Thursday, 24 March 2016

Is there a constant that reverses Jensen's inequality?



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](e1)E[X]. Here (e1) 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+1log(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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