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We consider the upper bounds on tail probabilities of self-normalized deviations $$ \\mathbf{P} \\Big( \\max_{1\\leq k \\leq n} \\sum_{i=1}^{k} |\\xi_i|\\big/ \\big(\\sum_{i=1}^{n} |\\xi_i|^\\beta \\big)^{1/\\beta} \\geq x \\Big) $$ for $x>0$ and $\\beta >1.$ Our bound is the best that can be obtained from the Bernstein inequality under the present assumption. 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