How to show that, $(x+y)^p\leq x^p+y^p$, where for $0\leq p\leq 1,x\geq 0, y\geq0?$ Any suggestion how to prove it?   Thanks in advance.
39825
If $p,q$ are positive quantities and $0 \leq m\leq 1$ then Prove that $$(p+q)^m \leq p^m+q^m$$    Trial: For $m=0$, $(p+q)^0=1 &lt; 2= p^0+q^0$  and for $m=1$, $(p+q)^1=p+q =p^1+q^1$.   So, For $m=0,1$ the inequality is true.How I show that the inequality is also true for $0 &lt; m &lt; 1$.   Please help.
6480
If $x,y$ are positive reals such that $x+y = 2$ show that $x^3y^3(x^3+y^3) \leq 2$.   I thought about working backwards on this one. We can try to prove that $x^3y^3(x^3+y^3) \leq 2$ is true by first realizing that $x^3+y^3 = (x+y)(x^2-xy+y^2)$ and substituting in for $x+y = 2$. We then get $x^3y^3(x^2-xy+y^2) \leq 1$. How do I prove this is true?
62621
Hy all! I'm having trouble finding a proof for the following problem:  Show that $a^x+b^x+c^x&gt;(a+b+c)^x$, if $a,b,c&gt;0$ and $0&lt;x&lt;1$ (over the real numbers).  This inequality has been torturing me for long hours the least. I couldn't find anything on Google. This kind of equation isn't made for search engines tbh.  Anyway, I really thinked about it a lot but didn't make any significant progress. It would be great to see a proof without the really high-end Mathematics. Any ideas to start with?
55235
For any real numbers $a$ and $b$ and $1 \leq p &lt; \infty$, prove that   $$|a+b|^p \leq 2^p \{ |a|^p +|b|^p \}$$  This inequality is given in the the book Real Analysis by Royden, Chapter $7$, page $136$. I don't understand how the author comes to this inequality. Can anyone provide some hints?
76950
can anyone help me show that if $f\in L^p(a, b)$ then $$ \lim_{t\to 0^{+}}\int_{a}^b|f(x+t)-f(x)|^p\ dx=0.$$ Thanks, any help will be useful..
87446
Let $x\in\mathbb{C}^n$. Do the following inequalities hold? $$\lVert x\lVert_2\leq\sqrt{\lVert x\lVert_1\lVert x\lVert_\infty}\leq\frac{1+\sqrt{n}}{2}\lVert x\lVert_2.$$ I think the first inequality is the Holder inequality with $p=1$ and $q=\infty$.
46332
OK guys I have this problem:  For $x,y,p,q&gt;0$ 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?
67900
I've been having a look at how Lieb and Loss (in their textbook on Analysis) prove Hanner's inequalities and have been trying to get a handle of the geometric intuition involved. In doing so I've been trying to prove that:  $(a+b)^p + (a-b)^p \geq 2a^p +p(p-1)a^{p-2}b^2$  For any $1\leq p\leq 2$ and $0&lt;b&lt;a$.  This is exercise 4 in chapter two of the aforementioned book. Can anyone give me any tips (or some geometric intuition)?  Also can anyone give me any explanation of why uniform convexity is important, and what it might mean geometrically?  Edit: for the latter question the following is useful but I would appreciate any more comments.
28282
Assuming that $e&lt;y&lt;x$, prove that $ x^y &lt; y^x$.  I think this must be easy, but I can't work it out. Thanks in advance for any kind of help.
86253
Prove that $n&lt;3^n$ where $n \in \mathbb N$,  when $ n=1 $, I have proved it's true. And assumed when $n = p$ , $p&lt;3^p$ is true. Can any body help me in showing that it is true for $n =p+1$
64023
let $x,y,z,w&gt;0$,show that $$(x+y)^3+(y+z)^3+(z+w)^3+(w+x)^3\ge 8(x^2y+y^2z+z^2w+w^2x)$$  it seem use AM-GM inequality to solve it,But I can't it,Thanks
14394
Prove that  $(x_1+\dots+x_n)^2 \leq n(x_1^2 + \dots+x_n^2)$  for all positive integers n and all real numbers $x_1,....,x_n$  I am attempting a proof by induction but wasn't sure if i need the Cauchy-Schwarz Inequality or perhaps another way other than induction to prove this.  Proof   $n=1$ true  assume true for $n=k$  Now for $n= k+1$  $(x_1 + \dots +x_k + x_{k+1})^2 \leq \dots$
84973
We have $x, y, z \in \mathbb R$ such that $x+y+z=1$. Prove that $x^2+y^2+z^2 \geq \frac{1}{3}$. I am able to do this using the relationship between the power and arithmetic means. Is there a way to not use this relationship?
1761
I am looking for a proof that this inequality: $$(W_1+W_2+\cdots+W_n)^a \leq  W_1^a +\cdots +  W_n^a$$  is valid.  I have a power function $f(W)=W^a$ where $a$ is a real number, constant but usually $0\lt a\lt 1$. The $W_i$ are real positive numbers.  Then I want to check if $\left(\sum W_i\right)^a \leq \sum(W_i^a)$   For example for $n=2$ : $(x+y)^a \leq x^a + y^a$.  Thanks.
38991
I find it very strange that $$ e^x \geq x^e \, \quad \forall x \in \mathbb{R}^+.$$   I have scratched my head for a long time, but could not find any logical reason. Can anybody explain what is the reason behind the above inequality? I know this is math community, but I'd appreciate a more intuitive explanation than a technical one.
72180
This is a classic problem, but I haven't seen it on the site before.     Suppose $x$ and $y$ are real numbers that both lie in the interval   $(0,1)$. Prove  $$x^y+y^x&gt;1.$$   The following spoiler box contains a hint that indicates the solution I know. Does anyone know a more natural solution?       Use Bernoulli's inequality to show $$x^y\ge \frac{x}{x+y}.$$   Edit: I recently learned the source is the 1996 French math olympiad.
99514
Can anyone see how to prove the following?  If $x$ and $y$ are real numbers with $y\geq 0$ and $y(y+1) \leq (x+1)^2$ then $y(y-1) \leq x^2$.  It seems it is true at least according to Mathematica.