Some new results on the Fejér and Hermite-Hadamard inequalities (Q380208)

Hadamard-Hermite-Fejér inequalities are considered in case that the convexity of the involved function \(f\) is relaxed. The following theorems are proved.NEWLINENEWLINENEWLINETheorem 1. Suppose that \(p(x)\geq 0\) is symmetric about \(\frac{a+b}{2}\) and \(f:[a,b]\rightarrow \mathbb{R}\) is a twice differentiable function such that \(f''\) is bounded in \([a,b]\). Then NEWLINENEWLINE\[NEWLINE\begin{multlined} m\int_a^{(a+b)/2} \left(\frac{a+b}{2}-x\right)^2 p(x) dx\\ \leq \int_a^b f(x)p(x) dx -f\left(\frac{a+b}{2}\right) \int_a^b p(x) dx \leq M \int_a^{(a+b)/2} \left(\frac{a+b}{2}-x\right)^2 p(x) dx \end{multlined}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\begin{multlined} -M\int_a^{(a+b)/2} (x-a)(b-x)p(x) dx \\ \leq \int_a^b f(x)p(x) dx - \frac{f(a)+f(b)}{2} \int_a^b p(x) dx \leq -m \int_a^{(a+b)/2} (x-a)(b-x) p(x) dx \end{multlined}NEWLINE\]NEWLINE where \(m=\inf_{t\in [a,b]} f''(t)\), \(M=\sup_{t\in [a,b]} f''(t)\).NEWLINENEWLINETheorem 2. Let \(1<p<\infty\) and \(p(x)\geq 0\) be symmetric about \(\frac{a+b}{2}\) and \(f:[a,b]\rightarrow \mathbb{R}\) is a twice differentiable function such that \(f''\in L^p([a,b])\). Then NEWLINE\[NEWLINE\begin{multlined} \left| \int_a^b f(x)p(x) dx -f\left(\frac{a+b}{2}\right) \int_a^b p(x) dx \right|\\ \leq \frac{q}{2(q+1)} \|f''\|_p \int_a^{(a+b)/2} (a+b-2x)^{(1/q)+1} p(x) dx\end{multlined}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\begin{multlined} \left| \frac{f(a)+f(b)}{2} \int_a^b p(x) dx -\int_a^b f(x)p(x) dx \right|\\ \leq \frac{q}{2(q+1)} \|f''\|_p \int_a^{(a+b)/2} \left( (b-a)^{(1/q)+1}-(a+b-2x)^{(1/q)+1}\right) p(x) dx, \end{multlined}NEWLINE\]NEWLINE where \(q\) is defined to be \(p/(p-1)\).NEWLINENEWLINETheorem 3. Suppose that \(p(x)\geq 0\) is symmetric about \(\frac{a+b}{2}\) and \(f:[a,b]\rightarrow \mathbb{R}\) is a differentiable function satisfying \(f'(a+b-x) \geq f'(x)\), for all \(x\in [a,(a+b)/2]\). Then NEWLINE\[NEWLINE f\left(\frac{a+b}{2}\right) \int_a^b p(x) dx \leq \int_a^b f(x)p(x) dx \leq \frac{f(a)+f(b)}{2} \int_a^b p(x) dx NEWLINE\]NEWLINE holds.