  In this paper we discuss the relationship between the numerical range of an
extensive class of unbounded operator functions and the joint numerical range
of the operator coefficients. Furthermore, we derive methods on how to find
estimates of the joint numerical range. Those estimates are used to obtain
explicitly computable enclosures of the numerical range of the operator
function and resolvent estimates. The enclosure and upper estimate of the norm
of the resolvent are optimal given the estimate of the joint numerical range.
