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In this paper we define new means of Cauchy's type using some recently obtained results that refine the Jensen-Steffensen-Boas inequality for convex and superquadratic functions (4),(5). Applying so called exp-convex method established in (8),(9), we interpret results in the form of exponentially convex or (as a special case) logarithmically convex functions. We also present some related results which generalize results in (2).

On exponential convexity, Jensen-Steffensen-Boas Inequality, and Cauchy's means for superquadratic functions