It is much simpler to slur up on a horn than to slur down.
Most non-wind and non-brass players don’t instinctively understand this, and it can lead to scoring that is much more difficult than the composer realizes. For a brass player to slur upwards, pressure must be increased and the embouchure tightened. This is fairly simple, as a natural muscular tendency combines with an increase of enforced control. However, it can be quite difficult both conceptually and physically to reverse this. The embouchure has to be loosened and pressure decreased, and that’s extremely tricky to step down into while maintaining control.
In the following sample from Richard Strauss’s Ein Heldenleben, the first horn slurs upwards over the span of two octaves plus a third – essentially, a 17-note span. Then it plays another leap of a 5th up to high F. That all seems pretty impressive, but your average first horn player can do this standing on his head. What he’s really dreading is that downward slur from high F to middle D. That is, to put it in musical terms, a bloody nuisance. It’s fast, it’s exposed, and it’s very hard to negotiate clearly. If the player isn’t giving it his complete attention, he will choke it miserably. But perhaps that was Strauss’s intention – he is, after all, writing a tone poem about a “hero’s life” – which one might well assume would require extraordinary effort from the musicians to portray effectively.
A couple of notes here for score-readers: this part is written in “old notation” – so while the treble staff is transposing down a 5th, the bass staff is transposing up a 4th. Notes look ridiculously lower in bass staff, and are an octave lower than they’re written on the treble staff, which is why we all use “new notation” now.
The other thing is that a low Bb in old notation is concert Eb a 4th higher – a very low note indeed for a solo horn, which is why Strauss makes the note more secure with the addition of the second player.