Differential analysis: convex functions

Click For Summary
SUMMARY

The discussion centers on a specific demonstration in Section 2 of "A Course in Mathematical Analysis Volume 1" by D. J. H. Garling, focusing on the second inequality involving convex functions. The user expresses confusion regarding the justification of the inequality f'(a+) < f'(b-). After seeking clarification, the user ultimately resolves the issue independently. This indicates that the original concern may stem from a misunderstanding rather than an error in Garling's text.

PREREQUISITES
  • Understanding of convex functions and their properties
  • Familiarity with differentiation and derivatives
  • Knowledge of mathematical analysis concepts
  • Experience with reading advanced mathematical texts
NEXT STEPS
  • Study the properties of convex functions in detail
  • Review the concept of one-sided derivatives, specifically f'(a+) and f'(b-)
  • Examine other examples of inequalities in mathematical analysis
  • Explore additional resources on differentiation techniques in convex analysis
USEFUL FOR

Students of mathematical analysis, educators teaching convex functions, and anyone seeking to deepen their understanding of differentiation in the context of convexity.

madafo3435
Messages
55
Reaction score
15
TL;DR
I am reading A Course in Mathematical Analysis Volume 1 by D. J. H. Garling, and I am having trouble in the following demonstration of Section 2 Differentiation. part 4 of the test, the first part of the second inequality does not make sense, I do not understand its justification. I hoped they can help you understand the inequality, or how to do the test correctly, as this seems like a Garling mistake.
I am reading A Course in Mathematical Analysis Volume 1 by D. J. H. Garling, and I am having trouble in the following demonstration of Section 2 Differentiation. part 4 of the test, the first part of the second inequality does not make sense, I do not understand its justification. I hoped they can help you understand the inequality, or how to do the test correctly, as this seems like a Garling mistake.
1596250627422.png

1596250661655.png
 
Physics news on Phys.org
By the first part of the second inequality, do you mean the f'(a+) <f'(b-)? If not can you be a bit more specific about which part you don't understand?
 
  • Like
Likes   Reactions: madafo3435
Office_Shredder said:
By the first part of the second inequality, do you mean the f'(a+) <f'(b-)? If not can you be a bit more specific about which part you don't understand?
I have already solved it, thanks anyway
 

Similar threads

MHB Normed and Inner Product Spaces .... Garling, Corollary 11.3.2 ....
  • · Replies 2 ·
Replies
2
Views
2K
MHB How Do We Know the Annihilator Operation Is Associative?
  • · Replies 2 ·
Replies
2
Views
1K
MHB The Annihilator of a Set ....Remarks by Garling After Proposition 11.3.5 ....
  • · Replies 2 ·
Replies
2
Views
2K
MHB Pseudometrics .... Garling, Proposition 11.1.13 .... ....
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Question about convex property in Jensen's inequality
  • · Replies 4 ·
Replies
4
Views
3K
MHB Necessary and Sufficient Meaning of Isometries by D. J. H. Garling
  • · Replies 5 ·
Replies
5
Views
2K
MHB Partial Order .... Garling, pages 9-10, Volume I ,,,
  • · Replies 2 ·
Replies
2
Views
2K
MHB Orthogonal Projections .... Garling, Proposition 11.4.3 .... ....
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Bounded in Norm .... Garling, Section 11.2: Normed Spaces ....
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Metric Spaces .... the Uniform Metric .... Garling, Proposition 11.1.11
  • · Replies 2 ·
Replies
2
Views
2K