Majorization-type inequalities for ( <i>m</i> , <i>M</i> , <i>ψ</i> )-convex functions with applications

Abstract In 2001, S. S. Dragomir introduced a generalized class of convexity, the so-called <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>m</m:mi> <m:mo>,</m:mo> <m:mi>M</m:mi> <m:mo>,</m:mo> <m:mi>ψ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left(m,M,\psi ) -convex functions, which covers many other classes of convexity. In this article, we prove some useful characterizations of this generalized class of convex functions. We obtain majorization-type inequalities for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>m</m:mi> <m:mo>,</m:mo> <m:mi>M</m:mi> <m:mo>,</m:mo> <m:mi>ψ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left(m,M,\psi ) -convex functions, providing also applications to new estimates for some well-known mean inequalities.