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

For entire Dirichlet series of the form $F(z)=\sum\limits_{n=0}^{+\infty} a_{n}e^{z\lambda_n},\ 0\le\lambda_n\uparrow+\infty\ (n\to+\infty)$, we establish conditions under which the relation $$F(x+iy)=(1+o(1))a_{\nu (x,F)}{e}^{(x+iy){\lambda }_{\nu (x,F)}}$$ is true as $x\to +\mathrm{\infty }$ outside some set $E$ such that $\text{ h-meas }(E)=\int_{E}dh(x)<+\infty$uniformlyin$y\in\Bbb{R}$,where$h(x)$ is positive continuous function increasing to $+\mathrm{\infty }$ on $[0,+\mathrm{\infty })$ with non-decreasing to $+\mathrm{\infty }$ derivative.