I'm solving some problems to prepare for my phd qualifying exam on functional analysis and measure theory. I want to prove that given a measure space $(X,\mathcal{A},\mu)$ for every $0<p<\infty$, there exists a constant $C_p>0$ such that for every $f,g\in L_p(\mu)$ we have the inequality
$$\lVert f+g \rVert_{L^p} \le C_p(\lVert f \rVert_{L^p}+ \lVert g\rVert_{L^p}).$$
I also need to find the optimal value for $C_p$ when $p<1$.
By now, I figured out that the case $p>1$ is a weaker statement than Minkowsky inequality taking $C_p=1$.
I also found this article where it is stated that $\lVert f+g \rVert_p \le 2^\nu (\lVert f \rVert_p+ \lVert g \rVert_p)$, where $\nu=(1-p)/p$, and that it can be proven using the function $(1+x^p)/(1+x)^p$. I don't see how could I relate this function to the desired result. I've tried some ideas but I wasn't able to link this function with a relation with the integral. Also it would be very instructive for me to see the way we approach $L^p$ spaces when the usual tools are out of the picture.
Any hint/answer is appreciated, thanks in advance.
Note that if you can prove that $ A \leq (1+x^p)/(1+x)^p \leq B $ for $x>0$ and $A,B>0$ independent of $x$, then it follows, by replacing $x$ by $y/x$ (with $x,y>0$) and multiplying up that $$ A(x^p+y^p) \leq (x+y)^p \leq B (x^p+y^p) \tag{1} $$ Replacing $x$ by $\lvert f \rvert$ and $y$ by $\lvert g \rvert$, the right-hand inequality becomes $$ (\lvert f \rvert + \lvert g \rvert )^p \leq B (\lvert f \rvert^p + \lvert g \rvert^p) . $$ The triangle inequality gives the additional inequality $ \lvert f + g \rvert^p \leq (\lvert f \rvert + \lvert g \rvert )^p $ and integrating then gives $$ \lVert f+g \rVert_p^p \leq B( \lVert f \rVert_p^p + \lVert g \rVert_p^p ) , $$ and now we can use the left-hand side of (1), so $$ \lVert f+g \rVert_p^p \leq \frac{B}{A}( \lVert f \rVert_p + \lVert g \rVert_p )^p , $$ and taking $p$th roots will give the result.