  The aim of this paper is to investigate harmonic Stieltjes constants
occurring in the Laurent expansions of the function \[ \zeta_{H}\left(
s,a\right) =\sum_{n=0}^{\infty}\frac{1}{\left( n+a\right)
^{s}}\sum_{k=0}^{n}\frac{1}{k+a},\text{ }\operatorname{Re}\left( s\right) >1,
\] which we call harmonic Hurwitz zeta function. In particular evaluation
formulas for the harmonic Stieltjes constants $\gamma_{H}\left( m,1/2\right) $
and $\gamma_{H}\left( m,1\right) $ are presented.
