Independence in connected graphs

We prove that if image is a finite, simple, and undirected graph with image components and independence number image, then there exist a positive integer image and a function image with non-negative integer values such that image for image, image, and image. This result is a best possible improvement of a result due to Harant and Schiermeyer [J. Harant, I. Schiermeyer, On the independence number of a graph in terms of order and size, Discrete Math. 232 (2001) 131–138] and implies that image for connected graphs image of order image, average degree image, and independence number image.