On the class of almost subadditive weight functions

We answer a question from A. V. Abanin and P. T. Tien about so-called almost subadditive weight functions in the sense of Braun-Meise-Taylor. Using recent knowledge of a growth index for functions, crucially appearing in the ultraholomorphic setting, we are able to show the existence of weights such that there does not exist an equivalent almost subadditive function and hence almost subadditive weights are a proper subclass.