Let $f(\theta)$ be a fixed **positive** $2\pi-$periodic $C^1$ function on $\mathbb{R}$ with $$\int_0^{2\pi}f(\theta)\cos\theta d\theta=\int_0^{2\pi}f(\theta)\sin\theta d\theta=0,$$
Does for any $2\pi-$periodic $C^1$ function $\phi$ satisfy $\int_0^{2\pi}f(\theta)\phi(\theta)d\theta=0$, we have the following inequality?$$\int_0^{2\pi}\phi(\theta)^2d\theta\leq\int_0^{2\pi}(\phi'(\theta))^2d\theta$$

Remark: If $f(\theta)\equiv1$, then its the Classical Poincare Inequality.

Moreover, I have the following question:

Let $(M^n,g)$ be a compact Riemannian manifold with $\partial M=\emptyset$, suppose $\lambda_1$ is the first eigenvalue of the Laplace operator on $(M^n,g)$ and $\{\phi_1,\phi_2,\cdots,\phi_k\}$ are the first eigenfunctions. Let **$f:M\rightarrow\mathbb{R}^+$** be a fixed smooth function with $\int_M f\phi_i d\mu=0,i=1,2,\cdots,k$. Does for any smooth function $\phi:M\rightarrow\mathbb{R}$ satisfy $\int_M f\phi d\mu=0$, we have the following inequality? $$\lambda_1\int_M\phi^2d\mu\leq\int_M|\nabla\phi|^2d\mu.$$

Does anyone have ever know about this kind of problems? Thank you very much if you can give any useful guides.

*Many thanks if you can give any reference!*