The analytical expression for the self-Kerr nonlinear coefficient in a three-level V-type atomic medium is found in the presence of the Doppler effect. Based on the analytical results, we have analyzed the enhancement and control of the Kerr nonlinear coefficient under the condition of electromagnetically induced transparency. It is shown that the Kerr nonlinear coefficient is significantly enhanced around the resonant frequency of both the probe and coupling fields. Simultaneously, the magnitude and sign of the Kerr nonlinear coefficient are controlled with respect to the intensity and frequency of the coupling laser field. The amplitude of the Kerr nonlinear coefficient decreases remarkably as temperature increases (i.e., the Doppler width increases). The analytical model can find potential applications in photonic devices and can explain experimental observations of the Kerr nonlinear coefficient at different temperatures.