Let $f_t$ be analytic function in the unit disk of the form $f_t(z) = \sum a_ne^{iΘ_nt}z^n,$ where $t ∈ \mathbb{R}, Θ_n ∈ \mathbb{N},$ and $h$ be positive continuous increasing to $+∞$ on $(0,1)$ function such that $\int_{r_0}^1 h(r)dr = +∞, r_0 ∈ (0,1).$ If sequence $(Θ_n)_{n≥0}$ satisfies condition $Θ_{n+1}$/$Θ_n ≥ q > 1 (n ≥ 0),$ then for any analytic function $f_t$ there exists the set $E = E(δ,t) ⊂ (0,1)$ such that $\int_E h(r)dr < +∞$ and 

$\overline{\lim\limits_{r \to 1-0, \; {r \notin E}}} = {ln M_f(r) - ln μ_f(r) \over 2ln  h(r) + lnln\{h(r)μ_f(r)\}} ≤ {1\over 4} .$

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