The Hodge number $h^{1,1}$ of irregular algebraic surfaces

We prove a new inequality for the Hodge number h1,1 of irregular complex smooth projective surfaces of general type without irrational pencils of genus ≥ 2. More specifically we show that if the irregularity q satisfies q=2k+1 then h1,1≥4q−3 . This generalizes results previously known for q=3 and q=5 .