Entire functions that have the smallest deviation from zero with respect to the uniform norm with weight

P. L. Chebyshev solved the problem of finding a polynomial of degree $n$ with leading coefficient one that has the smallest deviation from zero with respect to the maximum norm. A similar problem can be solved for some classes of entire functions. We find the entire function of exponential type $\sigma$ such that for any nonzero entire function $Q$ of type less than $\sigma$ and of class $A$ we have
$$ \sup_\mathbb R\left|\frac{f_\sigma-Q}{\rho_m}\right|>\sup_\mathbb R\left|\frac{f_\sigma}{\rho_m}\right|. $$