Three Applications of Jensen’s Inequality
for Real Measurable Functions
Problems 25 and 26 from Chapter 3 of Rudin's Real and Complex Analysis
In this post, I'll give you the solution to two interesting problems from Rudin's Real and Complex Analysis textbook. While the post says 3 problems, the Problem 25 has 2 inequalities.
The proofs below are by me
Because it's really convinient to attach a pdf rather than type math here, I'll just add a pdf :D
For those that do not know abstract measures, you could take μ below to be the Lebesgue measure, it’s really the same. If you are not familiar with that, then you could just treat the stuff below as Riemann integration – it’s not that different! In fact, just consider any integral as integrating a positive function f on [0, 1] with respect to some variable dx (but written here as d mu)
And yes, this was made in word :O
I hope you enjoyed reading my proofs!!!
Credit: Rudin, Walter. Real and Complex Analysis: International Student Edition. McGraw-Hill, 1970.