How to prove that $|x^p-y^p| \leq p|x-y|(x^{p-1}+y^{p-1})$ when $x,y \geq 0$ and $p \geq 1$?
129331
Prove that $|x^p - y^p| \le p|x-y|(x^{p-1} + y^{p-1})$ provided that $1 \le p \lt \infty$ and $x, y \ge 0$
41300
Show $(x+y)^a > x^a + y^a$ for $x,y>0$ and $a>1$
129584
How prove $E|X|^p<\infty$ and $E|Y|^p<\infty$, if $E|X+Y|^p<\infty$
33332
A classic exponential inequality: $x^y+y^x>1$
137511
Trying to show that $\sup \{b^p:p\in Q,\;0<p<x\} = \inf\{b^q:q\in Q,\;x<q\}$
109778
Proving $(x-c)|(p(x)-p(c))$
88687
Proof of the inequality $||a+b|^p - |a|^p| \leq \epsilon |a|^p + C_{\epsilon} |b|^p$
46760
To show that $P(|X-Y| \leq 2) \leq 3P(|X-Y| \leq 1)$
25419
If $n\geq m$ then $(x^m+y^m)^{1/m} \ge (x^n+y^n)^{1/n}$
106767
$|p(z)| \leq M$ for $|z| \leq 1$ Show that $|p(z)| \leq M|z|^n$ for $|z| \geq 1$
72081
$\forall x,y>0, x^x+y^y \geq x^y + y^x$
61472
If $0<x<y$, then prove that $\sqrt{x} <\sqrt{y}$ and $x <\sqrt{xy} <y$
126165
Prove $|\arctan x - \arctan y|\leq |x-y|$
88036
Prove that if $p \equiv 3 \pmod{4}$ then $x^2+y^2 \not \equiv 0 \pmod{p}$
134348
When is $(p - 1)! + 1$ a power of $p$?
128558
Inequality between $\ell^p$-norms
66247
Proving inequality $\left( \frac {x} {y}\right) ^{x}\left( \frac {1-x} {1-y}\right) ^{1-x}\geqslant 1$