In this paper, we discuss monotonicity formulae of various entropy functionals under various rescaled versions of Ricci flow. As an application, we prove that the lowest eigenvalue of a family of geometric operators $-4\Delta + kR$ is monotonic along the normalized Ricci flow for all $k\ge 1$ provided the initial manifold has nonpositive total scalar curvature.

Writing $s = \sigma + it$ for a complex variable, it is proved that the modulus of the gamma function, $|\Gamma(s)|$, is strictly monotone increasing with respect to $\sigma$ whenever $|t| > 5/4$. It is also shown that this result is false for $|t| \leq 1$.

Let $f\in L_{2\pi }$ be a real-valued even function with its Fourier series $% \frac{a_{0}}{2}+\sum_{n=1}^{\infty }a_{n}\cos nx,$ and let $S_{n}(f,x) ,\;n\geq 1,$ be the $n$-th partial sum of the Fourier series. It is well known that if the nonnegative sequence $\{a_{n}\}$ is decreasing and $\lim_{n\rightarrow \infty }a_{n}=0$, then% \begin{equation*} \lim_{n\rightarrow \infty }\Vert f-S_{n}(f)\Vert _{L}=0 \text{ if and only if }\lim_{n\rightarrow \infty }a_{n}\log n=0. \end{equation*}% We weaken the monotone condition in this classical result to the so-called mean value bounded variation (MVBV) condition. The generalization of the above classical result in real-valued function space is presented as a special case of the main result in this paper, which gives the $L^{1}$% -convergence of a function $f\in L_{2\pi }$ in complex space. We also give results on $L^{1}$-approximation of a function $f\in L_{2\pi }$ under the MVBV condition.