Mayer came away a 6-3, 6-4 winner over No. 7-seeded Alexandr Dolgopolov in just more than an hour. He had a two breaks in each set.
Tursunov ousted eighth-seeded Jeremy Chardy 6-5, 7-6 (7-1), taking the second-set tiebreaker after a seesaw first set in which there were five breaks.
No. 3-seeded Nicolas Almagro gained a spot in the third round with a 6-3, 6-3 decision over Tobias Kamke. Almagro dropped just eight points -- four in one game in which his serve was broken -- on serve in the match.
Also in the second round, No. 9-seeded Benoit Paire raced past Albert Montanes 6-1, 6-4, losing 10 points in nine games in which he served.
Carlos Berlocq, coming off his first WTA tournament title Sunday, out-battled Matthias Bachinger 6-7 (5-7), 6-1, 6-2 in a first-round match. Berlocq didn't lose his serve after the first set.
The formal name of the tournament is the bet-at-home Open-German Tennis Championships.