Entire Functions that have the Smallest Deviation from Zero with Respect to the Uniform Norm with a Weight

P. L. Chebyshev solved the problem of finding a polynomial of degree n with leading coefficient one that has the least deviation from zero with respect to the maximum norm. In the case of entire functions, a similar problem can be solved for some classes. We find an entire function of exponential type σ such that

$$ \underset{\mathbb{R}}{ \sup}\left|\frac{f_{\sigma }-Q}{\rho_m}\right|>\underset{\mathbb{R}}{ \sup}\left|\frac{f_{\sigma }}{\rho_m}\right| $$

for any nonzero entire function Q of type less than σ of class C.