Questions on proving and manipulating inequalities.

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estimate an equality on sine function

I want to prove the following: For any given $\epsilon>0$, there exists a $\delta>0$ such that for any fixed $0<\theta<\frac{\pi}{2}$ with $\frac{\pi}{2}-\theta<\delta$, there exists ...
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An upper bound for $\sum_{n,m} \left(|a_n\phi_n|^2+|a_m\phi_m|^2\right) \left|K_{n-m}\right|$.

Let us consider $a_n, \phi_n, K_{n}$ complex sequences. Let $$\sum_{n,m} \left(|a_n\phi_n|^2+|a_m\phi_m|^2\right) \left|K_{n-m}\right|$$ where $\left|K_{n}\right|\leq \gamma$ and $\gamma>0$. Can we ...