My question is, were horn bows used in (mainly Western) Europe in the middle ages? If so, in what period, where, by whom and how (infantry or cavalry)?
I know all sorts of people used them in the Hungarian Kingdom throughout the period: Hungarians, Cumans, Székelys etc. They were all light cavalry, horse archers.
I have seen some archers with hornbow-looking bows in the Morgan Bible, and a 14th century Italian miniature, used by western looking people. I know this alone doesn't mean anything.
So any ideas? :) Speculation is also welcome.

Reflexbogen 71-113. pp, publisher: Angelika Hörnig.

You can find it on google books. :)

https://www.academia.edu/1429936/Eastern_archery_in_Birka_s_Garrison

The uralic (finno-ugric) two-wood recurve bow was used in Northern Europe (there are finds from Russia, Finland, Sweden and Norway). It has a composite structure (two different woods glued together, bark wrapping, siyah's...) but it doesn't normally have horn parts.
It is very similar in function and looks to hornbows and is related to it. Most noticeable difference is that the two-wood bow is much longer and less recurved.

Thank you all!

Zoltán (vagy Henrik?), I'll have to refresh my German for that. :D

I read the Birka paper, though I thought most agreed,.that the items described belonged to mercenaries from the East?

Didn't know about the two-wood bows. Do you have any more info on those?

Makes sense that Scandinavia would have their own version; the horn bow certainly made it through Russia, particularly with the Mongols, but the Huns definitely had them, and they made it as far as Germany. If I was to guess, I'd say that the traditional compsite bow was certainly brought into Europe, repeatedly by many different groups, though it apparently only caught on in a limited fashion.

Since we have reliable evidence that the Huns, and the Hungarians, and the Mongols definitely used this bow, and all of them conquered, traded with, or fought with Western Europe and Russia from the fifth century right through to the 19th. There's no evidence that composite bows were manufactured in Europe, but I'm sure the bows made it through either trade or conquest, as well as people from the cultures mentioned traveling and working in Europe.

One explanation for the bow never catching on in Europe is that a horn and sinew bow is fairly tricky to store and use, and is very sensitive to humidity. For someone not brought up with the tradition of how to care for this technology, the superior performance could have been outweighed by fiddly-ness and unfamiliarity.

Scythians and the Greek used composite bows well before the Huns in Europe. Cretan composite bow archers were succesful merrcenaries even in Alexanders time. Also Romans are well known to have used composite bows everywhere in the Empire. There are Roman composite bow finds from England.

In medieval times composite bows were popular in Sicily which was for a time under muslim rule.
But the crossbow and longbow were just more popular. European crossbow prods were often composite so the technology was known and used.

Composite bows might not have been very popular for war in medieval Western Europe, but bows with a distinctly recurved look appear with some regularity in hunting scenes and these might have been of composite construction at least some of the time. And of course composite prods were fairly common for crossbows in Italy -- possibly due to the influence of Arab settlers and raiders and their descendants in Sicily and southern Italy, since some of the early Norman lords seem to have recruited archers from among their local Arab subjects.

If you have to guess a price ratio between longbow and composite bow of the same power (initial ballistic energy in joules) what should it be ?

Too many variables. When and where and in what circumstances? A composite bow might intuitively seem to be more expensive due to the multiple steps and sheer amount of labour needed just to get it into action, but a longbow made of high-quality wood might get really expensive in a place where the wood had to be imported from far away.