In the present paper, among other things, we prove that if f (fifi) = 0, /'(O) = 1) is regular and odd starlike in \z\ < 1, then Re f(z)/s"(z,f) > 1/2, |r| < 1, where s"(z,f) denotes the nth partial sum of /, n = 1,2,3,..., thus generalising the known result: Re/(z)/r > 1/2, \z\ < 1. As an application, we show that each partial sum of an odd convex function is close-to-convex in \z\ < 1. 00 n = 0