We examine the onset of synchronization transition in a star network of Kuramoto phase oscillators in the presence of inertia and a time delay in the coupling. A direct correlation between the natural frequencies of the oscillators and their degrees is assumed. The presence of time delay is seen to enhance the onset of first order synchronization. The star network also exhibits different synchronization transitions depending on the value of time delay. An analytical prediction to observe the effect of the time delay is provided and further supported by simulation results. Our findings may help provide valuable insights into the understanding of mechanisms that lead to synchronization on complex networks.