Lets first understand the hysteresis in the LTE -
Hysteresis: There can be a delay between action and reaction of a measurement and handover takes place. So there is a small error included in the measurement reporting to minimise the effect of the delay. In short measured neighbor look worse than measured to make sure it is really stronger before the UE decides to send a measurement report to initiate a handover.
Check the following http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6482433
Offset: Each UE evaluates the Event A3 condition every time a new averaged measurement sample is available. The evaluated condition is the entering condition of Event A3: whether the RSRP measured from a neighbouring cell becomes an offset better than the RSRP measured from the serving cell.
Now lets see the A3 Event
Event A3 is triggered when a neighbouring cell becomes better than the serving cell by an offset. The offset can be either positive or negative. The event is triggered when the following condition is true:
MEAS(neigh) + O(neigh,freq) + O(neigh,cell) - Hyst > MEAS(serv) + O(serv,freq) + O(serv,cell) + Offset
MEAS(neigh) + O(neigh,freq) + O(neigh,cell) + Hyst < MEAS(serv) + O(serv,freq) + O(serv,cell) + Offset