Some new Fejér type inequalities for ( <i>h</i> , <i>g</i> ; <i>α</i> - <i>m</i> )-convex functions

Abstract The study of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>h</m:mi> <m:mo>,</m:mo> <m:mi>g</m:mi> <m:mo>;</m:mo> <m:mspace width="1.42271pt"/> <m:mi>α</m:mi> <m:mo>−</m:mo> <m:mi>m</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left(h,g;\hspace{1.42271pt}\alpha -m) -convex functions extends the classical concept of convexity to more generalized forms, which provide flexible tools for analysis. The aim of this article is to yield the generalization of the Fejér type inequalities for various classes of convex functions, such as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>h</m:mi> <m:mo>,</m:mo> <m:mi>g</m:mi> <m:mo>;</m:mo> <m:mspace width="1.42271pt"/> <m:mi>α</m:mi> <m:mo>−</m:mo> <m:mi>m</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left(h,g;\hspace{1.42271pt}\alpha -m) -convex and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>h</m:mi> <m:mo>,</m:mo> <m:mi>g</m:mi> <m:mo>;</m:mo> <m:mspace width="1.42271pt"/> <m:mi>m</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left(h,g;\hspace{1.42271pt}m) -convex functions.