We give lower bounds on the number of effective divisors of degree $\le g-1$ with respect to the number of places of certain degrees of an algebraic function field of genus $g$ defined over a finite field. We deduce lower bounds for the class number which improve the Lachaud - Martin-Deschamps bounds and asymptotically reaches the Tsfasman-Vladut bounds. We give examples of towers of algebraic function fields having a large class number.