Two dimensional Berezin-Li-Yau inequalities with a correction term
  We improve the Berezin-Li-Yau inequality in dimension two by adding a positive correction term to its right-hand side. It is also shown that the asymptotical behaviour of the correction term is almost optimal. This improves a previous result by Melas. 
Hynek Kovarik, Semjon Vugalter, Timo Weidl