A bound for the growth in a half-strip of a function represented by a Dirichlet series

For a function defined by a Dirichlet series that converges in a right half-plane, we introduce the $R$-order $\rho$ in the half-plane and the $R$-order $\rho_s$ in a half-strip $S=\{s=\sigma+it:\sigma>0,\ |t|<a\}$. Under certain restrictions on the width of the half-strip, we obtain the inequalities $\rho_s\leqslant\rho\leqslant\rho_s+q$, where $q$ is defined by a sequence of powers. The two extreme inequalities are sharp.
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