# Questions tagged [holder-inequality]

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

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### Prove $\sum\limits_{cyc}\,\frac{a}{\sqrt{b(\,a+ b\,)}}\geqq \sum\limits_{cyc}\,\frac{a}{\sqrt{b(\,c+ a\,)}}$ with $a,\,b,\,c> 0$

Let $a,\,b,\,c$ be positive numbers. Prove that $$\sum\limits_{cyc}\,\frac{a}{\sqrt{b(\,a+ b\,)}}\geqq \sum\limits_{cyc}\,\frac{a}{\sqrt{b(\,c+ a\,)}}$$ I tried Holder and $\lceil$ https://...
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### Prove that $\sum\limits_{cyc}\,\frac{a^{\,2}}{bc+ a}\geqq \sum\limits_{cyc}\,\frac{a}{\sqrt{2\,bc+ 2}}$ [on hold]

Let $a,\,b,\,c$ be positive numbers. Prove that $$\sum\limits_{cyc}\,\frac{a^{\,2}}{bc+ a}\geqq \sum\limits_{cyc}\,\frac{a}{\sqrt{2\,bc+ 2}}$$ I tried Holder Inequality (it's only the hint to get you ...
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### Norm of function in $L^p$ as a sup of integrals

Let $p, q$ be positive real numbers that satisfy $\frac{1}{p} + \frac{1}{q} = 1.$ I want to show that, if $\color{magenta}{\mathbf{f \in L^p}}$ $||f||_p = \sup \displaystyle \int f \cdot g$, where ...
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### How to apply Holder's inequality to this problem?

I want to show the following: On a set of real numbers that has finite measure, if $a > b$ and $||f_n - f||_a \rightarrow 0$, then $||f_n - f||_b \rightarrow 0$ as well. Perhaps I should be using ...
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### show this inequality to $\sum_{cyc} \frac {a^3b}{(3a+2b)^3} \ge \sum_{cyc} \frac {a^2bc}{(2a+2b+c)^3}$

Let $a,b$ and $c$ be positive real numbers. Prove that $$\sum_{cyc} \frac {a^3b}{(3a+2b)^3} \ge \sum_{cyc} \frac {a^2bc}{(2a+2b+c)^3}$$ This problem is from Iran 3rd round-2017-Algebra final exam-P3,...
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### An application of Holder's Inequality

Suppose $1\leq p,q\leq \infty$ and $1/p+1/q=1$. Let $f\in\mathcal{L}^p(E)$. Show that $f=0$ a.e. if and only if \begin{align*} \int_E f\cdot gdm=0 \end{align*} for all $g\in \mathcal{L}^q(E)$. ...
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### Prove that $({a\over a+b})^3+({b\over b+c})^3+ ({c\over c+a})^3\geq {3\over 8}$

Let $a,b,c$ be positive real numbers. Prove that $$\Big({a\over a+b}\Big)^3+\Big({b\over b+c}\Big)^3+ \Big({c\over c+a}\Big)^3\geq {3\over 8}$$ If we put $x=b/a$, $y= c/b$ and $z=a/c$ we get $xyz=1$ ...
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### $\frac{a}{\sqrt{a^2 + 8bc}} +\frac{b}{\sqrt{b^2 + 8ac}} + \frac{c}{\sqrt{c^2 + 8ab}} \ge \frac{1}{\sqrt{a^3+b^3+c^3 + 24abc}}$ is true?

In one of the solutions of a problem in this site: https://artofproblemsolving.com/wiki/index.php?title=2001_IMO_Problems/Problem_2 It is used the following: If $a,b,c$ are positive real numbers ...
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### $L_p$ Norms and Holder's Inequality question

Suppose that $-∞ < a < b < ∞$ and $1 < p < q < ∞$. Let L_p[a,b] = \{ f :\Bbb R \to \Bbb R : \left( \int_a^b\left|f\left(x\right)\right|^p~\mathrm dx\right)^{\frac{1}{p}} < ∞ \}....