The minimum modulus of gap power series and h-measure of exceptional sets

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For entire Dirichlet series of the form F(z)=∑ₙ₌₀⁺∞ aₙeᶻλⁿ,0≤λₙ+∞(n→+∞), we establish conditions under which the relation F(x+iy)=(1+o(1))aν₍ₓ,F₎e(x+iy)λν₍ₓ,F₎ is true as x→+∞ outside some set E such that h-meas (E)=∫ₑdh(x)<+∞ uniformly in y∈R, where h(x) is positive continuous function increasing to +∞ on [0,+∞) with non-decreasing to +∞ derivative.