The title problem is considered under the assumption that both the surface temperature of the cylinder and the mainstream velocity at the outer edge of the boundary layer vary linearly with the axial distance x from the leading edge. Self-similar solutions are given in exact analytic form. Their domain of existence in the plane (f(w),lambda) of the suction/injection parameter f(w) and the mixed convection parameter lambda is determined. It is shown that in general for both the aiding (lambda > 0) as well as the opposing (lambda < 0) flow regimes multiple solutions bifurcating from certain branching curves of the plane (f(w),lambda) occur. The particular case of the dual solutions is discussed in the paper in detail. (c) 2005 Elsevier Ltd. All rights reserved.