lemma 11.68 Jensen's inequality for the logarithm

Rests on

• depends_on theorem 7.35 Mean value theorem ¶
  • depends_on proposition 7.29 Linearity ¶
    • depends_on definition 7.11 Sum of functions ¶
      • depends_on definition 7.9 Real function ¶
        • depends_on definition 3.43 Map ¶
    • depends_on equation 7.14 eq:ana-derivh ¶
    • depends_on proposition 7.6 Algebra of limits ¶
      • depends_on definition 7.4 Convergence ¶
        • depends_on definition 7.2 Absolute value ¶
      • depends_on proposition 7.3 Triangle inequality ¶
        • depends_on definition 7.2 Absolute value ¶ ↺
        • proves proof ch:05-real-analysis@proof-1 ¶
      • proves proof ch:05-real-analysis@proof-3 ¶
    • proves proof ch:05-real-analysis@proof-13 ¶
  • depends_on theorem 7.34 Rolle ¶
    • depends_on lemma 7.33 Fermat: interior extremum ¶
      • depends_on definition 7.26 Derivative of a function at a point ¶
        • depends_on definition 7.16 Limit ¶
        • depends_on definition 7.9 Real function ¶ ↺
      • depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
        • depends_on definition 7.16 Limit ¶ ↺
        • depends_on equation 7.7 eq:ana-limit-left ¶
        • depends_on equation 7.5 eq:ana-limit-right ¶
        • proves proof ch:05-real-analysis@proof-6 ¶
      • proves proof ch:05-real-analysis@proof-17 ¶
    • depends_on theorem 7.24 Extreme value theorem ¶
      • depends_on axiom 7.1 Completeness of $\R$ ¶
      • depends_on proposition 7.22 Sequential characterization ¶
        • depends_on definition 7.20 Continuity at a point ¶
        • depends_on definition 7.4 Convergence ¶ ↺
        • proves proof ch:05-real-analysis@proof-7 ¶
      • depends_on theorem 7.7 Bolzano–Weierstrass ¶
        • depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
        • depends_on corollary A.46 Monotone convergence ¶
        • proves proof ch:05-real-analysis@proof-4 ¶
      • proves proof ch:05-real-analysis@proof-9 ¶
    • proves proof ch:05-real-analysis@proof-18 ¶
  • proves proof ch:05-real-analysis@proof-19 ¶
• proves proof ch:09-probability-statistics@proof-28 ¶

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