Serena Williams US Open 2016
The serves were working fine as Serena Williams eased into the second round of the US OpenReuters

Serena Williams swatted away doubts over her fitness with an authoritative performance against Ekaterina Makarova in the first round of the US Open on Tuesday, while Andy Murray showed why he is the favourite for the title with an assured win over Lukas Rosol.

With a shoulder injury thought to be putting Serena's chance at history and retaining her No.1 ranking in jeopardy, the American allayed those fears in some style, easing to a 6-3, 6-3 win over the Russian in just an hour and three minutes.

At the Rio Olympics, where she crashed out to Elina Svitolina, Serena was struggling to get her serves in properly, but at Flushing Meadows, under the New York night sky, there were no such issues, as the world number one fired in serve after serve to overwhelm Makarova, who was expected to give the American a real fight, but was never allowed into the match.

"I was pleased with my serve because I haven't been hitting a lot of serves at all," Serena told reporters after her first round win. "In practice, none of them were going in, so I was definitely excited about that.

"I knew today I needed to be focused because I've played her. She's gotten to the semifinals. She goes deep in majors. She knows how to play big matches on big courts. She's not intimidated. I knew I had to really come out today. It was my only option really."

In the other women's singles matches on Day 2, there were wins for Simona Halep, Serena's sister Venus Williams and Agnieszka Radwanska.

While Halep cruised past Kirsten Flipkens on the opening match of the day at the Arthur Ashe Stadium, Venus had to pull out all the stops to find a way past Kateryna Kozlova in the final match of the morning session on the showpiece court.

Halep, seeded five, eased past Flipkens 6-0, 6-2, while Venus had to use all her experience in a 6-2, 5-7, 6-4 win.

Radwanska, one of the contenders for the US Open title, had no worries at all, cruising past American Jessica Pegula 6-1, 6-1.

Ana Ivanovic and Eugenie Bouchard, however, did not enjoy themselves on court on Tuesday, suffering first round defeats to Denisa Allertova and Keterina Siniakova respectively.

In the men's singles, Murray had little trouble in beating Rosol, an opponent he has had a word or two with in the past. The No.2 seed is seen by many as the favourite for the title, and Murray showed no signs to suggest otherwise, easing to a 6-3, 6-2, 6-2 victory in an hour and 52 minutes.

His most likely opponent in the semifinal, Stan Wawrinka was given a difficult first round opponent in the shape of Fernando Verdasco, but the Swiss had enough firepower to dispatch the Spaniard 7-6 (7-4), 6-4, 6-4.

Another of the contenders – Kei Nishikori – though was pushed into four sets by Benjamin Becker. Seemingly in cruise mode after taking the first two sets 6-1, 6-1, Becker came back by taking the third 6-3, but the Japanese found his range again in the fourth to move into the second round.

The Rio Olympics silver medallist and former champ Juan Martin Del Potro again showed signs of his old self with a 6-4, 6-4, 7-6 (7-3) win over his Argentinian compatriot Diego Schwartzman. The man who finds himself in the news usually for all the wrong reasons – Nick Kyrgios – kept his nose and his tennis clean in a 6-4, 6-4, 6-4 victory over Aljaz Bedene.

Fellow Australian Bernard Tomic, however, could not, going down to Damir Dzumhur 6-4, 6-3, 4-6, 7-6 (7-0), while losing his cool during the match as well, with a few profanities thrown at a heckler.