# Questions tagged [holder-inequality]

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

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### Application of Arzelà-Ascoli theorem

I am trying to understand the wikipedia article on the Arzelà-Ascoli theorem, which can be found here: https://en.wikipedia.org/wiki/Arzel%C3%A0%E2%80%93Ascoli_theorem More specifically, I am trying ...
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### Let $f, g :(0,1) \to \mathbb{R}\,\,$ s.t. $\lVert g \rVert_2 =3$ and $\lVert f \rVert_1 = e\,\,$, show that $|\int_{0}^{1} g \sqrt{\log(f)}dx| \le 3$

Let $g\in L^2((0,1))$ s.t. $\lVert g \rVert_2 =3\,\,\,$ and $f \ge1, f \in L^1((0,1))$ s.t. $\lVert f \rVert_1 = e\,\,$, prove that: $$\Biggl|\int_{0}^{1} g \sqrt{\log(f)}\,dx \Biggr|\,\,\le\,\,3$$ ...
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### Algebraic inequality $\sum \frac{x^3}{(x+y)(x+z)(x+t)}\geq \frac{1}{2}$

The inequality is $$\frac{x^3}{(x+y)(x+z)(x+t)}+\frac{y^3}{(y+x)(y+z)(y+t)}+\frac{z^3}{(z+x)(z+y)(z+t)}+\frac{t^3}{(t+x)(t+y)(t+z)}\geq \frac{1}{2},$$ for $x,y,z,t>0$. It originates from a 3-D ...
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### The convergence of $\int_0^{\infty} \sin(x)(x-[x])/x^{\alpha}$

I need to evaluate the following integral $$\int_0^{\infty} \sin(x)(x-[x])/x^{\alpha}dx$$ where $\alpha\in (0,1)$ and $[x]$ is the floor function. Without $x-[x]$, we can evaluate the integral easily ...
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### Application of Hölder to prove $L^q \subset L^p$ for $1\leq p\leq q<\infty$ with Lebesgue measure

I'm working on the following problem and got stuck. Any help would be really appreaciated. Let $a,b\in\mathbb{R}$ and $1\leq p \leq q < \infty$. Show that for any $f\in L^q([a,b])$ the following ...
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### $\frac{a^3}{b+c}+\frac{b^3}{a+c}+\frac{c^3}{a+b}\geq \frac{3}{2}$ for $a,b,c\in\mathbb{R}^+$ with $abc=1$

Suppose that $a,b,c$ are positive reals such that $abc=1$. Prove that $$\frac{a^3}{b+c}+\frac{b^3}{a+c}+\frac{c^3}{a+b}\geq \frac{3}{2}.$$ Hint: Use Titu's lemma. My approach: I am trying to use Titu'...
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### Holder inequality of Schatten norm for p=q=2 and Hermitian operators

I was reading some Linear Algebra contents and then I encountered this problem, I tried to prove this by using for example Holder inequality for Schatten norms, But I didn't succeed. The question is : ...
### Show that $\int_0^1 f^3(x) dx + \frac{4}{27} \ge \left( \int_0^1 f(x) dx \right)^2$, where $f',f'' >0$
Let $f :[0,1] \to [0,\infty)$, $f$ is twice differentiable, $f'(x) >0$, $f''(x) >0$ for all $x\in [0,1]$. Prove that $$\int_0^1 f^3(x) dx + \frac{4}{27} \ge \left( \int_0^1 f(x) dx \right)^2.$$ ...