<s>[INST] For $a,b>0$, $c>1$, and $n=3,4,\dots$ is $|ca^n+(-b)^n|\leq(ca^2+b^2)^{n/2}$ always true? Let $a,b>0$, $c>1$, and $n=3,4,\dots$, then is
\begin{equation}
|ca^n+(-b)^n|\leq(ca^2+b^2)^{n/2},
\end{equation}
true for all possible combinations of $a$, $b$, $c$, and $n$?
 [/INST] Hint: prove first that $\,\big(c+x^n\big)^2 \le \big(c+x^2\big)^n\,$ for $\,c \ge 1, x \ge 0, n \ge 2\,$, using for example the binomial expansion, noting that $\,\color{red}{c^2} \le \color{red}{c^n}\,$ and $\,\color{blue}{2cx^n}\,$ is no larger than either of the two blue terms on the RHS depending on whether $\,x \ge 1\,$ (when $\,x^n \le x^{2n-2}\,$) or $\,x \le 1\,$ (when $\,x^n \le x^{2}\,$):
$$\require{cancel}
\color{red}{c^2} + \color{blue}{2cx^n} + \cancel{x^{2n}} \le \color{red}{c^n} + \color{blue}{n \cdot c^{n-1}x^2}  + \ldots + \color{blue}{n \cdot c x^{2n-2}} + \cancel{x^{2n}}
$$
Then, use the above for $\,x=b/a\,$, multiply the inequality by $\,a^{2n}\,$ etc.
</s>