Disclosed in accordance with the present invention are processes for the preparation of hydroperoxides and/or carbinols by liquid phase oxidation of secondary alkyl-substituted naphthalenes with molecular oxygen, wherein the oxidation reaction is carried out by dissolving in the reaction mixture containing the secondary alkyl-substituted naphthalenes at least 0.5 ppm in terms of metal, based on the starting secondary alkyl-substituted naphthalenes, of at least one compound of metal selected from the group consisting of palladium and gold, said metal compound being soluble in the reaction mixture of the secondary alkyl-substituted naphthalenes. When compared with the prior art processes using catalysts of non-homogeneous system, therefore, not only the amount of catalyst used can be minimized but also the rate of oxidation can be enhanced, and hence there can be obtained in a short period of time hydroperoxides and/or carbinols in high yields which are the oxidation products in accordance with the present invention.According to the present invention, the desired hydroperoxide can be obtained in high yields as well as in high purity from the hydroperoxides and/or carbinols obtained in the manner as mentioned above.