Jansen, 28, converted 36 of his 38 save opportunities last year and went 2-1 with a 2.41 ERA in 54 games over 52 1/3 innings. He has 142 saves since debuting for the Dodgers in 2010. This was Jansen's final year of arbitration eligibility.
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Turner, 31, hit .294 with a .370 on-base percentage and a .491 slugging percentage in 439 plate appearances in 2015. He signed as a free agent before the '14 season after playing previously with the Orioles and Mets.
Grandal, 27, made his first All-Star Game appearance in 2015, and he finished the season with a .234/.353/.403 slash line and a career-best 16 home runs in 426 plate appearances.
Avilan, 26, went a combined 2-5 with a 4.05 ERA in 73 games with the Braves and the Dodgers. After joining Los Angeles as part of a three-team deal with Atlanta and Miami, Avilan went 0-1 with a 5.17 ERA in 23 games over 15 2/3 innings.
This week, the Dodgers also signed lefty reliever Chris Hatcher ($1.065 million) and outfielder Scott Van Slyke ($1.225 million) to one-year contracts.
Thomas Harding is a reporter for MLB.com. Follow him on Twitter @harding_at_mlb, listen to podcasts and like his Facebook page. This story was not subject to the approval of Major League Baseball or its clubs.