Very recently a new data structure, called a min-max heap, was presented for implementing the double-ended priority queue. A min-max heap on n keys is constructed in O(n) time; the minimum and maximum keys are found in constant time, and the operations of deleting the minimum, deleting the maximum and inserting a new key into the heap are performed in O(log n) time. In addition, the data structure can be stored implicitly, i.e. in an array of n elements without using any additional pointers. In this paper, we present lower bound results on the number of comparisons required, in the worst case, for the operations i) to construct a min-max heap on a given set of keys; ii) to convert a min-max heap into a max-min heap; and iii) to merge two min-max heaps into one min-max heap. New upper bounds for the convert and merge operations are also derived. It is found that the main difference between traditional heaps and min-max heaps lies in the time needed to perform the merge operation. While traditional heaps can be merged efficiently, it is shown that min-max heaps are not sublinearly mergeable. Even the seemingly simple task of converting a min-max heap into a max-min heap cannot be performed in less than linear time.