This paper is based on the work [JDE, 2024, 396, 147-171], which gives that the general cubic polynomial differential system has an isochronous global center at the origin if and only if the system can be reduced to the form \(\dot{x}=-y+ax^{2}\) , \(\dot{y}=x-2axy+2a^{2}x^{3}\) . In this paper we discuss the monotonicity and the convexity of period function for the system \(\dot{x}=-y+ax^{2}\) , \(\dot{y}=x-2axy+bx^{3}\) , a more general form with a free coefficient b of the cubic term. We prove that the period function is strictly decreasing for \(b>2a^{2}\) , isochronous for \(b=2a^{2}\) and strictly increasing for \(b<2a^{2}\) . Furthermore, we prove that the period function is convex for \(b\ne 2a^{2}.\)
