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The static kinks and their low energy excited states in scalar field theories with are studied in a function-series method. We give a universal formal solution, and a universal approach to find the low energy excited states as well. Excellent agreement between the FS solution and the exact answer is found. This confirms our predicted results of the energies of the excited quantum states by considering small oscillations about the kink motion.

In this article using analytic method and mathematical induction the authors prove the inequality(bx+γ−ax+γ)/(bx−ax)≥(x+γ)[(a+b)/2]γ/x,where γ≥1,x≥1,0