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Let $n \geq 3$ and consider two sequences of strictly monotone functions $\{\mu_l(t)\}_{l=1}^{n}$ and $\{\lambda_l(t)\}_{l=1}^n$ on the interval $[-1,1]$ with $\mu_l(0)=0$ and $\lambda_l(0)=1$ for all $l=0,\ldots,n$.

Let $$I(\epsilon)=\Im \left( \int_{-1}^{1} \Pi_{l=1}^n(\mu_l(t)-i\epsilon \lambda_l(t))^{-\frac{1}{2}}\right).$$

Here, $\Im \,a$ denotes the imaginary part of $a \in \mathbb C$.Is it true that $$I(\epsilon)\lesssim \epsilon$$?

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  • $\begingroup$ which branch of $z^{-1/2}$ do you consider? $\endgroup$
    – Fedor Petrov
    Commented Apr 20, 2019 at 8:18
  • $\begingroup$ We are only interested in values of square root oer a curve in complex plane that does not cross the origin. You freely choose a value for the square root at some reference point $t$ and then let the parameter $t$ to denote a way to measure polar angle continuosly so that the square root function becomes smooth for all $t$ above. $\endgroup$
    – Ali
    Commented Apr 20, 2019 at 10:15

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