Inequalities of Jensen's type for Lipschitzian mappings (Q2782928)

Let \(g:[a,b]\rightarrow \mathbb{R}\) and \(p:[a,b]\rightarrow \mathbb{R}\) be given functions such that \(g\) is continuous and \(p\) is positive and integrable on \( [a,b]\) . Denote NEWLINE\[NEWLINE P=\int_{a}^{b}p\left( x\right) dx,\;\overline{g}=\frac{1}{P} \int_{a}^{b}p( x) g(x) dx. NEWLINE\]NEWLINE Further, let \(f:I\rightarrow \mathbb{R}\) be an \(L\)-Lipschitzian mapping and let \(g([a,b])\subset I\) . The authors prove that for such functions it is valid the inequality NEWLINE\[NEWLINE \left|\frac{1}{P}\int_{a}^{b}p(x)f(g(x)) dx- f\left( \frac{1}{P} \int_{a}^{b}p(x)g(x) dx\right) \right|\leq \frac{1}{P}\int_{a}^{b}p(x)\left|g(x)-\overline{g}\right|dx. NEWLINE\]NEWLINE Moreover, if \( g(x)\in [m,M]\subseteq I\) \(\forall x\in [a,b], \) then we have NEWLINE\[NEWLINE \left|\frac{M-\overline{g}}{M-m}f(m)+\frac{\overline{g}-m}{M-m}f(M)-\frac{1}{P}\int_{a}^{b}p(x)f(g(x)) dx\right|NEWLINE\]NEWLINE NEWLINE\[NEWLINE \leq \frac{2L}{M-m}\cdot \frac{1}{P}\int_{a}^{b}p(x)(M-g(x))(g(x)-m) dx. NEWLINE\]NEWLINE These inequalities generalize the results of \textit{S. S. Dragomir, Y. J. Cho} and \textit{S. S. Kim} [J. Math. Anal. Appl. 245, No. 2, 489-501 (2000; Zbl 0956.26015)].