We present here a manifestly gauge invariant calculation of vacuum polarization to fermions in the presence of a constant Maxwell and a constant Kalb-Ramond field in four dimensions. The formalism is a generalisation of the one used by Schwinger in his famous paper on gauge invariance and vacuum polarization. We get an explicit expression for the vacuum polarization induced effective Lagrangian for a constant Maxwell field interacting with a constant Kalb-Ramond field. In the weak field limit we get the coupling between the Maxwell field and the Kalb-Ramond field to be $(tilde{H}.tilde{F})^2$, where ${tilde H}_{mu}= {1over {3!}}epsilon_{mualphabetalambda}H^{alphabetalambda}$ and $tilde F$ is the dual of $F_{mu u}$.
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