Prove the following inequality??

Question

Prove that \begin{equation} (1+k)^{p}<\frac{(1-k^{p+1})}{(1-k)}, \end{equation} Here k,p $\in$ (0,1). I have proved this ineqality for p=1/2. The proof is as follows. \begin{eqnarray} (1+k)^{0.5}<\frac{(1-k^{1.5})}{(1-k)} \\ (1+k)^{0.5}<\frac{(1-\sqrt{k})(k+1+\sqrt{k})}{(1-\sqrt{k})(1+\sqrt{k})}\\ (1+k)^{0.5}(1+\sqrt{k})<(k+1+\sqrt{k})\\ \text{Geometric Mean}<\text{Arithmatic Mean} \end{eqnarray}, However I am not able to show this inequality for p $\in$ (0,1) with k lying between 0 to 1. I have checked the inequality to hold numerically. Please have a look at it.

Answer 1

You can indeed show it using Bernouilli's inequality. It states that for $x > -1$ and $0 < r < 1$, $(1 + x)^r < 1 + rx$. The given inequality can be rewritten as $$ (1 - k)(1 + k)^p < 1 - k^{p + 1}. $$ Applying Bernouilli's inequality to the LHS, we get $$ (1 - k)(1 + k)^p < (1 - k)(1 + pk) = 1 - k - pk^2 + pk. $$ If we can show that this is strictly less than $1 - k^{p + 1}$ or, equivalently, that $$ k^p < 1 + pk - p, $$ we are done. This follows however from $k^p = (1 + (k - 1))^p < 1 + p(k - 1)$ by Bernouilli's inequality.