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<p/> <p>For <inline-formula> <graphic file="1029-242X-2009-493759-i1.gif"/></inline-formula>, the symmetric function <inline-formula> <graphic file="1029-242X-2009-493759-i2.gif"/></inline-formula> is defined by <inline-formula> <graphic file="1029-242X-2009-493759-i3.gif"/></inline-formula>, where <inline-formula> <graphic file="1029-242X-2009-493759-i4.gif"/></inline-formula> and <inline-formula> <graphic file="1029-242X-2009-493759-i5.gif"/></inline-formula> are positive integers. In this article, the Schur convexity, Schur multiplicative convexity and Schur harmonic convexity of <inline-formula> <graphic file="1029-242X-2009-493759-i6.gif"/></inline-formula> are discussed. As applications, some inequalities are established by use of the theory of majorization.</p>