The elastic fields of an infinite plane with an inhomogeneous inclusion due to the non-uniform distribution of temperature on the inclusion is studied, where the inclusion is shaped by the hypocycloid curve and has different properties from those of the matrix, except for the same elastic shear modulus. This is called the inhomogeneous thermoelastic problem. By means of the knowledge of complex variable functions, the closed-form solution for the general case of inhonmogeneous thermoelastic problems is firstly solved. Then through the Riemann conformal mapping, the exterior of the inclusion is mapped onto the exterior of the unit circle. Furthermore, in virtue of the features of analytical functions, and combining the Cauchy-type integrals with the Faber polynomials, we obtain the explicit analytical formulae of the K-M potentials inside and outside the inclusion with the region of inclusion affected by the temperature of a polynomial distribution. The stress fields are calculated from the potentials and illustrated in the cases for different polynomial distributions of temperature. It is found that the internal stress field is in good agreement with the finite element results, and the same as the reported solutions in the literature when the inclusion is elliptical. On top of that, the new formulae are of more generality and applicability.