On an upper bound for central binomial coefficients and Catalan numbers

Recently, Agievich proposed an interesting upper bound on binomial coefficients in the de Moivre-Laplace form. In this article, we show that the latter bound, in the specific case of a central binomial coefficient, is larger than the one proposed by Sasvari and obtained using the Binet formula for the Gamma function. In addition, we provide the expression of the next-order bound and apply it to Catalan numbers $C_n$. The bounds are very close to the exact value, the difference decreasing with $n$ and with the order of the upper bound.