A Poincare-Type Inequality and its generalization 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!
 A: Disclaimer : I initially misunderstood the question and rewrote the answer accordingly.

Regarding question 1 : the positivity condition implies in particular that the first Fourier coefficient of $f$ is nonzero, call it $f_0$. The orthogonality condition on $\phi$ reads : 
$$\phi_0 f_0 + \langle \phi - \phi_0, f - f_0\rangle = 0, \ \ (*)$$ 
i.e.
$$\phi_0 = - \frac{1}{f_0^2} \langle \phi - \phi_0, f - f_0\rangle $$
where $\langle.,.\rangle$ denotes the standard $L^2$ inner product.
Call $R(\phi) = \frac{\int |\nabla \phi|^2 dx}{\int |\phi|^2 dx}$. We have, for all $\phi$ satisfying $(*)$ :
$$R(\phi)^{-1} = 1 + \frac{1}{f_0^2} \frac{|\langle \phi - \phi_0, f - f_0\rangle|^2}{\|(\phi - \phi_0)'\|_{L^2}^2}. $$
Using Cauchy-Schwarz inequality and the usual Poincaré inequality on $\phi - \phi_0$, we get : 
$$R(\phi)^{-1} \leq 1 + \frac{\|f-f_0\|_{L^2}^2}{f_0^2} \ \ (**) $$
and thus a Poincaré inequality on $\phi$. What is actually necessary and sufficient here is that $f$ should have a nonvanishing mean (otherwise said, $f$ should not be orthogonal in $L^2$ to the constant function).
Regarding the second question, you can't possibly expect $\lambda_1$ as a constant in any case, because this constant highly depends on the mutual importance of the constant coefficient of $f$ and the the other ones, as $(**)$ shows. Whether you can recover a uniform constant instead of $\lambda_1$ or not depends on a claim like "a positive periodic function has a uniform fraction of its energy concentrated in the constant mode". The latter is false in general, peaked functions giving counterexamples. Modifying them so as to ensure orthogonality to some eigenfunctions of the Laplacian is another task.
