Math 2.0

Real functions Proof

Proof that the logarithm is strictly concave

Builds on Natural logarithmDerivative of the logarithmDifferentiable functions are continuousMean value theoremConcave function

Proof

We apply the mean value theorem on two adjacent intervals.

  1. Let x≠yx \ne y be in (0,∞)(0, \infty) and let t∈(0,1)t \in (0, 1). If we swap xx with yy and tt with 1−t1 - t, both sides of the claimed inequality stay the same. So we may assume x<yx < y.
  2. Let z=tx+(1−t)yz = t x + (1 - t) y. Then z−x=(1−t)(y−x)>0z - x = (1 - t)(y - x) > 0 and y−z=t(y−x)>0y - z = t (y - x) > 0. So 0<x<z<y0 < x < z < y.
  3. The natural logarithm is differentiable on (0,∞)(0, \infty) with derivative 1/x1/x. So it is also continuous there.
  4. By the mean value theorem on [x,z][x, z], there is c1∈(x,z)c_1 \in (x, z) with ln⁡z−ln⁡x=z−xc1=(1−t)(y−x)c1\ln z - \ln x = \frac{z - x}{c_1} = \frac{(1 - t)(y - x)}{c_1}.
  5. By the mean value theorem on [z,y][z, y], there is c2∈(z,y)c_2 \in (z, y) with ln⁡y−ln⁡z=y−zc2=t(y−x)c2\ln y - \ln z = \frac{y - z}{c_2} = \frac{t (y - x)}{c_2}.
  6. Since 0<c1<z<c20 < c_1 < z < c_2, we have 1c1>1c2\frac{1}{c_1} > \frac{1}{c_2}.
  7. Since t+(1−t)=1t + (1 - t) = 1, we can write ln⁡z−tln⁡x−(1−t)ln⁡y=t(ln⁡z−ln⁡x)−(1−t)(ln⁡y−ln⁡z).\ln z - t \ln x - (1 - t) \ln y = t (\ln z - \ln x) - (1 - t)(\ln y - \ln z).
  8. By steps 4 and 5, the right side equals t(1−t)(y−x)(1c1−1c2)t (1 - t)(y - x)\big(\frac{1}{c_1} - \frac{1}{c_2}\big).
  9. Each factor is positive, by step 6 and since 0<t<10 < t < 1 and x<yx < y. So ln⁡(tx+(1−t)y)>tln⁡x+(1−t)ln⁡y\ln(t x + (1 - t) y) > t \ln x + (1 - t) \ln y.
  10. This holds for all x≠yx \ne y in (0,∞)(0, \infty) and all t∈(0,1)t \in (0, 1). So ln⁡\ln is strictly concave on (0,∞)(0, \infty).

Idea

The point zz splits [x,y][x, y] into two pieces. By the mean value theorem, the slope of the graph over each piece is a value of 1/x1/x inside that piece. Since 1/x1/x is strictly decreasing, the right slope is smaller than the left slope, so the graph bends downward and lies strictly above the chord.

Reading path

To understand this page from the ground up, read these 272 pages in order. The foundations come first. The path takes the proofs Proof of the Cauchy–Hadamard theorem, Proof that a function with nonnegative second derivative is convex, Proof of the Weierstrass M-test, Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the formula for the derivative of a polynomial, Proof of the commutativity of addition by induction, Proof of term-by-term differentiation, Proof of the epsilon-delta form of continuity, Proof of the power rule by induction, Proof that absolutely convergent series converge, Proof of the sum of a finite geometric series, Proof that convergent sequences are Cauchy, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the chain rule with an auxiliary difference quotient, Proof of the Cauchy criterion for series, Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem, Proof that the exponential series has infinite radius of convergence, Proof that a function with nonnegative derivative is increasing, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the root test, Proof of the double negation law by a truth table, Proof that the logarithm is the inverse of the exponential, Proof that power series converge uniformly on smaller intervals inside the radius of convergence, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof of the derivative of the logarithm by the inverse function rule, Proof that the exponential function is positive, Proof of the uniqueness of limits, Proof of strong induction from ordinary induction, Proof that power series are differentiable inside the radius of convergence, Proof of the limit of a product of sequences, Proof that uniform convergence of derivatives allows term-by-term differentiation, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of Pascal's rule from the recursive definition, Proof that a differentiable function is convex iff its derivative is increasing, Proof of the linearity of differentiation, Proof of the sequential characterization of limits, Proof that the exponential function is strictly increasing, Proof that Cauchy sequences are bounded, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of the comparison test, Proof of the existence of the infimum by negation, Proof of the continuity of the inverse of a strictly monotone continuous function, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof that 1 + x is at most exp x, Proof of the factorial formula for binomial coefficients by induction, Proof that a function with positive derivative is strictly increasing, Proof of the intermediate value theorem by the supremum, Proof of the binomial theorem by induction, Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms, Proof that differentiable functions are continuous, Proof that one-sided limits of a monotone function exist, Proof of the epsilon-delta form of the limit of a function, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the Archimedean property in an Archimedean ordered field, Proof that a composition of continuous functions is continuous, Proof of the exponential of a sum by the Cauchy product, Proof that a decreasing sequence bounded below converges, Proof of the term test, Proof that the exponential function is its own derivative, by term-by-term differentiation, Proof that the reals are Archimedean from the completeness axiom, Proof of the ratio test, Proof that convergent sequences are bounded, Proof of the product rule, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof of the ratio formula for the radius of convergence, Proof that a series of nonnegative terms converges iff its partial sums are bounded, Proof of the mean value theorem from Rolle's theorem, Proof that the Cauchy product of absolutely convergent series converges absolutely to the product, Proof that slopes of a convex function are increasing, Proof that the exponential function is strictly convex, Proof of the constant multiple rule.

  1. Binomial coefficient definition
  2. Construction of the reals from Cauchy sequences concept
  3. Convex function definition
  4. Definition by recursion concept
  5. Element definition
  6. Factorial definition
  7. Group concept
  8. Integers object
  9. Limit point of a set definition
  10. Local maximum definition
  11. Mathematical induction concept
  12. Natural numbers object
  13. Ordered field definition
  14. Peano axioms concept
  15. Proposition concept
  16. Rational numbers object
  17. Real numbers object
  18. Sequence definition
  19. Set bounded above definition
  20. Strictly monotone function definition
  21. Proof of Pascal's rule from the recursive definition proof
  22. Concave function definition
  23. Proof that slopes of a convex function are increasing proof
  24. Strictly convex function definition
  25. Set definition
  26. Ring definition
  27. Ring of integers statement
  28. Limit of a function definition
  29. Local minimum definition
  30. Archimedean ordered field definition
  31. Successor function definition
  32. Conjunction definition
  33. Disjunction definition
  34. Implication definition
  35. Negation definition
  36. Predicate definition
  37. The reals form a field statement
  38. Real sequence definition
  39. Subsequence definition
  40. Infimum definition
  41. Supremum definition
  42. Pascal's rule statement
  43. Slopes of a convex function are increasing statement
  44. Ordered pair definition
  45. Commutative ring definition
  46. Continuity at a point definition
  47. Derivative definition
  48. One-sided limit definition
  49. Addition of natural numbers definition
  50. Proof of the Archimedean property in an Archimedean ordered field proof
  51. The rationals form a field statement
  52. The successor function is injective statement
  53. Zero is not a successor statement
  54. Biconditional definition
  55. Truth table concept
  56. Modus ponens concept
  57. Existential quantifier definition
  58. Universal quantifier definition
  59. Bounded sequence definition
  60. Cauchy sequence definition
  61. Geometric sequence definition
  62. Limit of a sequence definition
  63. Monotone sequence definition
  64. Sequence tending to infinity definition
  65. Series definition
  66. Completeness axiom statement
  67. Proof of the factorial formula for binomial coefficients by induction proof
  68. Cartesian product definition
  69. Continuous function definition
  70. Differentiable function definition
  71. One-sided derivative definition
  72. Archimedean property statement
  73. Order on the natural numbers definition
  74. Proof of the commutativity of addition by induction proof
  75. Subset definition
  76. The rationals form an ordered field statement
  77. Propositional logic concept
  78. Mathematical proof concept
  79. Uniqueness quantifier definition
  80. Limit superior definition
  81. Proof of the sum of a finite geometric series proof
  82. Convergent sequence definition
  83. Pointwise convergence definition
  84. Sequential continuity definition
  85. The sequence r^n tends to infinity for r > 1 example
  86. Cauchy product definition
  87. Partial sum definition
  88. Power series definition
  89. Factorial formula for binomial coefficients statement
  90. Proof that a composition of continuous functions is continuous proof
  91. Derivative function definition
  92. Binary relation definition
  93. Order on the reals definition
  94. Commutativity of addition statement
  95. Proof of strong induction from ordinary induction proof
  96. Tautology definition
  97. Uniform convergence definition
  98. Sum of a finite geometric series statement
  99. Radius of convergence definition
  100. Convergent series definition
  101. Proof of the binomial theorem by induction proof
  102. Higher derivative definition
  103. Antisymmetric relation definition
  104. Reflexive relation definition
  105. Transitive relation definition
  106. Interval definition
  107. Strong induction statement
  108. Function definition
  109. Uniform convergence on a set definition
  110. Logical equivalence definition
  111. Sum of a series definition
  112. Series of functions definition
  113. Absolute convergence definition
  114. Binomial theorem statement
  115. Second derivative definition
  116. Preorder definition
  117. Proof of the epsilon-delta form of continuity proof
  118. Proof of the epsilon-delta form of the limit of a function proof
  119. Proof of the well-ordering principle from strong induction proof
  120. Proof of the double negation law by a truth table proof
  121. Partial order definition
  122. Epsilon-delta definition of continuity statement
  123. Monotone function definition
  124. Epsilon-delta definition of a limit statement
  125. Well-ordering principle statement
  126. Double negation law statement
  127. Total order definition
  128. Composition of continuous functions is continuous statement
  129. Proof by contradiction concept
  130. Proof that every sequence has a monotone subsequence, by peaks proof
  131. Proof that the reals form an ordered field, with the Cauchy sequence construction proof
  132. Every sequence has a monotone subsequence statement
  133. The reals form an ordered field statement
  134. Absolute value definition
  135. Proof of the approximation property of the supremum proof
  136. Proof of the existence of the infimum by negation proof
  137. Proof of the triangle inequality for real numbers proof
  138. Proof that the absolute value is multiplicative proof
  139. Proof of the epsilon-N definition of the limit of a sequence proof
  140. Approximation property of the supremum statement
  141. Existence of the infimum statement
  142. Triangle inequality statement
  143. Absolute value is multiplicative statement
  144. Epsilon-N definition of the limit of a sequence statement
  145. Proof of the intermediate value theorem by the supremum proof
  146. Proof that one-sided limits of a monotone function exist proof
  147. Proof that the reals are Archimedean from the completeness axiom proof
  148. Proof of the limit of a sum of sequences proof
  149. Proof of the squeeze theorem for sequences proof
  150. Proof of the uniqueness of limits proof
  151. Proof that limits preserve non-strict inequalities proof
  152. Intermediate value theorem statement
  153. Proof that differentiable functions are continuous proof
  154. Proof that convergent sequences are bounded proof
  155. Proof that convergent sequences are Cauchy proof
  156. The reals are Archimedean statement
  157. The sequence 1/n tends to zero example
  158. Proof that Cauchy sequences are bounded proof
  159. Proof of the term test proof
  160. Proof of the monotone convergence theorem for sequences proof
  161. Differentiable functions are continuous statement
  162. Proof of the continuity of the inverse of a strictly monotone continuous function proof
  163. Proof that there is a natural number n with 1/n below any positive epsilon proof
  164. Limit of a sum of sequences statement
  165. Uniqueness of limits statement
  166. Limits preserve non-strict inequalities statement
  167. Convergent sequences are bounded statement
  168. Proof of the ratio formula for the radius of convergence proof
  169. Squeeze theorem statement
  170. Convergent sequences are Cauchy statement
  171. Cauchy sequences are bounded statement
  172. Monotone convergence theorem for sequences statement
  173. Proof of the limit of a product of sequences proof
  174. There is a natural number n with 1/n below any positive epsilon statement
  175. One-sided limits of a monotone function exist statement
  176. Term test statement
  177. Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
  178. Proof that a decreasing sequence bounded below converges proof
  179. The sequence r^n tends to zero for |r| < 1 example
  180. Continuity of the inverse of a strictly monotone continuous function statement
  181. Proof of the sequential characterization of limits proof
  182. Proof that continuity is equivalent to sequential continuity proof
  183. Decreasing sequence bounded below converges statement
  184. Limit of a product of sequences statement
  185. Geometric series example
  186. A series of nonnegative terms converges iff its partial sums are bounded statement
  187. Continuity is equivalent to sequential continuity statement
  188. Proof that the Cauchy product of absolutely convergent series converges absolutely to the product proof
  189. Proof of the root test proof
  190. Sequential characterization of limits statement
  191. Proof of the Cauchy–Hadamard theorem proof
  192. Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms proof
  193. Proof of the ratio test proof
  194. Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
  195. Proof of the comparison test proof
  196. Proof of the chain rule with an auxiliary difference quotient proof
  197. Proof of the product rule proof
  198. Proof of the linearity of differentiation proof
  199. Comparison test statement
  200. Cauchy product of absolutely convergent series statement
  201. Bolzano–Weierstrass theorem statement
  202. Exponentials dominate powers in sequences statement
  203. Chain rule statement
  204. Product rule statement
  205. Linearity of differentiation statement
  206. Ratio test statement
  207. Root test statement
  208. Proof of the boundedness theorem by Bolzano–Weierstrass proof
  209. Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem proof
  210. Derivative of an inverse function statement
  211. Proof of the constant multiple rule proof
  212. Proof of the power rule by induction proof
  213. Boundedness theorem statement
  214. Ratio formula for the radius of convergence statement
  215. Cauchy–Hadamard theorem statement
  216. Cauchy criterion for sequences statement
  217. Constant multiple rule statement
  218. Power rule statement
  219. Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
  220. Uniformly Cauchy sequence of functions definition
  221. Proof of the Cauchy criterion for series proof
  222. Extreme value theorem statement
  223. Proof of the formula for the derivative of a polynomial proof
  224. Proof of Fermat's theorem on stationary points proof
  225. Cauchy criterion for series statement
  226. Derivative of a polynomial statement
  227. Proof that absolutely convergent series converge proof
  228. Fermat's theorem on stationary points statement
  229. Proof of Rolle's theorem proof
  230. Absolutely convergent series converge statement
  231. Rolle's theorem statement
  232. Proof of the Weierstrass M-test proof
  233. Proof of the mean value theorem from Rolle's theorem proof
  234. Weierstrass M-test statement
  235. Mean value theorem statement
  236. Proof that power series converge uniformly on smaller intervals inside the radius of convergence proof
  237. Proof that a differentiable function is convex iff its derivative is increasing proof
  238. Proof that a function with nonnegative derivative is increasing proof
  239. Proof that uniform convergence of derivatives allows term-by-term differentiation proof
  240. Proof that a function with positive derivative is strictly increasing proof
  241. Power series converge uniformly on smaller discs inside the radius of convergence statement
  242. A function with nonnegative derivative is increasing statement
  243. A differentiable function is convex iff its derivative is increasing statement
  244. Uniform convergence of derivatives allows term-by-term differentiation statement
  245. A function with positive derivative is strictly increasing statement
  246. Proof that the exponential series has infinite radius of convergence proof
  247. Proof that a function with nonnegative second derivative is convex proof
  248. Proof of term-by-term differentiation proof
  249. The exponential series has infinite radius of convergence statement
  250. A function with nonnegative second derivative is convex statement
  251. Term-by-term differentiation statement
  252. Exponential function definition
  253. Proof of the exponential of a sum by the Cauchy product proof
  254. Proof that power series are differentiable inside the radius of convergence proof
  255. Exponential of a sum statement
  256. Power series are differentiable inside the radius of convergence statement
  257. Proof that the exponential function is positive proof
  258. Proof that the exponential function is its own derivative, by term-by-term differentiation proof
  259. The exponential function is positive statement
  260. The exponential function is its own derivative statement
  261. Proof that the exponential function is strictly increasing proof
  262. The exponential function is strictly increasing statement
  263. Proof that the exponential function is strictly convex proof
  264. Proof that 1 + x is at most exp x proof
  265. Natural logarithm definition
  266. The exponential function is convex statement
  267. 1 + x is at most exp x statement
  268. Proof that the logarithm is the inverse of the exponential proof
  269. Logarithm is the inverse of the exponential statement
  270. Proof of the derivative of the logarithm by the inverse function rule proof
  271. Derivative of the logarithm statement
  272. Proof that the logarithm is strictly concave proof

Where this leads

25 pages build on this one.

Further reading

  1. Web https://proofwiki.org/wiki/Real_Function_is_Strictly_Convex_iff_Derivative_is_Strictly_Increasing, Sufficient Condition
About this page
Tag8320TK08GR
URIm2://8320TK08GR
Kindproof
WritingW0 stub
MSC26A51
ProvesThe logarithm is strictly concave
UsesNatural logarithm, Derivative of the logarithm, Differentiable functions are continuous, Mean value theorem, Concave function
Changed4 November 2026

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