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Extension of Berezin-Lieb Inequalities

The Berezin-Lieb inequalities provide upper and lower bounds for a partition function based on phase space integrals that involve the Glauber-Sudarshan and Husimi representations, respectively. Generalizations of these representations have recently been introduced by the present authors, and in this article, we extend the use of these new representations to develop numerous analogs of the Berezin-Lieb inequalities that may offer improved bounds. Several examples illustrate the use of the new inequalities. Although motivated by problems in quantum mechanics, these results may also find applications in time-frequency analysis, a valuable cross fertilization that has been profitably used at various times in the past.