A revised Inverse Care Law

Tudor Hart's Inverse Care Law[1] states "The availability of good medical care tends to vary inversely with the need for it in the population served. This ... operates more completely where medical care is most exposed to market forces, and less so where such exposure is reduced."
This was certainly the case in 1971, to the extent that a minister for health even as late as 2001 thanked the Asian doctors and GPs that work in so many of our deprived areas, where white doctors were unwilling to work. I believe that there is much better provision of health care services in deprived areas. Yes it's far from perfect, but we're a long way forwards from where we were in 1971.
However the poor aren't getting healthier.
There could be any number of reasons for this. Cheap food of dubious nutritional value. Lower expectation of health. Lower self-responsibility for exercise and activity. others?
A revised Inverse Care Law
I'd like to propose an alternative Inverse Care Law. Simply stated:
"Those who need most, ask least"
Those who have the greatest need of health care and support, are also the ones with the lowest expectation that they are entitled to it. The rich and middle-class make full use of the health services available, including GP, A&E, urgent care centres, hospital, NHS Direct; health activities that may not be vital for continuing life but that make them feel better. The 'worried well' and 'only slightly sick'. Whereas some in deprived areas assume that it is normal to take time off work due to sickness every month, to have irritable bowel, to be tired the whole time, to get old. They don't ask for help, because they don't believe help is available
References
[1]http://en.wikipedia.org/wiki/Inverse_care_law