I need to find the widest class I can for functions that obey the following inequality:
$$\left| \int\limits_{-1}^{1} x f(x) dx \right| \le \left| \int\limits_{-1}^{1} f(x) dx \right|$$
It is clear that all functions $f$ that don't change sign within the interval do:
$\displaystyle \left| \int\limits_{-1}^{1} x f(x) dx \right| \le \int\limits_{-1}^{1} \left|x\right| \left|f(x)\right| dx \le \int\limits_{-1}^{1} \left|f(x)\right| dx = \left| \int\limits_{-1}^{1} f(x) dx \right|$,
And so do even functions:
$$\left| \int\limits_{-1}^{1} x f(x) dx \right| = \left| \int\limits_{-1}^{0} x f(x) dx + \int\limits_{0}^{1} x f(x) dx \right| = 0 \le \left| \int\limits_{-1}^{1} f(x) dx \right|$$
But is there a wider class?