Estimate for the amplitude of the limit cycle of the Liénard equation (Q2358830)

Consider the Liénard system \[ {dx\over dt}= y,\quad {dy\over dt}=- f(x)y- g(x) \] under the conditions introduced by \textit{A. Liénard} [Ann. Physique, X. Ser. 10, 5--69 (1928; JFM 54.0806.05)] implying the existence of a unique stable limit cycle \(\Gamma\). Under some additional conditions on \(f\) and \(g\), one of them reads \[ \lim_{x\to+\infty}\, \int^x_0 g(s)F(s)\,ds=+\infty, \] where \(F(s)= \int^s_0 f(u)\,du\), the author proves that the amplitude of \(\Gamma\) is bounded from above by the number \(\alpha\) which is uniquely defined by \[ \int^\alpha_0 g(s)F(s)\,ds= 0. \] This result is applied to the van der Pol equation and yields an estimate which is better than known estimates in the literature.