The owners are **** scared of relegation. From what I've seen of Leicester they're not playing for Ranieri so as harsh as it is the owners haven't got an awful lot of choice. Ranieri is a good manager when things are going well but really struggles to turn things around when things are going badly. Last season's title win does make the sacking look ridiculous but at the same time this season's form alone justifies the sacking 100%.
I'll say it now and hope it comes true. I'd take us winning the Premier League and a relegation battle the following season, any time it's on offer. And I'd want to keep the manager too!