C}_+) \cap L^1(\mathbb{R})$ with $\widehat{f}(x)\ne 0$, $x\in{\mathbb{R}_-}$.