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The aim ofthis article is to improve Noether's classical inequality: K 2p^ -4, for minimal complex algebraic surfaces of general type, by using the irregularity of the surface. We show the following inequalities: * K^2p^•^^q—4}, with equality oniy for the product of two curves of genus 2. * K Ipy if the surface is irregular. They are derived from the following inequalities: K 3p^ — 2 q + 9, if the canonical System is composed ofa pencil; Ki3p,•^q—^, if the canonical map is birational (this resuit is due to F. JONGMANS. 1947); K^2py+4(q-4},ifthe canonical map is ofdegree 2, with equality oniy for the product of two curves, one of them of genus 2, the other one of genus at least 2; Ki.3py—4, if the canonical map is of degree 3 and the surface is irregular. Finally, an appendice written by A. BEAUVILLE is devoted to the proofof the inequality p^2q—4. Introduction On sait depuis M. Noether qu'une surface minimale de type general verifie l'inegalite K 2py 4; de plus les surfaces avec K = 2py 4 et K = 2pg 3 sont regulieres ([Hl] et [H2]). On doit donc s'attendre a une inegalite plus fine si l'on fait intervenir l'irregularite q de la surface. Le but de cet article est de montrer que toute surface complexe minimale de type general verifie l'inegalite :

Inégalités numériques pour les surfaces de type général. Appendice : «L'inégalité $p_g \ge 2q-4$ pour les surfaces de type général» par A. Beauville