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This work is devoted to the study of the integrability of Mellin transforms under some sufficient and necessary conditions in the space $ X_{c}^{p} $ X c p of Lebesgue measurable functions f on $ \mathbb {R}_+=(0,+\infty) $ R + = (0 , + ∞) such that \[ \int_{0}^{+\infty}|u^{c}f(u)|^{p}\frac{{\rm d}u}{u} ∫ 0 + ∞ | u c f (u) | p d u u < ∞ , c ∈ R (1 < p ≤ 2). These results are inspired by a Móricz-type result in classical Fourier analysis, which generalizes a well-known theorem of Titchmarsh. [ABSTRACT FROM AUTHOR]