Finding Arbitrage in two Puts

A European Put Option on a non-dividend paying stock with strike price 80 is currently priced at 8 and a put option on the same stock with strike price 90 is priced at 9. Is there an arbitrage opportunity existing in these two Options?

I know we have to used the fact that Put Options values are convex with respect to their Strike Prices and could use the equation $P(\lambda K) < \lambda P(K)$? But, in the solution book that I have, they take $\lambda$ to be 8/9 and I don't know why this is.

Answer

Let $K_1=0$, $K_2=80$, and $K_3=90$. Then

$$\begin{align*} K_2 = 1/9 \, K_1 + 8/9 \, K_3. \end{align*}$$ Moreover, $$\begin{align*} Put(K_2) &= Put(1/9 \, K_1 + 8/9 \, K_3)\\ &< 1/9 \, Put (K_1) + 8/9\, Put(K_3)\\ &= 8/9 \, Put(K_3). \end{align*}$$ Taking $K=K_3$ and $\lambda = 8/9$, we have that $$Put(\lambda K) < \lambda Put(K).$$