Minimal Fourier majorants in $L^p$

Denote the coefficients in the complex form of the Fourier series of a function $f$ on the interval $[-\pi, \pi)$ by $\hat f(n)$. It is known that if $p = 2j/(2j-1)$ for some integer $j>0$, then for each function $f$ in $L^p$ there exists another function $F$ in $L^p$ that majorizes $f$ in the sense that $\hat F(n) \ge |\hat f(n)|$ for all $n$, and for which $\|F\|_p \le \|f\|_p$. When $j > 1$, the existence proofs for such small majorants do not provide constructions of them, but there is a unique majorant of minimal $L^p$ norm. We modify previous existence proofs to say more about the form of that majorant.