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Approximate convexity with respect to a subfield

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Abstract

Let \({\mathbb{F}}\) be a subfield of \({\mathbb{R}}\) and X be a linear space over \({\mathbb{F}}\). Let \({ D\subseteq X }\) be a nonempty \({\mathbb{F}}\)-convex set, \({ D^*:=D-D:=\{x-y : x,y\in D\} }\), and \({\alpha \colon {D^* \rightarrow \mathbb{R}}}\) be a nonnegative even function. The function \({ f \colon {D\rightarrow \mathbb{R}}}\) is called \({(\alpha,\mathbb{F})}\)-convex, if it satisfies the inequality

$$\begin{aligned}{f(tx+(1-t)y)\leq tf(x)+(1-t)f(y) + t\alpha((1-t)(x-y))+(1-t)\alpha(t(y-x))}\end{aligned}$$

for all \({x,y\in D}\) and for all \({t\in \mathbb{F}\cap [0,1]}\). In this paper we characterize \({(\alpha,\mathbb{F})}\)-convex functions by comparison of modified difference ratios and support properties. If \({\alpha}\) satisfies some additional conditions, we obtain the differentiability of \({(\alpha,\mathbb{F})}\)-convex functions in the appropriate sense.

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Correspondence to Z. Boros.

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This research was (partially) carried out in the framework of the Center of Excellence of Mechatronics and Logistics at the University of Miskolc.

Research of Z. Boros has been supported by the Hungarian Scientific Research Fund (OTKA) grant K-111651.

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Boros, Z., Nagy, N. Approximate convexity with respect to a subfield. Acta Math. Hungar. 152, 464–472 (2017). https://doi.org/10.1007/s10474-017-0701-y

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  • DOI: https://doi.org/10.1007/s10474-017-0701-y

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