caa a22 4500 606200614 CHVBK 20210128100944.0 cr unu---uuuuu 210128e20150601xx s 000 0 eng 10.1007/s00025-015-0451-5 doi (NATIONALLICENCE)springer-10.1007/s00025-015-0451-5 Inequalities of the Hermite-Hadamard Type Involving Numerical Differentiation Formulas [Elektronische Daten] [Andrzej Olbryś, Tomasz Szostok] We observe that the Hermite-Hadamard inequality written in the form $$f\left(\frac{x+y}{2}\right)\leq\frac{F(y)-F(x)}{y-x}\leq\frac{f(x)+f(y)}{2}$$ f x + y 2 ≤ F ( y ) - F ( x ) y - x ≤ f ( x ) + f ( y ) 2 may be viewed as an inequality between two quadrature operators $${f\left(\frac{x+y}{2}\right)}$$ f x + y 2 $${\frac{f(x)+f(y)}{2}}$$ f ( x ) + f ( y ) 2 and a differentiation formula $${\frac{F(y)-F(x)}{y-x}}$$ F ( y ) - F ( x ) y - x . We extend this inequality, replacing the middle term by more complicated ones. As it turns out in some cases it suffices to use Ohlin lemma as it was done in a recent paper (Rajba, Math Inequal Appl 17(2):557-571, 2014) however to get more interesting result some more general tool must be used. To this end we use Levin-Stečkin theorem which provides necessary and sufficient conditions under which inequalities of the type we consider are satisfied. The Author(s), 2015 Hermite-Hadamard inequality nationallicence differentiation formulas nationallicence convex functions nationallicence Olbryś Andrzej Institute of Mathematics, Bankowa 14, 40-007, Katowice, Poland aut Szostok Tomasz Institute of Mathematics, Bankowa 14, 40-007, Katowice, Poland aut Results in Mathematics Springer Basel 67/3-4(2015-06-01), 403-416 1422-6383 67:3-4<403 2015 67 25 https://doi.org/10.1007/s00025-015-0451-5 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00025-015-0451-5 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Olbryś Andrzej Institute of Mathematics, Bankowa 14, 40-007, Katowice, Poland aut NATIONALLICENCE 700 1- Szostok Tomasz Institute of Mathematics, Bankowa 14, 40-007, Katowice, Poland aut NATIONALLICENCE 773 0- Results in Mathematics Springer Basel 67/3-4(2015-06-01), 403-416 1422-6383 67:3-4<403 2015 67 25