New criteria for the monotonicity of the ratio of two Abelian integrals

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:465

Article

Liu, Changjian¹  Chen, Guoting²  Sun, Zhongqin³ 

[1] Sun Yat Sen Univ, Sch Math Zhuhai, Zhuhai 519086, Peoples R China

[2] Univ Lille 1, UMI? CNRS 8524, Lab Paul Painleve, UFR Math, F-59655 Villeneuve Dascq, France

[3] Soochow Univ, Sch Math, Suzhou 215006, Peoples R China

关键词: Abelian integrals;    Number of zeros;    Planar systems;   

DOI  :  10.1016/j.jmaa.2018.04.074

来源: Elsevier

【 摘 要 】

New criteria to determine the monotonicity of the ratio of two Abelian integrals are given. When two Abelian integrals have the forms integral(Gamma h) f(1)(x)ydx and integral(Gamma h) f(2)(x)ydx or the forms integral(Gamma h) f(1)(x)/y dx and integral(Gamma h) f(2)(x)/y dx and Gamma(h) are ovals belonging to the level set {(x, y)vertical bar H(x, y) = h}, where H(x, y) has the form y(2)/2 + Psi(x) or phi(x)y(2)/2 + Psi(x), we give new criteria, which are defined directly by the functions which appear in the above Abelian integrals, and prove that the monotonicity of the criteria implies the monotonicity of the ratios of the Abelian integrals. The new criteria are applicable in a large class of problems, some of which simplify the existing proofs and some of which generalize known results. (C) 2018 Elsevier Inc. All rights reserved.

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