I'm trying to prove that the function $f:\mathbb{R}\to \mathbb{R}$, $$ f(x) = |x|^p \quad ,\quad p \geq 1$$ is convex. By using the definition of a convex function and simplifying a bit, I arrived in the following inequality $$ |a+b|^p \leq |a|^p+|b|^p $$ where a and b are real numbers.
If I can prove that this holds, then $f$ is convex. However, I'm a bit lost here. I'm aware of the triangle inequality, but that only proves it for $p=1$. Is there such thing as a triangle inequality to the power of $p$? If not, any other suggestions would appreciated.
No it does not hold for a,b greater than 1. Let b=ca. LHS is (1+c)^p a^p whereas RHS is (1+c)a^p . The reason it doesn't work is because x^p is increasingly increasing so it is more increasing between a and a+b than it is between 0 and b or 0 and a.