Reverses of Féjer's Inequalities for Convex Functions

Let $f$ be a convex function on $I$ and $a,$ $bin I$ with $a<b.$ If $p:% left[ a,bright] rightarrow lbrack 0,infty )$ is Lebesgue integrable and symmetric, namely $pleft( b+a-tright) =pleft( tright) $ for all $tin % left[ a,bright] ,$ then we show in this paper that begin{align*} 0& leq frac{1}{2}int_{a}^{b}leftvert t-frac{a+b}{2}rightvert pleft( tright) dtleft[ f_{+}^{prime }left( frac{a+b}{2}right) -f_{-}^{prime }left( frac{a+b}{2}right) right]  \ & leq int_{a}^{b}pleft( tright) fleft( tright) dt-left( int_{a}^{b}pleft( tright) dtright) fleft( frac{a+b}{2}right)  \ & leq frac{1}{2}int_{a}^{b}leftvert t-frac{a+b}{2}rightvert pleft( tright) dtleft[ f_{-}^{prime }left( bright) -f_{+}^{prime }left( aright) right] end{align*} and begin{align*} 0& leq frac{1}{2}int_{a}^{b}left[ frac{1}{2}left( b-aright) -leftvert t-frac{a+b}{2}rightvert right] pleft( tright) dtleft[ f_{+}^{prime }left( frac{a+b}{2}right) -f_{-}^{prime }left( frac{a+b}{% 2}right) right]  \ & leq left( int_{a}^{b}pleft( tright) dtright) frac{fleft( aright) +fleft( bright) }{2}-int_{a}^{b}pleft( tright) fleft( tright) dt \ & leq frac{1}{2}int_{a}^{b}left[ frac{1}{2}left( b-aright) -leftvert t-frac{a+b}{2}rightvert right] pleft( tright) dtleft[ f_{-}^{prime }left( bright) -f_{+}^{prime }left( aright) right] . end{align*}.