Malesani had replaced Gasperini just three games ago, but drew all of his matches in charge.
Gasperini himself was brought in to replace Guiseppe Sannino at the start of the season at Palermo, who are bottom of Serie A.
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Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.
Newcastle United are staring down the barrel of relegation to the Championship for the second time in the past eight years.