I need to find $\lambda_{n}(t)$ such that it gives $0$ for $t = 0$ and $p + \dfrac{1}{n}$ for $t = p$ and $1$ for $t = 1$. The first case is fulfilled but I can't seem to find a function that would give me this for the second case. Thanks so much
$$\lambda_{n}(t) = \begin{cases} (1+\dfrac{1}{nt})t, & t \in [0, p)\\ (1-\dfrac{1}{nt})t + \dfrac{2}{n}, & t \in [p,1] \end{cases} $$ It also needs to be stricly increasing.
Your second function can only be increasing if $p + \frac{1}{n} \leq 1$
You can always use the straight line function between $(p, p+\frac{1}{n}) \rightarrow (1,1)$
$$\frac{\lambda_n-1}{p+\frac{1}{n} - 1} = \frac{t-1}{p - 1} $$ $$\lambda_n(t) = \frac{1-(p+\frac{1}{n})}{1-p}(t-1) + 1$$