If we count the palindromes among first $n$ numbers in all natural bases $b>1$, but ignore the one-digit-palindromes (ones that fill up the black triangle on the picture), Which base $b$ will contain the most palindromes? The $x$-axis represents the numbers, and the $y$-axis repr...

The is the standard version of Young's inequality $$ab \leq \frac{a^p}{p}+\frac{b^q}{q}$$ for $a,b,p,q >0$ and $$\frac{1}{p}+\frac {1}{q}=1$$ But there is another formula called generalization of Young inequality. Why it's a generalization? let $f$ denote a real-valued, continuous and strictl...

According to wikipedia, "let $f$ denote a real-valued, continuous and strictly increasing function on [0, c] with c > 0 and f(0) = 0. Let $f^{−1}$ denote the inverse function of $f$. Then, for all a ∈ [0, c] and b ∈ [0, f(c)]," $$ab \le \int_0^a f(x)\,dx + \int_0^b f^{-1}(x)\,dx$$ I am wonderi...

Recall that the convolution of two functions is given by $$f*g(y)=\int f(x)g(y-x)dx.$$ The well known inequality known as Young's inequality, say that $$\|f*g\|_r\leq\|f\|_p\cdot\|g\|_q $$ provided $\frac 1p + \frac 1q = 1 + \frac 1r$ and $1\le p,q,r\le\infty$. Obvious implications is that $...

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...

Let $x, y, z \geqslant 0$ and let $p, q, r > 1$ be such that $$ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 1. $$ How can one show that under these hypotheses we have $$ xyz \leqslant \frac{x^p}{p} + \frac{y^q}{q} + \frac{z^r}{r} $$ with equality if and only if $x^p = y^q = z^r$, using twice the s...

I need to prove that if $1 < p < \infty$ and $a, b \geqslant 0$ then $$ ab \leqslant \frac{a^{p}}{p} + \frac{b^{q}}{q}$$ where $\frac 1p+\frac 1q=1$. I fix $b$ and maximize the function $f(a) = ab - \frac{a^{p}}{p}$, but the maximum I find is $b^{q}$ with $q = \frac{1}{p-1}$. I have no idea how...

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...

I am trying to prove Young's Inequality by considering the function $$h(u) = \frac{u^p}{p} + \frac{C^q}{qu^q}$$ for $C,u>0$ and $p,q >1$. We also require $$\frac{1}{p}+\frac{1}{q}=1$$ so that $pq=p+q$. I starting by using straightforward calculus to show that $h$ has a global minimum at $u=C^{\fr...

Part of an exercise to prove Holder's inequality in Rudin involves proving Young's Inequality... That is, given $\frac{1}{p}+\frac{1}{q} = 1$, prove $$ab \leqslant \frac{a^p}{p} + \frac{b^q}{q}.$$ Here's my attempt at a proof: Let $$f(x) = \frac{x^p}{p} + \frac{b^q}{q} -bx$$ then, $$f...

I am referring to the inequality: Young's inequality The standard version for increasing functions. I read the article of Young and also a generalization of this claim in Hardy, Littlewood and Polya's Inequalities. But I don't see that Young proves rigorously his claim, and in Zygmund's Trigono...

Can you change the name of a tag? I recently came across the tag sage, which I think should be renamed to sagemath to be more clear. Is it possible to change the name of a tag? If so, how?

What do I need to do to rename a tag? I see some people post a question here and ask for a tag to be renamed. Is there some other way?