Abstract
In this paper we study the properties of the Lebesgue constant of the conjugate transforms. For conjugate Fejer means we will find necessary and sufficient condition on $$t$$ for which the estimation $$E\left|\widetilde{\sigma}_{n}^{\left(t\right)}f\right|\leq cE\left|f\right|$$ holds. We also prove that for dyadic irrational $$t$$ , $$L\log L$$ is the maximal Orlicz space for which the estimation $$E\left|\widetilde{\sigma}_{n}^{\left(t\right)}f\right|\leq c_{1}+c_{2}E\left(\left|f\right|\log^{+}\left|f\right|\right)$$ is valid.
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