On a Lower Bound for the Jensen Inequality

We present an improvement to the Jensen inequality for certain convex functions φ on (0\,∞). This implies finding a nonzero lower bound for the Jensen gap. In particular, just as the Jensen inequality becomes an equality when φ is linear, for the new inequality we obtain an equality also when φ is quadratic. At the heart of the paper is an apparent new technique for improving classical inequalities based on nonlinear functions.