# Questions tagged [holder-inequality]

Proving or manipulations with inequalities by using Hölder's inequality.

421 questions
Filter by
Sorted by
Tagged with
1 vote
75 views

### Prove that $(1+x)^k/k + (1-x)^m/m\geq 1/k +1/m$ without calculus

Note: this has been edited to make the question more general. I want to show that $(1+x)^k/k + (1-x)^m/m$ is minimized at $x=0$ when $k,m\geq 1$ and $-1\leq x \leq1$. Of course, I could take the first ...
• 234
1 vote
31 views

• 83
57 views

I'm trying to prove Holder's inequality using that in a measure space $(X,\mu)$ for every $h:X\to \mathbb{C}$ measurable, $0<\theta<1$ and $0<p<r<q<\infty$, with $$\frac{1}{p}(1-\... • 1,035 0 votes 0 answers 28 views ### Lower bound on the difference between the product of l1 and l2 norm I've just occurred to a question that supposes we have two positive vectors A and B. Then what can we say about ||A||_{l1}||B||_{l1} - ||A||_{l2}||B||_{l2}, i.e. \sum a_i \sum b_i - (\sum a_i^2)... 2 votes 1 answer 32 views ### Two implications of an operator that preserves positivity on L2 Suppose (X,\mathcal{A},m) is a probability space. Let A be an operator on function spaces. I am currently taking a course in measure theory, and in the lecture slides the following two remarks are ... 1 vote 1 answer 45 views ### Scalar Product with Telescope Sum I have a basic analysis problem that I don't manage to solve: Let (a_n)_{n \in \mathbb{N}}\subseteq \mathbb{R} be a strictly positive sequence with a_n \overset{n \to \infty}{\longrightarrow}\infty... • 364 1 vote 0 answers 26 views ### Is the following estimate of this integral correct? Let \epsilon>0 and suppose that U_t^\epsilon, V_t^\epsilon, \Gamma^{2\epsilon}_t are some real functions of time (depending on \epsilon) defined on [0,T], T < \infty and assume that ... • 71 0 votes 0 answers 61 views ### Array power-means / generalisation of Hölder inequality Now cross-posted to MO. If you have a comment or answer, please post it there instead. Summary: I have a generalisation of Hölder's inequality that I can neither prove nor find references to. ... 1 vote 0 answers 77 views ### Is d(x, y)={(\sum_{i=1}^{n}{|{x_i} -{y_i}|}^p})^{1/p} , x, y\in\mathbb{R^n}, 0<p<1 metric on \mathbb{R^n}? X=\mathbb{R^n} Define , d:X×X\rightarrow\mathbb{R} by d(x, y)={(\sum_{i=1}^{n}{|{x_i} -{y_i}|}^p})^{1/p} , x, y\in\mathbb{R^n}, 0<p<1 Question: Is d a metric on \mathbb{R^n}? ... • 4,246 2 votes 1 answer 68 views ### Is true \int\limits_{-\infty}^{\infty} f(t)\,g(t)\,dt \leq \sqrt{\int\limits_{-\infty}^{\infty} |f(t)|^2 dt}\,\cdot \sup\limits_t|g(t)|? Does holds \int\limits_{-\infty}^{\infty} f(t)\,g(t)\,dt \leq \sqrt{\int\limits_{-\infty}^{\infty} |f(t)|^2 dt}\,\cdot \sup\limits_t|g(t)| true for every real valued functions f(t),\, g(t)? I want ... • 879 1 vote 0 answers 67 views ### Prove \sum_{k=1}^n a_kb_k \leq \left (\sum_{k=1}^n a_k^p \right)^{1/p} \left (\sum_{k=1}^n b_k^{q} \right)^{1/q} without Hölder's inequality I have proved this version by applying Hölder's inequality . Could you confirm if my proof is correct? I also would like to ask for a non-measure-theoretic proof. For p,q \ge 0 such that 1/p+1/q=1... • 10.2k 2 votes 2 answers 91 views ### A generalized version of Hölder's inequality I've found this version from this Wikipedia page. I've re-written the proof to make my understanding clear. Could you confirm if my attempt is correct? Let (X, \mathcal A, \mu) be a \sigma-... • 10.2k 0 votes 1 answer 28 views ### Hölder condition for the function f(x)=x^{\frac{1}{n}} [closed] I think my algebra is not very good.. I'm trying to show that f:[0,\infty)\to[0,\infty) defined as f(x)=x^{\frac{1}{n}} ,n\in\mathbb{N} satisfies the Hölder condition |f(x)-f(y)|\leq c|x-y|^{\... • 343 1 vote 0 answers 91 views ### Baby rudin theorem 8.18 \Gamma(x) = \int_0^\infty t^{x-1}e^{-t}dt. Here is the Hölder's inequality: (1/p) + (1/q) = 1. |\int_a^bfg d\alpha| \leq {\int_a^b |f|^p d\alpha}^{1/p} {\int_a^b |g|^q d\alpha}^{... 1 vote 1 answer 60 views ### Show E(|X|) \ge \frac{1}{\sqrt{E(X^4)}} We are asked to show the following: E(|X|) \ge \frac{1}{\sqrt{E(X^4)}} Given that E(X^2) = 1 and E(X^4) < \infty Using Holder's inequality I can show that E(X^4) \le 1 assuming X,Y = X^2,... • 458 1 vote 0 answers 68 views ### It is true that L^p(\mathbb{R}^n)\subset \mathscr{S}'(\mathbb{R}^n), 1\leq p\leq\infty? Remember that \mathscr{S}' is the space of tempered distributions. In a certain text they suggest that this statement is true, and that he uses Holder's inequality to prove it. The question is, how ... • 355 1 vote 2 answers 97 views ### Show the d_p(x,y) is not a metric on \mathbb{R}^2 Question: For 0\lt p\lt 1 and x, y \in\mathbb{R}^2 with x=(\xi_1, \xi_2), y = (\eta_1, \eta_2), let d :\mathbb{R}^2 \times \mathbb{R}^2 \rightarrow \mathbb{R}  Be defined as d_p(x, y) = (|\... 2 votes 3 answers 156 views ### Prove that \sqrt[3]{a} + \sqrt[3]{b} + \sqrt[3]{c} \ge ab + bc + ca For all positive a,b,c  satisfying a+b+c = 3,Prove:$$ \sum_{cyc} \sqrt[3]{a} \ge \sum_{cyc} ab  This is a hard problem and I tried it myself, but it's really hard without using advanced ...
For positive numbers $a$, $b$, $c \geq 0$ and $a+b+c=1$ show that: $\sqrt[3]{4+17a^2b}+\sqrt[3]{4+17b^2c}+\sqrt[3]{4+17c^2a}+10 \Big(\frac{1}{27}-abc \Big) \geq 5$ I tried to use $AM-GM$ with \$\frac{...