We consider an energy-efficient scheduling problem where n jobs J₁, J₂, . . ., J_n need to be executed such that the total energy usage by these jobs is minimized while ensuring that each job is finished within its deadline. The processor executing these jobs can be in either active or sleep state at any point of time. The speed of the processor in an active state can be arbitrary. If the processor is running at a speed s≥0, the required power P(s) is assumed to be equal to s^α+g where α>1, g>0 are constants. On the other hand, the required power is zero when the processor is in the sleep state. However, L>0 amount of wake-up energy is needed to wake-up the processor from the sleep state to the active state. In this paper, we work in an online setting where a job is known only at its arrival time, along with its processing volume and deadline. In such a setting, the currently best-known algorithm by Han et al. (2010) [3] provides a competitive ratio max{4, 2+α^α} of energy usage. We present a new online algorithm SqOA which provides a competitive ratio max{4, 2+(2-1/α)^α2^α-1} of energy usage. For α≥2.34, the competitive ratio of our algorithm is better than that of any other existing algorithm for this problem.