Let X be a normed space, V be an open convex subset of X and let Q : ½0; 1Þ ! ½0; 1Þ be a given function.By the result of Tabor and Tabor, we know that under respective conditions on Q, if f is Q-midconvex and locally bounded above at a point then there exists a continuous function w : ½0; 1 ! R such that f ðtx þ ð1 2 tÞyÞ # tf ðxÞ þ ð1 2 tÞf ðyÞ þ wðtÞQðkx 2 yjkÞ for x; y [ V; t [ ½0; 1:In this paper we determine the smallest function w satisfying the above inequality. The required conditions on Q are such that the functions wðtÞ ¼ t p , p [ ½1; 2 satisfy them. As the main tool we use de Rham theorem.
Under some additional assumptions we determine solutions of the equationwhere f : R → R is Lebesgue measurable or Baire measurable, M : R → R and • : R 2 → R. All three functions are unknown.
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