An algebraic inequality for $\delta,q\in(0,1)$

Question

Let $\delta,q\in(0,1)$. Then the following inequality holds: $$ (1-\delta)(\delta^{-q}-1)>\frac{q}{1-q}(\delta^{\frac{1-q}{2}}-1)^2. $$

I am facing difficulty how to proceed for the proof. Any hint would be very helpful.

Thanks.

Answer 1

Edited question

This appears to depend upon first proving the following lemma.

If $0<D<1$ and $0<Q$, then $$\frac{1-D^Q}{1-D}<\frac{2Q}{D}.$$

Once this is done then let $Q=\frac{1-q}{2q},D=\delta^q.$ The inequality we are required to prove is then $$2Q(1-\delta)(1-D)>D(1-D^Q)^2$$ which is true since $1-\delta>1-\delta^\frac{1-q}{2}=1-D^Q$ and $2Q(1-D)>D(1-D^Q).$

Proof of Lemma For fixed $Q$ the graph of $D^Q$ is concave and so $\frac{1-D^Q}{1-D}$ is less than the gradient of the curve. The gradient is $QD^{Q-1}<\frac{Q}{D}.$

Answer 2

For any two given values of $\delta,q$, all of the quantities $(1-\delta),(\delta^{-q}-1),q,1-q$ and $(\delta^{\frac{1-q}{2}}-1)^2$ are positive.

Therefore all that is required is for $c$ to be positive and less than the fixed positive number $$\frac{(1-\delta)(\delta^{-q}-1)(1-q)}{q(\delta^{\frac{1-q}{2}}-1)^2}.$$