Due to the performance capability I was catching up to Aircraft Y on the climb. Since I was catching up with Aircraft Y I called on the radio to the aircraft that I was going to side step to the right to provide room and continue my climb beyond 2,300 feet before making my left crosswind departure. My left crosswind turn was initiated at 3,500 feet. I proceeded to head to my practice area and after I was done I came back to do three takeoffs and landings. The potential issue, which was brought to my attention, was that the Aircraft Y pilot felt that we had a “near miss.” As a pilot in command I operated the aircraft safely providing separation of 500 feet of more and communicating with the pilot on the radio with my location and intentions.
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