If $\frac{a}{b}<\frac{a'}{b'}<\frac{a''}{b''}$, with all values positive, then $\frac{a}{b}<\frac{ab+a’b’+a’’b’’}{b^2+b’^2+b’’^2}<\frac{a’’}{b’’}$
Inequality from Challenge and Thrill of Pre-college Mathematics