Real part of cycle integrals and conjectures of Kaneko

We prove two of Kaneko’s conjectures on the “values” $\mathrm{v}\mathrm{a}\mathrm{l}(w)$ of the modular $j$ function at real quadratic irrationalities: we prove the lower bound $\mathrm{R}\mathrm{e}(\mathrm{v}\mathrm{a}\mathrm{l}(w))\ge \mathrm{v}\mathrm{a}\mathrm{l}\left(\frac{1+\sqrt{5}}{2}\right)$ for all real quadratics $w$ and the upper bound $\mathrm{R}\mathrm{e}(\mathrm{v}\mathrm{a}\mathrm{l}(w))\le \mathrm{v}\mathrm{a}\mathrm{l}(1+\sqrt{2})$ for all Markov irrationalities $w$. These results generalize to the “values” at quadratic irrationalities of any weakly holomorphic modular function $f$ such that $f(e^{it})$ is real, non-negative and increasing for $t\in [\pi \mathrm{/}3,\pi \mathrm{/}2]$.