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In this paper, we investigate nonlinear the perturbed nonlinear Schrödinger's equation (NLSE) with Kerr law nonlinearity given in [Z.Y. Zhang, et al., Appl. Math. Comput. 216 (2010) 3064] and obtain exact traveling solutions by using infinite series method (ISM), Cosine-function method (CFM). We show that the solutions by using ISM and CFM are equal. Finally, we obtain abundant exact traveling wave solutions of NLSE by using Jacobi elliptic function expansion method (JEFEM).

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We investigate the Schur harmonic convexity for two classes of symmetric functions and the Schur multiplicative convexity for a class of symmetric functions by using a new method and generalizing previous result. As applications, we establish some inequalities by use of the theory of majorization, in particular, and we give some new geometric inequalities in the n-dimensional space.