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This paper studies the optimum design of antennas. The objective function to be maximized is the ratio of gain to quality factor (Q) (i.e., the product of gain and bandwidth) in a specified direction. The theoretical upper bounds for the ratio of gain to Q are first rederived by using the IEEE standard definition of antenna Q. The ratio of gain to Q in a specified direction may be considered as a linear functional of the current distribution, and once it is maximized, an eigenvalue equation can be obtained from the variational principle. This eigenvalue equation can then be solved, yielding an optimum current distribution that maximizes the ratio of gain to Q in the specified direction. A number of numerical examples for small antennas have been presented to demonstrate how the theoretical upper bounds for the ratio of gain to Q can be approached by optimizing the current distributions as well as antenna geometries.