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We get a new inequality on the Hodge number $h^{1,1}(S)$ of fibred algebraiccomplex surfaces $S$, which is a generalization of an inequality of Beauville.Our inequality implies the Arakelov type inequalities due to Arakelov,Faltings, Viehweg and Zuo, respectively.

A new inequality on the Hodge number $h^{1,1}$ of algebraic surfaces