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In the paper the ideas of relative Nevanlinna L∗-order and relative Nevanlinna L∗-lower order of an analytic function with respect to an entire function in the unit disc U = {z : |z| < 1} are introduced. Hence, we study some comparative growth properties of composition of two analytic functions in the unit disc U on the basis of relative Nevanlinna L∗-order and relative Nevanlinna L∗-lower order.