The zeta-invariant inequalities for smooth real functions on the circle

Let $ain C^\infty({\mathbb S})$ be a real-valued smooth function, and let $Z_k(a)$ denote the zeta-invariants defined by the Fourier-coefficient formulas above. For $k\geq 2$, the conjectured inequalities are the inequalities denoted by (3.1) in the source.

Zeta-invariant inequalities. Inequalities (3.1) hold for every real function $a\in C^\infty({\mathbb S})$.

These inequalities generalize the explicitly known formula for $Z_1(a)$ and are supported by extensive numerical experiments. For $k\geq 2$, the conjecture remains open.