Modulus of Continuity and Convergence of Fejér Means of Vilenkin-Fourier Series in the Variable Martingale Hardy Space \(H_{p(\cdot )}\)

The main aim of this paper is to investigate the weighted maximal operator \({\sup }_{k\in \mathbb {N}}\left (\left \vert \sigma _{k}f\right \vert /\log ^2(k+1)\right )\) of Fejér means of Vilenkin-Fourier series and prove that the it is bounded from the variable martingale Hardy space \(H_{p(\cdot )}\) to the Lebesgue space \(L_{p(\cdot )},\) where \(p(\cdot )\geq \underline p\geq 1/2.\) By using this result we get sufficient condition for the modulus of continuity which is guarantee for the convergence of Fejér means in the variable martingale Hardy space \(H_{p(\cdot )}.\) We also show that this condition is sharp. As a consequence we obtain some new and well-know results.