In this paper we introduce new refinements of both the discrete and the
classical Jensen’s inequality. First, we give the weighted version of a
recent cyclic refinement. By using this result, we obtain new refinements of
the classical Jensen’s inequality. We investigate m-exponential convexity of some
functionals coming from the new refinements. To apply our
results we define some new mixed symmetric means, generalized means, and
Cauchy means, and study their properties.
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