A power series ∑_k=0^∞akx ^nk with radius of convergence equal 1 is called a (p,A)-lacunary one if n_k ≥ Ak^p, A > 0, 1 < p < ∞. It is proved that if 1 < p < 2 and f(x) is a (p,A)-lacunary series that satisfies the condition |f(x)|exp ( B(1 - x)^{- 1/p - 1} + ε(1 - x) ^{-1/p - 1}/(|log (1 - x)| + 1)_x→1-0→ 0, where B = (p - 1)(π/p)^{p/p - 1}.1/A^1/(p-1).1/|cos πp/2|1/(p-1), for some ε > 0, then f ≡0. We construct a (p,A)-lacunary series f ₀ such that |f₀ (x)|exp ( B(1 - x)^-1/p-1 + C₀(1 - x)^-1/p-1/(|log(1 - x)|² + 1)) _x→1-0 → 0 with a constant C₀ = C ₀(p,A) > 0. Bibliography: 4 titles.