We show that an inequality related to Newton's inequality provides one more relation between skewness and kurtosis. This also gives simple and alternative proofs of the bounds for skewness and kurtosis.
"Skewness, kurtosis and Newton's inequality." Rocky Mountain J. Math. 45 (5) 1639 - 1643, 2015. https://doi.org/10.1216/RMJ-2015-45-5-1639