Articles of young inequality

Young's inequality without using convexity

I was doing some problems from Rudin’s Principles of Mathematical Analysis and came across a problem in which he asks you to prove Hölder’s inequality via Young’s inequality: If $u$ and $v$ are nonnegative real numbers, and $p$ and $q$ are positive real numbers such that $\displaystyle \frac{1}{p}+\frac{1}{q}=1$, then $\displaystyle uv \leq \frac{1}{p}u^p+\frac{1}{q}v^q$. I’m familiar […]

Equality in Young's inequality for convolution

I am trying to understand how sharp Young’s inequality for convolution is. The inequality says $||f \ast g||_r \leq ||f||_p ||g||_q$ where as $1/p+1/q = 1+1/r$. Actually, there are a couple of papers (for example: Sharpness in Young’s inequality for convolution) talking about the case of $p, q>1$, and the answer is that we can […]

Prove that $xy \leq\frac{x^p}{p} + \frac{y^q}{q}$

OK guys I have this problem: For $x,y,p,q>0$ and $ \frac {1} {p} + \frac {1}{q}=1 $ prove that $ xy \leq\frac{x^p}{p} + \frac{y^q}{q}$ It says I should use Jensen’s inequality, but I can’t figure out how to apply it in this case. Any ideas about the solution?

Purely “algebraic” proof of Young's Inequality

Young’s inequality states that if $a, b \geq 0$, $p, q > 0$, and $\frac{1}{p} + \frac{1}{q} = 1$, then $$ab\leq \frac{a^p}{p} + \frac{b^q}{q}$$ (with equality only when $a^p = b^q$). Back when I was in my first course in real analysis, I was assigned this as homework, but I couldn’t figure it out. I […]

Geometric interpretation of Young's inequality

Is there a geometric interpretation of Young’s inequality, $$ab \leq \frac{a^{p}}{p} + \frac{b^{q}}{q}$$ with $\dfrac{1}{p}+\dfrac{1}{q} = 1$? My attempt is to say that $ab$ could be the surface of a rectangle, and that we could also say that: $\dfrac{a^{p}}{p}=\displaystyle \int_{0}^{a}x^{p-1}dx$, but them I’m stuck.