Defending champions Andy Murray, Serena Williams coast at U.S. Open

(CNN) -- Andy Murray became the first member of the so-called Big Four in men's tennis to flinch at the U.S. Open. He lost a set to Argentina's Leonardo Mayer on Friday.

But the defending men's champion in New York had an easier time in sweltering conditions Sunday, dispatching German Florian Mayer 7-6 6-2 6-2 to move into the fourth round.

Women's defending champion Serena Williams progressed, too, defeating Sloane Stephens 6-4 6-1 in a rematch of their thriller at the Australian Open.

Even though Murray's serve wasn't fully firing, he didn't get broken and struck 42 winners -- double his unforced error count.

It was especially impressive given Mayer's ability to throw off opponents with his change of pace and quirky forehand.

Mayer, ranked as high as 18th and now the world No. 47, needed to claim the opening set to have a chance of upsetting the Scot.

Once Murray took it in a tiebreak, and convincingly, Mayer's shoulders slumped and he proceeded to win four games the rest of the way.

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Murray was happy he didn't have to spend too long on court -- under two hours -- in the draining conditions.

"It cooled down a little bit towards the end but the first and second sets it was very hot," he said in an interview on court afterwards.

"It was very tough conditions and he's a very tricky opponent," Murray continued. "He plays strange shots and he takes your time away. It's tough to get into a rhythm.

"I'm really glad to come through in three sets."

Murray's next foe is Uzbekistan's Denis Istomin, who topped steady Italian Andreas Seppi in five sets.

Looking further ahead, Murray could meet the dangerous Tomas Berdych in the quarterfinals and the Czech had an even easier time than Murray on Sunday.

Berdych conceded a mere five games to Frenchman Julien Benneteau.

The 31-year-old Williams, bidding to become the oldest U.S. Open women's winner in the Open era, captured eight of the last nine games to see off Stephens and book a spot in the quarterfinals.

Stephens sent Williams packing in the quarterfinals at the Australian Open, although the latter was injured in the match, and some regard the 20-year-old as the future of American tennis.

The pair were friends, fell out when Stephens criticized Williams but are now back on good terms.

Williams next plays Spaniard Carla Suarez Navarro, who upset Germany's Angelique Kerber in a third-set tiebreak.