Positive-real logarithmic amplifier for short arcs ($\Phi (z)={\sum }_j\frac{{\delta }_j}{D}\frac{1+{\delta }_j}{1+{\delta }_j-z{e}^{-i{\theta }_j}}$)

Let finitely many arcs have centers ${\theta }_j$, half-lengths $0<{\delta }_j\le 1$ and $D={\sum }_j{\delta }_j$. Each summand has positive real part on the closed disk, and $\Phi (0)=1$. On the $j$th arc its contribution has real part at least $1/(4D)$. Thus the analytic logarithm $A=\log \Phi$ obeys $A(0)=0$, $|\mathrm{Im}A|<\pi /2$, and $\mathrm{Re}A\ge \log (1/(4D))$ throughout the arc union. Approximate $A$ by an analytic polynomial and modulate its bounded imaginary part; a partial Fourier sum isolates one half of the polynomial, producing a large value from a bounded function.