I am trying to prove or disprove the triangle inequality for the function:
$f(X) = \sqrt{\sum\limits_{k=1}^{d}\frac{|X_{k}|^{2}}{k}}$ ; where $X \in R^{d}$.
I tried whatever I could, and my calculations are shown below. I want to know if we can take it any further from there or is there some other way to find the result.
$$f(X + Y) = \sqrt{\sum\limits_{k=1}^{d}\frac{|X_{k} + Y_{k}|^{2}}{k}}$$ According to Minkowski Inequality, $\sqrt{a^{2}} + \sqrt{b^{2}} \geq \sqrt{a^{2} + b^{2}}$ $$ \implies f(X + Y) \leq \sum\limits_{k=1}^{d}\sqrt{\frac{|X_{k} + Y_{k}|^{2}}{k}} = \sum\limits_{k=1}^{d}\frac{|X_{k} + Y_{k}|}{k} \\ \leq \sum\limits_{k=1}^{d}\frac{|X_{k}|}{\sqrt{k}} + \sum\limits_{k=1}^{d}\frac{|Y_{k}|}{\sqrt{k}} = \sum\limits_{k=1}^{d}\sqrt{\frac{|X_{k}|^{2}}{k}} + \sum\limits_{k=1}^{d}\sqrt{\frac{|Y_{k}|^{2}}{k}} $$
If we further try to solve by applying Minkowski Inequality again, we will get $\geq$ and that would mean that we are not reaching any meaningful conclusion...