This paper develops systematic approaches to obtain f -divergence inequalities, dealing with pairs of probability measures defined on arbitrary alphabets. Functional domination is one such approach, where special emphasis is placed on finding the best possible constant upper bounding a ratio of f -divergences. Another approach used for the derivation of bounds among f -divergences relies on moment inequalities and the logarithmic-convexity property, which results in tight bounds on the relative entropy and Bhattacharyya distance in terms of χ 2 divergences. A rich variety of bounds are shown to hold under boundedness assumptions on the relative information. Special attention is devoted to the total variation distance and its relation to the relative information and relative entropy, including "reverse Pinsker inequalities," as well as on the E γ divergence, which generalizes the total variation distance. Pinsker's inequality is extended for this type of f -divergence, a result which leads to an inequality linking the relative entropy and relative information spectrum. Integral expressions of the Rényi divergence in terms of the relative information spectrum are derived, leading to bounds on the Rényi divergence in terms of either the variational distance or relative entropy.

The complete characterization of the capacity region of a two-user Gaussian interference channel is still an open problem unless the interference is strong. In this work, we derive an achievable rate region for this channel. It includes the rate region which is achieved by time/ frequency division multiplexing (TDM/ FDM), and it also includes the rate region which is obtained by time-sharing between the two rate pairs where one of the transmitters sends its data reliably at the maximal possible rate (i.e., the maximal rate it can achieve in the absence of interference), and the other transmitter decreases its data rate to the point where both receivers can reliably decode its message. The suggested rate region is easily calculable, though it is a particular case of the celebrated achievable rate region of Han and Kobayashi whose calculation is in general prohibitively complex. In the high power regime, a lower bound on the sum-rate capacity (i.e., maximal achievable total rate) is derived, and we show its superiority over the maximal total rate which is achieved by the TDM/ FDM approach for the case of moderate interference. For degraded and one-sided Gaussian interference channels, we rely on some observations of Costa and Sato, and obtain their sum-rate capacities. We conclude our discussion by pointing out two interesting open problems.

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