  In this paper we study the properties of the Lebesgue constant of the
conjugate transforms. For conjugate Fej\'er means we will find necessary and
sufficient condition on $t$ for which the estimation $E\left\vert \widetilde{%
\sigma }_{n}^{\left( t\right) }f\right\vert \lesssim E\left\vert f\right\vert $
holds . We also prove that for dyadic irrational $t$, $L\log L $ is maximal
Orlicz space for which the estimation $E\left\vert \widetilde{% \sigma
}_{n}^{\left( t\right) }f\right\vert \lesssim 1+E\left( \left\vert f\right\vert
\log ^{+}\left\vert f\right\vert \right) $ is valid.
