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Fourth seed Martic made a good start against American opponent Alycia Parks on Wednesday, only to lose 2-6 7-6 (7-3) 6-2.
Sixth seed Sherif was more convincingly beaten as she fell to Czech teenager Linda Noskova 6-3 6-2.
Noskova will face Camila Osorio in the quarter-finals after the Colombian came out on top in a tight clash with Jule Niemeier 7-6 (7-5) 7-5
In the two remaining first-round matches, Jasmine Paolini defeated Rebeka Masarova 6-3 6-2, while Belgian Alison Van Uytvanck will face top seed Caroline Garcia in the last 16 after beating Tamara Korpatsch 6-2 6-4.
At the Thailand Open, it was not much better for seeded players as Anna Kalinskaya and Linda Fruhvirtova both exited on Thursday (AEDT).
Fourth seed Kalinskaya was overwhelmed 6-0 in the opening set against Lesia Tsurenko, only to win a tie-break in the second.
The unseeded Ukrainian ultimately won out, though, sealing a 6-0 6-7 (3-7) 6-4 victory to reach the quarter-finals.
Eighth seed Fruhvirtova also managed to level after falling a set behind, and also could not find the win as the 17-year-old was beaten by Tamara Zidansek 6-3 4-6 6-4.
The Slovenian will face Zhu Lin in the last eight after her straight-sets win over Jang Su-jeong, but there was one win for a seed as Xinyu Wang (seventh seed) beat Mirjam Bjorklund 6-3 6-3.