A converse to the neo-classical inequality with an application to the Mittag-Leffler function. (English) Zbl 1526.33004
Summary: We prove two inequalities for the Mittag-Leffler function, namely that the function \(\log E_\alpha (x^\alpha)\) is sub-additive for \(0 < \alpha < 1\), and super-additive for \(\alpha >1\). These assertions follow from two new binomial inequalities, one of which is a converse to the neo-classical inequality. The proofs use a generalization of the binomial theorem due to Hara and Hino (Bull London Math Soc 2010). For \(0 < \alpha < 2\), we also show that \(E_\alpha (x^\alpha)\) is log-concave resp. log-convex, using analytic as well as probabilistic arguments.
MSC:
| 33E12 | Mittag-Leffler functions and generalizations |
| 26D15 | Inequalities for sums, series and integrals |
| 60G52 | Stable stochastic processes |
Keywords:
Mittag-Leffler function; binomial coefficient; inequality; log-convexity; stable subordinatorReferences:
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