Intereting Posts

Equidecomposability of a Cube into 6 Trirectangular Tetrahedra
Calculus and Category theory
How can I prove the convergence of a power-tower?
Effect the zero vector has on the dimension of affine hulls and linear hulls
Understanding definition of tensor product
Solve for $\theta$: $a = b\tan\theta – \frac{c}{\cos\theta}$
Can we determine uniform continuity from graphs?
Piecewise function plot in Matlab
If $f$ is differentiable at $x = x_0$ then $f$ is continuous at $x = x_0$.
Product of independent random variables
The meaning of notation $\subset\subset$ in complex analysis
Proving a particular subset of $R^n$ is closed
A gamma function inequality
How is Cauchy's estimate derived?
Is $V$ a simple $\text{End}_kV$-module?

How to prove such inequality: $|x|^p+|y|^p\geq |x+y|^p$ for $0<p\leq 1$ and $x,y \in \mathbb{R}$?

- Is $ \pi $ definable in $(\Bbb R,0,1,+,×, <,\exp) $?
- Is this a perfect set?
- If $A$ and $B$ are sets of real numbers, then $(A \cup B)^{\circ} \supseteq A^ {\circ}\cup B^{\circ}$
- Understanding the relationship of the $L^1$ norm to the total variation distance of probability measures, and the variance bound on it
- Why does this limit exist $x^{x}$
- True Definition of the Real Numbers
- Is it mathematically wrong to prove the Intermediate Value Theorem informally?
- Proof regarding Robin's inequality (RI).
- Uniform convergence of the sequence $f_n(x)=f(x+1/n)$ for uniformly continuous $f$
- Convergence Problem.

For $p=1$, we can prove this using the fact that $x^p$ is convex.

If $x$ or $y$ is $0$, then the result is obvious. For $0<p<1$, $x^p$ is concave on $(0, \infty)$. So for $x,y > 0$, $$\frac{y}{x+y}0^p + \frac{x}{x+y}(x+y)^p \le x^p $$$$\frac{x}{x+y}0^p + \frac{y}{x+y}(x+y)^p \le y^p $$

So for $x,y>0$$$x^p + y^p \ge (x+y)^p$$

And hence, (assuming the result with $p=1$) $\forall x,y \in \mathbb R$, $$|x+y|^p \le(|x| + |y|)^p \le|x|^p + |y|^p$$

- What is a general solution to a differential equation?
- First-order nonlinear ordinary differential equation
- The standard role of intuitive numbers in the foundations of mathematics
- The sum of three consecutive cubes numbers produces 9 multiple
- Integral of $\sqrt{1-x^2}$ using integration by parts
- Properties of automorphism group of $G={Z_5}\times Z_{25}$
- Probability the three points on a circle will be on the same semi-circle
- Asymptotic (divergent) series
- Formula for $\sum_{k=0}^n k^d {n \choose 2k}$
- Is $\{ \sin n^m \mid n \in \mathbb{N} \}$ dense in $$ for every natural number $m$?
- A finite ring is a field if its units $\cup\ \{0\}$ comprise a field of characteristic $\ne 2$
- Where we have used the condition that $ST=TS$, i.e, commutativity?
- Weak closure of $\{\sqrt n e_n|n\in \mathbb N\}$ and metrizability of weak topology
- Convexity of a complicated function
- Limit of $\lim_{x\rightarrow 1}\sqrt{x-\sqrt{x-\sqrt{x-\sqrt{x-…}}}}$