I would like to say
Please feel free to contact me.
I know it is probably
気軽に連絡してください。
But will it be wrong to use 気楽に in this case?
気楽に連絡してください。
What is the difference in nuance between 気軽 and 気楽?
Japanese Language Stack Exchange is a question and answer site for students, teachers, and linguists wanting to discuss the finer points of the Japanese language. It only takes a minute to sign up.
Sign up to join this community「気軽{きがる}に連絡{れんらく}してください。」
is just a natural, everyday phrase that "everyone" uses very often. I do not think anyone would argue about that.
Today, when I saw the following sentence in your question,
「気楽{きらく}に連絡してください。」
I had to stop and think about it, which is more often than not a bad sign. Would I ever say it? No, I would not;That I know for sure. Would anyone I personally know say it? No, I do not think so and that is why I had to stop and really think. So, would anyone ever say it? I suppose, yes, (even though I know I will not follow suit.)
While I might not call the second sentence using 「気楽」 "incorrect", I will at least call it "pretty strange".
「気軽に + Verb describing a specific action」
is a common and valid usage expressing the idea of performing a specific action without hesitation, formality, pressure, etc. 「連絡する (to contact a person)」 is a specific action requiring making a phone call, writing an e-mail or letter, etc.
Many of us would, however, not often express these ideas using 「気楽に」. When we use 「気楽に」, we usually refer to "spending time or living with a relaxed state of mind, easy manners", etc. These are not specific actions.
"Make yourself at home.", "Take it easy.", "Sit back and relax.", etc. are the ideas behind the expressions such as 「気楽にする」、「気楽に過{す}ごす」、「気楽に生{い}きる」, etc. Again, these are different from performing specific actions.
It's not really appropriate to use 気楽 here. It implies one is unguarded and sounds not necessarily constructive or positive unless it refers to kinds of leisure.
気軽's translations: freely, readily, cheerfully.
気楽's translation: comfortably.
「気軽」は、ある行動をするときに、負担や障害が少ないということです。常に行動を伴います。
気軽 is when doing something, the burden is low. Always involving acts.
「気楽」は気分が楽な状態です。つまり、心配や心の負担がない状態です。行動を伴う必要はありません。
気楽 is when feeling good. Anyway, thn mental burden is low. It does not need involving acts.
何かあったら連絡してください。
instead?