Convexity and robustness of the R\'{e}nyi entropy

We study convexity properties of R'{e}nyi entropy as function of $\alpha >0$ on finite alphabets. We also describe robustness of the R'{e}nyi entropy on finite alphabets, and it turns out that the rate of respective convergence depends on initial alphabet. We establish convergence of the disturbed entropy when the initial distribution is uniform but the number of events increases to $\mathrm{\infty }$ and prove that limit of R'{e}nyi entropy of binomial distribution is equal to R'{e}nyi entropy of Poisson distribution.