caa a22 4500 605508542 CHVBK 20210128100637.0 cr unu---uuuuu 210128e20150601xx s 000 0 eng 10.1007/s00010-014-0286-2 doi (NATIONALLICENCE)springer-10.1007/s00010-014-0286-2 Szostok Tomasz Institute of Mathematics, University of Silesia, Bankowa 14, 40-007, Katowice, Poland aut Ohlin's lemma and some inequalities of the Hermite-Hadamard type [Elektronische Daten] [Tomasz Szostok] Using the Ohlin lemma on convex stochastic ordering we prove inequalities of the Hermite-Hadamard type. Namely, we determine all numbers $${a,\alpha,\beta\in[0,1]}$$ a , α , β ∈ [ 0 , 1 ] such that for all convex functions f the inequality $$af(\alpha x+(1-\alpha )y)+(1-a)f(\beta x+(1-\beta) y)\leq \frac{1}{y-x} \int\limits_{x}^yf(t)dt$$ a f ( α x + ( 1 - α ) y ) + ( 1 - a ) f ( β x + ( 1 - β ) y ) ≤ 1 y - x ∫ x y f ( t ) d t is satisfied and all $${a,b,c,\alpha\in(0,1)}$$ a , b , c , α ∈ ( 0 , 1 ) with a+b+c=1 for which we have $$af(x)+bf(\alpha x+(1-\alpha)y)+cf(y)\geq\frac{1}{y-x} \int\limits_{x}^yf(t)dt$$ a f ( x ) + b f ( α x + ( 1 - α ) y ) + c f ( y ) ≥ 1 y - x ∫ x y f ( t ) d t . The Author(s), 2014 Convex functions nationallicence Hermite-Hadamard inequalities nationallicence Aequationes mathematicae Springer Basel 89/3(2015-06-01), 915-926 0001-9054 89:3<915 2015 89 10 https://doi.org/10.1007/s00010-014-0286-2 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/s00010-014-0286-2 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Szostok Tomasz Institute of Mathematics, University of Silesia, Bankowa 14, 40-007, Katowice, Poland aut NATIONALLICENCE 773 0- Aequationes mathematicae Springer Basel 89/3(2015-06-01), 915-926 0001-9054 89:3<915 2015 89 10