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


60E15 Inequalities; stochastic orderings
Full Text: DOI


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