  We study optimal lower and upper bounds for Widom factors $W_{\infty,n}(K,w)$
associated with Chebyshev polynomials for the weights $w(x)=\sqrt{1+x}$ and
$w(x)=\sqrt{1-x}$ on compact subsets of $[-1,1]$. We show which sets saturate
these bounds. We consider Widom factors $W_{2,n}(\mu)$ for $L_2(\mu)$ extremal
polynomials for measures of the form $d\mu(x)=(1-x)^\alpha (1+x)^\beta
d\mu_K(x)$ where $\alpha+\beta\geq 1$, $\alpha,\beta\in\mathbb{N}\cup \{0\}$
and $\mu_K$ is the equilibrium measure of a compact regular set $K$ in $[-1,1]$
with $\pm 1\in K$. We show that for such measures the improved lower bound
$[W_{2,n}(\mu)]^2\geq 2S(\mu)$ holds. For the special cases
$d\mu(x)=(1-x^2)d\mu_K(x)$, $d\mu(x)=(1-x)d\mu_K(x)$, $d\mu(x)=(1+x)d\mu_K(x)$
we determine which sets saturate this lower bound and discuss how saturated
lower bounds for $[W_{2,n}(\mu)]^2$ and $W_{\infty,n}(K,w)$ are related.
