Anett Kontaveit and Barbora Krejcikova both breezed through at the Hamburg European Open, while Yulia Putintseva eased into the Palermo Ladies Open second round.
Top seed Kontaveit overcame world number 61 Irina Bara in straight sets in her first-round clash and third favourite Krejcikova followed suit in her second-round match against Poland's Magdalena Frech.
Kontaveit will meet Sweden's Rebecca Peterson in the next round as she looks to set up a quarter-final meeting in Germany with eighth seed Andrea Petkovic, who defeated Misaki Doi 6-4 6-3 on Tuesday.
There was no such luck for ninth seed Elena Gabriela Ruse, though, after she lost 6-0 6-4 to last week's Budapest Grand Prix winner Bernarda Pera.
Swiss Joanne Zuger will next be the next challenge for Pera as she aims to reach a quarter-final clash against Katerina Siniakova, who triumphed 6-1 6-1 over Brazil's Laura Pigossi.
Putintseva is ranked as the second favourite in Italy and progressed with ease from her first-round clash against world number 147 Jaimee Fourlis, winning 6-3 6-1.
Seventh seed Anna Bondar had little difficulty beating Clara Burel 6-4 6-4, though Caroline Garcia and Nuria Parrizas Diaz faced more difficult tasks in their respective openers.
France's Garcia, the fifth favourite in Palermo, responded to a first-set scare to defeat compatriot Chloe Paquet, while eighth seed Parrizas Diaz also lost her opening set before downing Asia Muhammad.
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