## 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

### Keywords:

convex function; Jensen’s inequality
Full Text:

### References:

 [1] Agnew, R.A., Inequalities with application in economic risk analysis, J. appl. probab., 9, 441-444, (1972) · Zbl 0239.90011 [2] Cartwright, D.I.; Field, M.J., A refinement of the arithmetic mean — geometric Mean inequality, Proc. amer. math. soc., 71, 36-38, (1978) · Zbl 0392.26010 [3] Chao, M.T.; Strawderman, W.E., Negative moments of positive random variables, J. amer. statist. assoc., 67, 429-431, (1972) · Zbl 0238.60008 [4] Eckberg, A.E., Sharp bounds on laplace—stieltjes transforms, with applications to various queueing problems, Math. oper. res., 2, 135-142, (1977) · Zbl 0405.60017 [5] Karlin, S.; Studden, W.J., Tchebycheff systems with applications in analysis and statistics, (1966), Wiley New York · Zbl 0153.38902 [6] Skibinsky, M., The range of the (n + 1)th moment for distributions on [0, 1], J. appl. probab., 4, 543-552, (1967) · Zbl 0189.18803
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.