Let $\varphi(x)$ and $\psi(x)$ be two complex-valued continuous functions on $[a,b]$, and let $f(x)$ be a complex-valued continuously differentiable function on $[a,b]$. Suppose that $|f(x)|$ has an absolute maximum at an interior point, say $\xi$, of the interval. Prove or disprove \begin{equation}\label{eq3} \lim_{n\to\infty}\frac{\int_a^b\varphi(x)[f(x)]^ndx}{\int_a^b\psi(x)[f(x)]^ndx}=\frac{\varphi(\xi)}{\psi(\xi)}. \end{equation}

Remark 1: This is true for $f(x)\in C^2$, by Laplace's method.

Remark 2: Micheal has given a counter example, which makes the use of the fact $f'(\xi)\neq0$. This is a good example. Now, if we further assume that $f'(\xi)=0$, and also we assume that $\phi\neq0$ and $\psi(\xi)\neq0$. how about now? I believe this will be more difficult.

Remark 3: Sorry for my question style. Well, this is not just a home work. It's an open problem, when one try to prove a theorem of Chung and Erd\"os 1951. That theorem is essentially said that the ratio of two coefficients in Fourier series of $f(x)^n$ will tend to 1. Where the assumption on $f$ could be translated as $f\in C^1[-\pi,\pi]$, $f(0)=1$, $f'(0)=0$ and $|f(x)|<1$ for $x\neq0$. This theorem will be a corollary if the limit is true here. So I make a further assumption in Remark 2 that $f'(\xi)\neq0$.

Thank you.

questions(and "research" questions at that), so demands like "prove or disprove" come across (usually unintentionally) as rude. $\endgroup$3more comments