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Extra resources for Convex functions and their applications
Accordingly, if a function f : I → R is convex in the neighborhood of each point of I, then it is convex on the whole interval I. 2 Proof. If f is convex, then clearly D f ≥ D2 f ≥ 0. 3. 2 Now, suppose that D f > 0 on I. If f is not convex, then we can ﬁnd a 2 point x0 such that D f (x0 ) ≤ 0, which will be a contradiction. In fact, in this case there exists a subinterval I0 = [a0 , b0 ] such that f ((a0 +b0 )/2) > (f (a0 )+ f (b0 ))/2. A moment’s reﬂection shows that one of the intervals [a0 , (a0 +b0 )/2], [(3a0 + b0 )/4, (a0 + 3b0 )/4], [(a0 + b0 )/2, b0 ] can be chosen to replace I0 by a smaller interval I1 = [a1 , b1 ], with b1 − a1 = (b0 − a0 )/2 and f ((a1 + b1 )/2) > (f (a1 ) + f (b1 ))/2.
As T. Popoviciu noticed in his book , if f is n-times diﬀerentiable, with f (n) ≥ 0, then f is n-convex. See  and  for a more detailed account on the theory of n-convex functions. Exercises 1. (An application of the second derivative test of convexity) (i) Prove that the functions log((eax −1)/(ex −1)) and log(sinh ax/ sinh x) are convex on R if a ≥ 1. √ √ (ii) Prove that the function b log cos(x/ b) − a log cos(x/ a) is convex on (0, π/2) if b ≥ a ≥ 1. 2. Suppose that 0 < a < b < c (or 0 < b < c < a, or 0 < c < b < a).
T→∞ (iii) Suppose that f ∈ L∞ (µ), f ≥ 0. Prove the convexity of the function t → t log Mt (f ; µ) on R. (iv) Notice that (tr − 1)/r decreases to log t as r ↓ 0 and apply the dominated convergence theorem of Lebesgue to conclude that lim Mr (f ; µ) = M0 (f ; µ) r→0+ for all f ∈ L1 (µ), f ≥ 0. 2. Infer from the integral form of the arithmetic-geometric-harmonic mean inequality that L(a, b) < I(a, b) < A(a, b) for all a, b > 0, a = b. Here L(a, b), I(a, b), A(a, b) are the logarithmic, the identric and respectively the arithmetic mean of a and b.