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As far as I know there is no exact relation between harmonic distortion and DC gain. Because harmonic distortion depends on several parameters. However as wonbef said, usually more DC-gain results in less harmonic distortion. Theoretically, one can express hamonic distortion factors by device paramters in an opamp. However, these expressions differ from one topology to another.
Professor Sansen from K. Leuven University in Belgium and Profs. Palumbo and Pennisi from Catania University in Italy have several years of experience on this field of study. For more reading you can refer to "Feedback Amplifier" written by Profs. Palumbo and Pennisi. There are some intuitve expressions extracted for some well-known opamp topologies based on low-frequency device parameters such as transconductance (gm).