Around the Fej\'er-Jackson inequality: Tight bounds for certain oscillatory functions via Laplace transform representations

The error of approximation of the $2\pi$-periodic sawtooth function $(\pi-x)/2$, $0\leq x<2\pi$, by its $n$-th Fourier polynomial is shown to be bounded by arccot$((2n+1)\sin(x/2))$. Related asymptotically tight inequalities with explicit constants are given for the integral of the Dirichlet kernel interpolated to non-integer values of frequency parameter and for the Taylor series remainder of the logarithmic function $\log(1-z)$ in the unit circle. The proofs are based on the Laplace transform representation of the Lerch Zeta function with $s=1$.