In this paper, we continues to investigate the properties of symmetric inequalities. We study the necessary condition for the extremum to be attained in a semi-algebraic system. From which, we partly solve the problem of the further generalization of Jensen inequality which the author brought up in the previous article. Namely, by using the sign of the (n+1)^th derivative while xing the value of the sum of power of n variables, we get a dimension-descending method. Further, we prove that the necessary and sucient condition for the symmetric inequality of degree m with n variables to hold on R+n is that it holds when the number of nonzero variables does not exceed max&{1, frac{m}{2}}&.

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Fix a number $g>1$, let $S$ be a close surface of genus $g$ and $\Teich(S)$ be the Teichm\"uller space of $S$ endowed with the Weil-Petersson metric. In this paper we show that the Riemannian sectional curvature operator of $\Teich(S)$ is non-positive definite. As an application we show that any twist harmonic map from rank-one hyperbolic spaces $H_{Q,m}=Sp(m,1)/Sp(m)\cdot Sp(1)$ or $H_{O,2}=F_{4}^{-20}/SO(9)$ into $\Teich(S)$ is a constant.