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Let S be a smooth minimal complex projective surface of maximal Albanesedimension. Under the assumption that the canonical class of S is ample and theirregularity of S, q(S), is greater or equal to 5 we show that  K^2>= 4\chi(S)+(10/3)q(S)-8, thus improving the well known Severi inequalityK^2>=4\chi(S). We also give stronger inequalities under extra assumptions onthe Albanese map or on the canonical map of S.

Severi type inequalities for irregular surfaces with ample canonical class