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In this paper we investigate the numerical range of  C *  bφ  m  C aφ  n   and  C aφ  n  C *  bφ  m   on the Hardy space where  φ  is an inner function fixing the origin and  a  and  b  are points in the open unit disc. In the case when | a | = | b | = 1 we characterize the numerical range of these operators by constructing lacunary polynomials of unit norm whose image under the quadratic form incrementally foliate the numerical range. In the case when  a  and  b  are small we show numerical range of both operators is equal to the numerical range of the operator restricted to a 3-dimensional subspace.

The Numerical Range of C * ψ C φ and C φ C * ψ