Entire Dirichlet series with monotonous coefficients and logarithmic h-measure

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Let F be an entire function represented by absolutely convergent for all z Dirichlet series of the form F(z) = ∑ₙ₌₀⁺∞ aₙeᶻλₙ,where a sequence (λₙ) such that λₙ(n≥0), λₙ=λₖ for any n=k and (∀ n≥ 0):0≤λₙ<β:={λⱼ:j≥0}≤ +∞. Let h be non-decrease positive continuous function on [0,+∞) and Φ increase positive continuous on [0,+∞) function. In this paper we find the condition on (μₙ) and (λₙ) such that the relation F(x+iy)=(1+o(1))aν₍ₓ, F₎e(x+iy)λν₍ₓ, F₎ holds as x→ +∞outside some set E of finite logarithmic h-measure uniformly in y.