Selecting parent chain considering substituent (prefix) functional groups
According to the usual nomenclature rules, two equivalent names are possible for the compound that is given in the question:  


6-amino-4-(2-hydroxyethyl)hexanoic acid
4-(2-aminoethyl)-6-hydroxyhexanoic acid


In order to identify the preferred name, we have to dig deeper in the current version of Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book), where we finally find two potentially relevant rules:


  P-45.2.3 The preferred IUPAC name is based on the senior parent structure that has the lower locant or set of locants for substituents cited as prefixes to the parent structure (other than ‘hydro/dehydro’ prefixes) in their order of citation in the name.


and


  P-45.5.1 The preferred IUPAC name is the name that is earlier in alphanumerical order (see P-14.5). Alphabetic letters are considered first in the order that they appear in the name; all Roman letters are considered before any italic letters, unless the latter are used as locants or are a part of a compound or composite locant, for example, ‘N’ and ‘4a’. Then, if still there is a choice, numerical locants are considered in the order of their appearance in the name.


Since P-45.2.3 comes first, the preferred name is 4-(2-aminoethyl)-6-hydroxyhexanoic acid and not 6-amino-4-(2-hydroxyethyl)hexanoic acid because the locant set ‘4,6’ is lower than ‘6,4’.



By way of comparison, a similar example given in the (corrected) Blue Book is 2-(2-bromoethyl)-4-chlorobutan-1-ol and not 4-bromo-2-(2-chloroethyl)butan-1-ol since the locant set ‘2,4’ is lower than ‘4,2’.

Remarkably, also according to the subordinate rule P-45.5.1, the preferred name would be 4-(2-aminoethyl)-6-hydroxyhexanoic acid and not 6-amino-4-(2-hydroxyethyl)hexanoic acid because ‘aminoethyl-hydroxy’ is earlier alphabetically than ‘amino-hydroxyethyl’. However, the preferred name for the example given in the Blue Book would have been 4-bromo-2-(2-chloroethyl)butan-1-ol and not 2-(2-bromoethyl)-4-chlorobutan-1-ol since ‘bromo-chloroethyl’ is earlier alphabetically than ‘bromoethyl-chloro’. This example was actually given in the Blue Book by mistake but it has been corrected as part of the official Corrections to Nomenclature of Organic Chemistry. IUPAC Recommendations and Preferred Names 2013.