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- W73209369 abstract "We present a new, in a sense direct, proof that the system of two functional inequalities $$biggl Vert F biggl(frac{x+y}{2} biggr) - frac{1}{y-x} int_{x}^{y} F(t),dt biggr Vert leqfrac{1}{y-x} int_{x}^{y} f(t),dt - f biggl(frac{x+y}{2} biggr) $$ and $$biggl Vert frac{F(x) + F(y)}{2} - frac{1}{y-x} int_{x}^{y} F(t),dt biggr Vert leqfrac{f(x) + f(y)}{2} -frac{1}{y-x} int_{x}^{y} f(t),dt $$ 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 Open image in new window holds true provided that x,y∈I, x≤y." @default.
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- W73209369 date "2012-01-01" @default.
- W73209369 modified "2023-09-27" @default.
- W73209369 title "On Vector Hermite-Hadamard Differences Controlled by Their Scalar Counterparts" @default.
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- W73209369 doi "https://doi.org/10.1007/978-3-0348-0249-9_12" @default.
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