The general Jensen's inequality states: φ(E[X])≤E[φ(X)]. I'm wondering if there is a constant c (function of φ), such that cφ(E[X])≥E[φ(X)]?
More specifically I wan't to show logE[eX]≤(e−1)E[X]. Here (e−1) would be the c mentioned above.
I've noticed that many other common inequalities have such 'reversing' constants (or at least a corresponding lower bound). Like log(1+x)≤x, and xx+1≤log(1+x).
Answer
For the particular problem about the exponential function, let X=−100, 0, or 100 each with probability 13. Then E(X)=0, but E(eX) is large.
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