This is maybe more  an open problem than  a question, since I have seriously thought about it  and  asked  several people  working on  algebraic surfaces with no success. I hope  somebody here can suggest an approach different from the standard arguments in surface theory.

BACKGROUND: let $X$ be a smooth minimal complex projective surface of general type. An  *irrational pencil* is a morphism with connected fibers $f\colon X\to B$, with $B$ a smooth curve of genus $b>0$.

For  $b>1$,  $X$ has at most finitely many pencils of genus $b$, having such a pencil is a topological property  and it is possible to bound explicitly the genus of a general fiber of $f$ in terms of $K^2_X$ (Arakelov' theorem).
 
For $b=1$, namely for *elliptic pencils*,  things are very different in general:  a surface can have infinitely many such pencils, the genus of the general fibers of these pencils can be unbounded, and  it is possible that a surface with an elliptic pencil deforms to a surface without elliptic pencils.

However, if $h^1({\mathcal O}_X)=1$, then the Albanese map $a\colon X\to Alb(X)$ is an elliptic pencil, and for fixed $K^2$ the genus of a general fiber of $a$ is bounded, since the moduli space of surfaces with fixed $K^2$ is quasiprojective.

QUESTION: can one give a bound  for the genus of the general fiber of the Albanese pencil of a minimal surface of general type $X$ with $h^1({\mathcal O}_X)=1$ in terms of $K^2_X$?
Such a bound would be very interesting in the fine classification of surfaces of general type.