1. ## proof question 2,1

it is given that
$\displaystyle 1\leq p< +\infty\\$
$\displaystyle \alpha ,\beta >0 \\$
$\displaystyle a,b\geq 0\\$

prove that
$\displaystyle (\alpha a+\beta b )^p\leq (\alpha +\beta )^p\left ( \frac{\alpha }{\alpha +\beta }a^p+\frac{\beta }{\alpha +\beta }b^p \right )$

hint: prove first that $\displaystyle f(t)=t^p$ is a convex
on this region $\displaystyle [0,+\infty)$
reminder: function f(t) is called convex on some region if for every b,a
and on
$\displaystyle 0\leq \lambda \leq 1\\$
we have
$\displaystyle f(\lambda a +(1-\lambda)b)\leq\lambda f(a)+(1-\lambda)f(b)$

my thoughts:
i know from calc1 that a function is convex if its second derivative is negative or something (i am not sure)

i dont know
how to prove that $\displaystyle f(t)=t^p$ is negative
its pure parametric thing

??

2. Originally Posted by transgalactic
it is given that
$\displaystyle 1\leq p< +\infty\\$
$\displaystyle \alpha ,\beta >0 \\$
$\displaystyle a,b\geq 0\\$

prove that
$\displaystyle (\alpha a+\beta b )^p\leq (\alpha +\beta )^p\left ( \frac{\alpha }{\alpha +\beta }a^p+\frac{\beta }{\alpha +\beta }b^p \right )$

hint: prove first that $\displaystyle f(t)=t^p$ is a convex
on this region $\displaystyle [0,+\infty)$
reminder: function f(t) is called convex on some region if for every b,a
and on
$\displaystyle 0\leq \lambda \leq 1\\$
we have
$\displaystyle f(\lambda a +(1-\lambda)b)\leq\lambda f(a)+(1-\lambda)f(b)$

my thoughts:
i know from calc1 that a function is convex if its second derivative is negative or something (i am not sure)

i dont know
how to prove that $\displaystyle f(t)=t^p$ is negative
its pure parametric thing

??

Yes....almost. A twice derivable function on some interval is convex (upwards) there iff its second derivative there is positive, and:

$\displaystyle f(t)=t^p \Longrightarrow f''(t)= p(p-1)t^{p-2}\geq 0$

Tonio

$\displaystyle f(t)=t^p \Longrightarrow f''(t)= p(p-1)t^{p-2}\geq 0$

i dont know anything about p ant t

4. Originally Posted by transgalactic
$\displaystyle f(t)=t^p \Longrightarrow f''(t)= p(p-1)t^{p-2}\geq 0$

i dont know anything about p ant t

Tonio

5. the only f(t) i see is just to explain whats convex by definition
and the formula presented is not a part of the question

6. Originally Posted by transgalactic
the only f(t) i see is just to explain whats convex by definition
and the formula presented is not a part of the question

Please do not take this as an offense, but from this and the other thread you started and the responses I got there from you, I think you should really begin to worry about learning SERIOUSLY the very basic facts of the subjects you're asking about. It seems like you don't have the slightest idea what you're talking about, you're completely lost... and yet you're the one who sent the question!

I, for one, am not here to teach from scratch a whole subject to anyone so that he/she will be able to understand solutions to his/her own questions.

you better study the basics, understand definitions and notation, and THEN, if you get stuck somewhere, you ask. That's, btw, the only way to learn something in mathematics, imo.

Tonio

Ps. The answer to this thread's question is already in my prior messages...and no: the $\displaystyle f(t)$ given in your question is not to explain convexity: you need it to approach the question and eventually reach a solution, and also p is given and from where t is taken...