Chebyshev polynomials corresponding to a vanishing weight

We consider weighted Chebyshev polynomials on the unit circle corresponding to a weight of the form $(z-1)^s$ where $s>0$. For integer values of $s$ this corresponds to prescribing a zero of the polynomial on the boundary. As such, we extend findings of Lachance, Saff and Varga, to non-integer $s$. Using this generalisation, we are able to relate Chebyshev polynomials on lemniscates and other, more established, categories of Chebyshev polynomials. An essential part of our proof involves the broadening of the Erd\H{o}s--Lax inequality to encompass powers of polynomials. We believe that this particular result holds significance in its own right.