Why is it easier to bend a note down on a brass instrument than it is to bend up? When playing a brass instrument, on a given harmonic, you can bend the pitch above (to a higher pitch) and below (to a lower pitch) where it sounds the best (while staying on the same harmonic, with the same length of tubing, i.e not changing fingering if playing trumpet or lengthening the slide on a trombone). However, it is much easier to bend the pitch below than it is to bend above. Also, you can bend the pitch a lot lower than you can bend it higher. You can bend the pitch almost down to the pitch of the harmonic below it, but you can even get close to bending the pitch up to almost the harmonic above it. For example, if playing a middle F on a trombone, you can bend it down to almost a Bb (the note of the harmonic below the F), without moving the slide, but when bending up, you can't get even close to the next harmonic (also a Bb) above it.  
So my question is: why is it easier, and why is there more room, to bend a pitch below where it rings the best than it is to bend above?
 A: The note produced by vibrating air in the instrument. Blowing air over the player's lips is what sets up the vibration. You can do this without an instrument. 
The instrument has a resonance frequency. Vibrations at that frequency get reinforced. The oscillating pressure acts on the lips and encourages them to vibrate at the resonance frequency. This makes it easiest to play the note that matches the resonance. 
Tension of the lips plays a part. By tensing his lips, a player can hit a resonance an octave up. By relaxing them, an octave down. 
He can also use this to make his lips more prone to vibrate at a frequency that doesn't match the resonance. This is bending the note. It is easier to bend a note down because this involves relaxing the lips. 
A: There is a nice paper on this topic:
Newton et al., "Predicting the playing frequencies of brass instruments,"     September 2014, Conference: Forum Acusticum, Krakow.
Brass instruments such as trumpets and trombones are known in acoustics as "lip reeds." Helmholtz defined two types of reeds, inward-striking and outward-striking. In the idealized case, one of these can only bend its pitch down below the acoustic resonance, the other only up. Newton studied this on the trombone in three ways: using numerical simulations, humans, and an artificial player. He found that the lips have to be treated as having two different degrees of freedom, one acting like an inward-striking reed and one like an outward-striking reed. So presumably the answer to your question is that the lips act more like an inward-striking reed.
