A question by Alexei Aleksandrov and logarithmic determinants

We construct an analytic function $f$ of Smirnov's class in the unit disk such that $\mathrm{Re}\,f$ vanishes almost everywhere on the unit circle and
$$ \liminf_{t\to\infty} t\operatorname{meas}\{\zeta:\,|\zeta|=1,\ |f(\zeta)|\ge t\}=0. $$
This answers negatively to the question posed by A. Aleksandrov. We also find new sufficient conditions for representations of functions of Smirnov's class by the Schwarz and Cauchy integrals. These conditions extend previous results by Aleksandrov.