Jensen-type inequalities for a second-order differential inequality condition

Abstract Extension for Jensen-type inequality and reverse of Jensen’s inequality under the assumption of second-order differential inequality <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mi>p</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> <m:mi>f</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>+</m:mo> <m:mi>q</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> <m:msup> <m:mi>f</m:mi> <m:mrow> <m:mi mathvariant="normal">′</m:mi> </m:mrow> </m:msup> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>+</m:mo> <m:msup> <m:mi>f</m:mi> <m:mrow> <m:mi mathvariant="normal">′</m:mi> <m:mi mathvariant="normal">′</m:mi> </m:mrow> </m:msup> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:math> $$ \begin{equation*}p(x)f(x)+q(x)f^{\prime}(x)+f^{\prime\prime}(x)\ge 0 \end{equation*}$$ are established, Hermite–Hadamard-type inequality is also proven. We get some refinements of power mean inequality as applications and further extend normalised Jensen functional under the assumption of second-order differential inequality as an improvement.