Denote by H gamma ( D ) the Hilbert space of holomorphic functions over the open unit disk D with the reproducing kernel K (omega) (( gamma )) (z) = 1 /(1-omega(-)z)(gamma) , gamma > 0. Let J (alpha) : J (alpha) f (z) = f (alpha z(-))(sic) for z E D and alpha is an element of partial derivative D, and let W (u,v) be the weighted composition operator. In this paper, we prove that if b is an element of R and b not equal k pi/2 , for any integer k, then (cos b+i sin bW (u,v) ) J (alpha) is a conjugation if and only if v ( z ) = c-z /1 -c(-)z , u ( z ) = +/- k (c) (( gamma )) ( z ), c is an element of D and c(-) = c alpha, where k(c) (( gamma )) (z) is the normalized reproducing kernel, or (cos b + i sin bW (u,v) ) J (alpha) =lambda J(alpha) for some lambda is an element of C with |lambda| = 1. We then derive a necessary and sufficient condition for a Toeplitz operator to be complex symmetric with respect to the conjugation (cos b + i sin bW (u,v) ) J (alpha) on H (gamma) ( D ) with gamma >= 1. Similarly, we also conduct a similar study for composition operators on H (gamma) ( D ) with gamma > 0.
