Abstract
We present a new, in a sense direct, proof that the system of two functional inequalities
and
is satisfied for functions F and f mapping an open interval I of the real line ℝ into a Banach space and into ℝ, respectively, if and only if F yields a delta-convex mapping with a control function f.
A similar result is obtained for delta-convexity of higher orders with detailed proofs given in the case of delta-convexity of the second order, i.e. when the functional inequality
holds true provided that x,y∈I, x≤y.
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Ger, R., Pečarić, J. (2012). On Vector Hermite-Hadamard Differences Controlled by Their Scalar Counterparts. In: Bandle, C., Gilányi, A., Losonczi, L., Plum, M. (eds) Inequalities and Applications 2010. International Series of Numerical Mathematics, vol 161. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0249-9_12
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DOI: https://doi.org/10.1007/978-3-0348-0249-9_12
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