What is Holder's Inequality?

Question

I have done some research on the internet about the inequality called Holder, but I've encountered some explanations in sites like Turkis Math Wikipedia or Wolfram but none of those explanations were helpful enough, I want to be able to define what Holder's Inequality is and solve inequality questions using Holder's, I need an elementary level explanation, I do not know integrals, I am not very well informed about series (at least in some level), I am a Turkish high school student in preparation for Maths Olympiad, so I know about Cauchy-Schwarz, AM-GM and all these other concepts, if I am not advanced enough to understand the definition of it or put it to application, what topics should I study first?

What are your suggestions?

Thank you:)

see here: https://services.math.duke.edu/~wka/math204/conv.12.4.pdf i hope this will help you! — Dr. Sonnhard Graubner, Sep 24 '17 at 14:49
Thank you:) There are of course parts that I am not sure of, or cannot understand completely, but now I have an idea about what to study first:) — Deniz Tuna Yalçın, Sep 24 '17 at 14:52

score 6 · Accepted Answer · edited Oct 11 '18 at 10:12

For Math Olympiad purposes you just need basic algebra to understand Hölder's inequality, so if you're familiar with means inequalities and Cauchy-Schwarz you're just fine.

Anyway, this is Hölder's inequality, as I know it. Let $p$, $q$ be positive real numbers such that $\frac{1}{p} + \frac{1}{q} = 1$. Also, let $a_1, \: \dots, \: a_n$ and $b_1, \: \dots, \: b_n$ be nonnegative real numbers. Then $$\sum_{i = 1}^n a_ib_i \le \left(\sum_{i = 1}^n a_i^p\right)^{1/p} \left(\sum_{i = 1}^n b_i^q\right)^{1/q}$$

What is Holder's Inequality?

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