  We prove that, for asymptotically bounded holomorphic functions in a sector
in $\mathbb{C}$, an asymptotic expansion in a single direction towards the
vertex with constraints in terms of a logarithmically convex sequence admitting
a nonzero proximate order entails asymptotic expansion in the whole sector with
control in terms of the same sequence. This generalizes a result by A. Fruchard
and C. Zhang for Gevrey asymptotic expansions, and the proof strongly rests on
a suitably refined version of the classical Phragm\'en-Lindel\"of theorem, here
obtained for functions whose growth in a sector is specified by a nonzero
proximate order in the sense of E. Lindel\"of and G. Valiron.
