  Monotonicity properties of the ratio $$ \log \frac{f(x+a_1)\cdots
f(x+a_n)}{f(x+b_1)\cdots f(x+b_n)}, $$ where $f$ is an entire function are
investigated. Earlier results for Euler's gamma function and other entire
functions of genus 1 are generalised to entire functions of genus $p$ with
negative zeros. Derivatives of order comparable to $p$ of the expression above
are related to generalised Stieltjes functions of order $p+1$. Our results are
applied to the Barnes multiple gamma functions. We also show how recent results
on the behaviour of Euler's gamma function on vertical lines can be sharpened
and generalised to functions of higher genus. Finally a connection to the
so-called Prouhet-Tarry-Escott problem is described.
