C182 pilot reported that after takeoff at a non-towered airport; another aircraft landed opposite direction under the pilot's aircraft.

Narrative

At approximately XA:45 local time on Day 0; I taxied to runway XX at ZZZ for a local flight. As I approached the departure end of XX; there were two aircraft that departed befor0e me on XX. I performed the actions of a preflight check list for about 3 minutes. I noted that there had been no radio traffic during the run-up. I scanned the pattern and noted no traffic. As I crossed the hold short line onto runway XX; I heard a King Air announce a straight in five mile final. I announced that I was taking runway XX with a departure to the East. As I climbed through about 500 feet above ground level; I noticed the King Air was beneath me landing on runway XY. I did not notice any evasive action taken by the King Air. I climbed normally and continued the flight. The conditions at the time were that there was a direct crosswind to XX/XY. There was a fly-in being held on the field with about 50 aircraft attending. All traffic during the day used runway XX as did all training flights being conducted. A helicopter was giving rides and using a right hand pattern for runway XX.This situation could have been avoided if I had noted that the King Air's landing announcement was to the opposite runway.