Endogenous Gentrification and Rates of Return to Education
@denesp comment is right to the point as regards the scope of this question. In street-talk "Gentrification" just means "things get more expensive" ("rightly" or "wrongly", that's another matter). The displacement of poor people will happen, for those who remain poor, where here "poor" is a relative concept "poor enough not to be able to afford the new cost levels".  

What I believe is interesting here is to ask :"Assume that the amenities of an area increase in value, and so rents and "cost of living" goes up. If this value-increase of amenities is also increasing the productivity/efficiency/earnings of the local population, which effect will dominate?" 

Will the income-increase of local residents be enough to cover the increased costs of living? 

Instinctively speaking, I would say "no", if the productivity-increase comes through education, because the phenomenon of price-increases in such areas happens much faster than the time needed for education to bear income fruits.