Suppose that $X$ is a random varible with moments $\mu_1'$, $\mu_2'$, $\mu_3'$, $\mu_4'$ and central moments $\mu_1=0$, $\mu_2=\text{Var}(X)$, $\mu_3$ and $\mu_4$. Define the cumulants \begin{aligned} k_1&=\mu_1',\\ k_2&=\mu_2'-(\mu_1')^2=\mu_2,\\ k_3&=\mu_3'-3\mu_1'\mu_2'+2(\mu_1')^3=\mu_3,\\ k_4&=\mu_4'-4\mu_1'\mu_3-3(\mu_2')^2+12(\mu_1')^2\mu_2'-6(\mu_1')^4=\mu_4-3\mu_2^2, \end{aligned} and standardized cumulants $\rho_r=\frac{k_r}{k_2^{r/2}}$. My text mentions the following inequality in passing $$ \rho_4\geq\rho_3^2-2. $$ I can prove this particular one but the text has a few others. What is a source of statements and proofs for similar inequalities?
My proof (please share if you have a different one): the required inequality is equivalent to $$ \frac{k_4}{k_2^2}\geq\frac{k_3^2}{k_2^3}-2\iff k_2k_4\geq k_3^2-2k_2^3. $$ Then I express $k$'s in terms of the $\mu$'s to rewrite the inequality as $$ k_2k_4\geq k_3^2-2k_2^3\iff\mu_2(\mu_4-3\mu_2^2)\geq\mu_3^2-3\mu_2^3\iff\mu_2\mu_4\geq\mu_3^2. $$ But the last inequality is true thanks to the Cauchy-Schwarz inequality $$ \mu_3^2=E[(X-\mu_1')(X-\mu_1')^2]^2\leq E[(X-\mu_1')^2]E[(X-\mu_1')^4]=\mu_2\mu_4. $$
