Sharp mean-variance bounds for Jensen-type inequalities.(English)Zbl 0705.60017

Suppose $$\phi$$ is a convex function and Y is a random variable such that $$\mu =E(Y)$$ and $$E(\phi(Y))$$ are finite. Then $$\phi(\mu)\leq E[\phi(Y)]$$ by Jensen’s inequality. Suppose that $$\sigma^ 2=Var Y$$ is finite. Let $$x_ 0$$ belong to the domain of $$\phi$$. If $$\mu=x_ 0$$, define $$a_ 0=x_ 0$$, and $$p_ 0=1$$; otherwise let $p_ 0=(\mu-x_ 0)^ 2/[\sigma^ 2+(\mu -x_ 0)^ 2],\;a_ 0=\mu +\sigma^ 2/(\mu-x_ 0).$ If $$Y\leq x_ 0$$ a.s. and if $$(\phi (x)-\phi (x_ 0))/(x-x_ 0)$$ is concave on $$[-\infty,x_ 0]\cap dom(\phi)$$, then it is shown that $(*)\;p_ 0\phi (a_ 0)+(1-p_ 0)\phi (x_ 0)\leq E[\phi(Y)].$ If, in addition, $$\phi$$ is convex, then the left side of (*) is greater than or equal to $$\phi(\mu)$$. Conditions for equality in (*) are discussed. Several special cases have been derived.
Reviewer: B.L.S.Prakasa Rao

MSC:

 6e+16 Inequalities; stochastic orderings
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References:

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