Intereting Posts

Correspondence of representation theory between $\mathrm{GL}_n(\mathbb C)$ and $\mathrm U_n(\mathbb C)$
Why should I care about fields of positive characteristic?
Can $f(x,y) = |x|^y$ be be made continuous?
Is $V$ a simple $\text{End}_kV$-module?
Image of unit ball dense under continuous map between banach spaces
Prove inequality $\frac a{1+bc}+\frac b{1+ac}+\frac c{1+ab}+abc\le \frac52$
Every space is “almost” Baire?
Limit Rule $\lim f(x)^{g(x)}$
How do you find the formula for an area of the circle through integration?
Show that $f(2n)= f(n+1)^2 – f(n-1)^2$
A $p$-group of order $p^n$ has a normal subgroup of order $p^k$ for each $0\le k \le n$
Determining whether a symmetric matrix is positive-definite (algorithm)
the converse of Pythagoras Theorem
Is there a 'nice' axiomatization in the language of arithmetic of the statements ZF proves about the natural numbers?
Hartshorne Problem 1.2.14 on Segre Embedding

I’m taking a course in Analysis in which the following exercise was given.

**Exercise** Let $(\Omega, \mathcal{F}, \mu)$ be a probability space. Let $f\ge 0$ be a measurable function.

- Using Jensen’s inequality, prove that for $1\le p < q$, $$\left(\int f^p\, d\mu\right)^{\frac{1}{p}}\le \left(\int f^q\, d\mu\right)^{\frac{1}{q}}.$$
- Deduce Hölder’s inequality from Jensen’s inequality and discuss the cases of equality.

The first part is standard and I had no problems with it. On the contrary, the second part is somewhat unclear. The standard proof of Hölder’s inequality uses Young’s inequality which may be proved by means of the convexity of the exponential function. So, strictly speaking, this is a way of “deducing Hölder’s inequality from Jensen’s”, but I don’t think this is what the examiner had in mind. More likely, one is supposed to look for a proof employing Jensen’s inequality in $(\Omega, \mathcal{F}, \mu)$, or perhaps applying directly the first point. But I have no idea on how to do this.

- with inequality $\frac{y}{xy+2y+1}+\frac{z}{yz+2z+1}+\frac{x}{zx+2x+1}\le\frac{3}{4}$
- An interesting inequality about the cdf of the normal distribution
- Prove that $\|a\|+\|b\| + \|c\| + \|a+b+c\| \geq \|a+b\| + \|b+c\| + \|c +a\|$ in the plane.
- An Inequality problem relating $\prod\limits^n(1+a_i^2)$ and $\sum\limits^n a_i$
- Proof by induction of Bernoulli's inequality $ (1+x)^n \ge 1+nx$
- Induction proof of $n^{(n+1) }> n(n+1)^{(n-1)}$

Thank you.

- Inductive Proof that $k!<k^k$, for $k\geq 2$.
- How to prove this polynomial inequality?
- Floor function inequality of multiplication
- How can I prove this monster inequality?
- Fastest way to check if $x^y > y^x$?
- variance inequality
- How do I prove $\frac 34\geq \frac{1}{n+1}+\frac {1}{n+2}+\frac{1}{n+3}+\cdots+\frac{1}{n+n}$
- Prove by mathematical induction that: $\forall n \in \mathbb{N}: 3^{n} > n^{3}$
- How would you prove that $2^{50} < 3^{33}$ without directly calculating the values
- Hardy's inequality again

As Mike suggests, take the measure $\nu:=\frac{g^q}{\int g^qd\mu}\cdot \mu$ (a probability measure) and $h:=\frac f{g^{q-1}}$. Then

$$\int fgd\mu=\int hg^qd\mu=\int g^qd\mu\cdot\int hd\nu\leqslant \int g^qd\mu

\left(\int h^pd\nu\right)^{1/p}=\left(\int g^qd\mu\right)^{1/q}\left(\int f^pd\mu\right)^{1/p}.$$

- Solve summation $\sum_{i=1}^n \lfloor e\cdot i \rfloor $
- Prove $x = \sqrt{\sqrt{3} + \sqrt{2}} + \sqrt{\sqrt{3} – \sqrt{2}}$ is irrational
- Calculating the probability of a coin falling on its side
- Does $1 + \frac{1}{x} + \frac{1}{x^2}$ have a global minimum, where $x \in \mathbb{R}$?
- Kernel of the homomorphism $\mathbb C → \mathbb C$ deﬁned by $x→t,y→ t^{2},z→ t^{3}$.
- Recurrent points and rotation number
- How find this $I=\int_{0}^{\infty}\frac{x\sin{(2x)}}{x^2+4}dx$
- Goat tethered in a circular pen
- Showing that $G$ is a group under an alternative operation.
- “weird” ring with 4 elements – how does it arise?
- Solving Cubic Equations (With Origami)
- Prove $(-a+b+c)(a-b+c)(a+b-c) \leq abc$, where $a, b$ and $c$ are positive real numbers
- What are some things we can prove they must exist, but have no idea what they are?
- An algebra of nilpotent linear transformations is triangularizable
- Theory $T$ that is consistent, such that $ T + \mathop{Con}(T)$ is inconsistent