The paper is concerned with the problem Wi-Fi and LTE-U networks sharing access to a band of communication channels, while also considering the issue of fairness in how the channel is being shared. As a criteria of fairness for such joint access, α-fairness and maxmin fairness with regards to expected throughput are explored as fairness metrics. Optimal solutions are found in closed form, and it is shown that these solutions can be either: (a) a channel on/off strategy in which access to the channels is performed sequentially, or (b) a channel sharing strategy, i.e., where simultaneous joint access to the channels is applied. A criteria for switching between these two type of optimal strategies is found, and its robustness on the fairness coefficient is established, as well as the effectiveness of the fairness coefficient to control the underlying protocol of the joint access to the shared resource is managed. Finally, we note that the approach that is explored is general, and it might be adapted to different problems for accessing a sharing resource, like joint sharing of voice and data traffic by cellular carriers.