Sixth seed Somdev defeated N Vijay Sundar Prashanth with a comfortable 6-2 6-3 scoreline, avenging his Pune Challenger defeat against local lad while fifth seed Myneni saw off the challenge of Sanam Singh 6-2 6-4.
Also moving to the next round was N Sriram Balaji, who knocked out seventh seed Hans Podlipnik-Castillo 6-2 6-4.
However, campaigns of Jeevan Nedunchezhiyan and Prajnesh Gunneswaran were over in the first round itself. Nedunchezhiyan, southpaw from Chennai, lost to second seed Briton James Ward 4-6 3-6 and Prajnesh lost tamely 3-6 0-6 to eight seed Jozef Kovalik.
"Each match at this level would be tough but I will put in my best efforts before my local fans who would be rooting for me," Ramkumar said.
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"I have played a lot of Challenger events last year and I am geared up for Chennai Open. I hope to finish around 120 rankings at the end of this season and if I get into top 100, it would be a bonus," he said.
Top seed and double defending champion Stanislas Wawrinka got a bye in the first round along with second seed Kevin Anderson.
strong favorite against her 87th-ranked compatriot King, a wild card she trounced 6-1 6-0 in their only prior meeting, in the second round of the 2014 US Open.
Sixth-seeded Venus Williams, whose seven Grand Slam titles include two US Opens, was eyeing a place in the last 32 against German Julia Goerges.
- Halep happy to be through -
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Fired-up fifth seed Simona Halep reached the third round with a 6-3 6-4 victory over Lucie Safarova in a battle of former French Open finalists.
Halep showed a flash of frustration after letting an early break get away in the opening frame, but she won the last three games to pocket the set and sealed the win when Safarova double-faulted on match point -- one of 11 breaks of serve in the contest.
"I know it's not nice," she said. "But I can't control my emotions. I'm trying, but I can't do it."
Halep called playing under the roof "amazing," although it didn't relieve all climate problems.
"I was a little cold," she said. "I felt the air conditioning."
Among those waiting out the rain, sixth-seeded Kei Nishikori of Japan had split two sets and was at 4-4 in the third against Russian qualifier Karen Khachanov, while eighth-seeded Dominic Thiem of Austria was level at 4-4 against Lithuanian Ricardas Barankais.
Women's fourth seed Agnieszka Radwanska of Poland and men's third seed Stan Wawrinka were in limbo with second-round matches scheduled on the Louis Armstrong stadium.