We prove some optimal logarithmic estimates in the Hardy space $H^{\infty }\left(G\right)$ with Hölder regularity, where $G$ is the open unit disk or an annular domain of $ℂ$. These estimates extend the results established by S. Chaabane and I. Feki in the Hardy-Sobolev space $H^{k,\infty }$ of the unit disk and those of I. Feki in the case of an annular domain. The proofs are based on a variant of Hardy-Landau-Littlewood inequality for Hölder functions. As an application of these estimates, we study the stability of both the Cauchy problem...