Bhambri, seeded 15th, overcame Canada's Bradely Schuner 1-6 6-3 6-4 in the opening round and will next take on Spain's Carlos Taberner.
Ramkumar, seeded 28th, edged out American Bradley Klahn 6-7(8) 7-6(3) 6-2 in his match to set up a second round clash against French man Gleb Sakharov.
Prajnesh made a good start but lost steam as the match progressed, losing 6-1 1-6 2-6 to Germany's Tobias Kamke.
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