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In the paper, by finding linear relations of differences between some means,the authors supply a unified proof of some double inequalities for boundingNeuman-S\'andor means in terms of the arithmetic, harmonic, and contra-harmonicmeans and discover some new sharp inequalities involving Neuman-S\'andor,contra-harmonic, root-square, and other means of two positive real numbers.

A unified proof of inequalities and some new inequalities involving Neuman-S\'andor mean