Rodgers has masterminded two consecutive domestic trebles since joining the Glasgow giants, but publicly voiced his frustration at not adding to his squad before a 1-1 draw at home to Greek champions AEK Athens in the Champions League third qualifying round in midweek.
Celtic now have a huge task in Athens on Tuesday to keep their chances of reaching the Champions League group stages alive.
And on the domestic front Hearts gave an early indication Rodgers's men won't have it easy as they target an eighth straight Scottish title.
Kyle Lafferty's fine strike from the edge of the box 11 minutes after half-time sent Craig Levein's men to the top of the table with maximum points from their opening two games.
"Of course we want quality players but there’s still improvement in this squad, this team, lots of young players so we’ll continue to work with and improve them," said Rodgers.
"I love my life here. Despite all reports, there’s talk of ultimatums with the board, I don’t work that way, my communication is very open, so that’s certainly not the case.
"And I’d read somewhere else about walking away. I’ve got three years left on my contract here and I love every day of my life at Celtic.
"All I want is for us to improve every aspect of the club. It’s a disappointing result for us."
Defeat for Celtic was also an early season boost for Steven Gerrard's Rangers, who host St. Mirren on Sunday.
Rodgers made five changes with the visit to Edinburgh coming just three days before Celtic travel to Athens.
The visitors struggled throughout a bad-tempered encounter, but did have the best chance of the first period when Leigh Griffiths's shot was cleared off the line by Michael Smith.
Lafferty has been linked with a return to former club Rangers, and showed why Gerrard might be interested with the only goal.
The Northern Ireland international scored three times against Celtic last season and repeated the trick when he met Uche Ikpeazu's cross at the end of a powerful run to drill the ball low past the helpless Craig Gordon.
Rodgers introduced first-team regulars James Forrest, Tom Rogic and club record signing Odsonne Edouard from the bench in the final quarter in a bid to salvage a point.
Australian international Rogic came closest when he fired inches wide from the edge of the area 15 minutes from time, but Celtic quickly ran out of ideas as Hearts held out for a deserved win.