The inequality is attributed to    
P.L. Chebyshev, *On approximate expressions of some integrals in terms of others, taken within the same limits*, Proc. Math. Soc. Kharkov **2**, 93–98 (1882).    
<sub>more often cited with the title in French: "Sur les expressions approximatives des integrales definies par les autres prises entre les mêmes limites"</sub>

This article is online <A HREF="http://kms.univer.kharkov.ua/pdf/1879_1883/1882_2/1882_2_1_Chebyshev.pdf">here</A>, in Russian. I am having some difficulty to locate the inequality. The equation should be the one below, with $R_1\geq 0$, hopefully someone who reads Russian can locate that statement.

<IMG SRC="https://ilorentz.org/beenakker/MO/Chebyshev.png"/>