I'm going through Carothers Real Analysis to study for my qualifying exams, and came across this unjustified statement, that Youngs inequality (for $p > 1$ and q such that $\frac{1}{p} + \frac{1}{q} = 1$ we have that for $a, b \geq 0$, $ab \leq \frac{a^p}{p} + \frac{a^q}{q}$), when $p=q=2$, reduces to the arithmetic geometric mean inequality $\sqrt{ab} \leq \frac{a+b}{2}$. I tried manipulating youngs inequality but cant seem to derive the arithmetic geometric mean inequality from it. How is it done?
How does Young’s inequality reduce to the arithmetic-geometric mean inequality
real-analysis
Related Solutions
First note that we have $$ab \leq \int_0^{a} f(x) dx + \int_0^{b} f^{-1}(x) dx $$ for any strictly increasing integrable function $f(x)$. The geometric interpretation is from looking at the area of the rectangle with coordinates $(0,0)$,$(a,0)$,$(a,b)$ and $(0,b)$ and comparing it with the areas given by the integrals. From the image it is also clear that the equality hold only when $b=f(a)$.
To get the Young's inequality, choose $f(x) = x^{p-1}$.
I have added the following picture for clarity.
The image was made using grapher and some post processing was done using LaTeXiT and preview on Mac OSX.
I think you will like "Berkeley Problems in Mathematics" by de Souza and Silva.
In 1977 the Mathematics Department at the University of California, Berkeley, instituted a written examination as one of the first major requirements toward the Ph.D. degree in Mathematics. Its purpose was to determine whether first-year students in the Ph.D. program had successfully mastered basic mathematics in order to continue in the program with the likelihood of success. Since its inception, the exam has become a major hurdle to overcome in the pursuit of the degree. The purpose of this book is to publicize the material and aid in the preparation for the examination during the undergraduate years. The book is a compilation of over 1,250 problems which have appeared on the preliminary exams in Berkeley over the last twenty-five years. It is an invaluable source of problems and solutions for every mathematics student who plans to enter a Ph.D. program. Students who work through this book will develop problem-solving skills in areas such as real analysis, multivariable calculus, differential equations, metric spaces, complex analysis, algebra, and linear algebra. The problems are organized by subject and ordered in an increasing level of difficulty.
(Also a link to Amazon so you can look at the reviews there.)
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Best Answer
Substitute $\sqrt a$, $\sqrt b$ for $a,b$, respectively.