A convolution inequality and the Krzyż conjecture

Using a sharp bound for convolutions of non-negative trigonometric polynomials with the Fejér kernel resembling a pointwise variant of Bernstein's inequality for polynomials, we prove new properties of extremal functions for the Krzyż conjecture. In particular, we resolve a 1993 conjecture of Peretz by showing that if $f$ is extremal for some coefficient, then $|f(0)|\leq 1/e$. Furthermore, we show that if equality holds in this estimate for every extremal function, then the Krzyż conjecture is true.