Cauchy-Schwarz says
$${\int }_{E}fg\mathrm{d}x\le {\left({\int }_E{f}^2\mathrm{d}x\right)}^{1/2}{\left({\int }_E{g}^2\mathrm{d}x\right)}^{1/2}$$
where ${\int }_E$ is a definite integral. Then the $\underline{\mathrm{A}\mathrm{M}?\mathrm{G}\mathrm{M}}$ says that for $a,b\ge 0$
$$\sqrt{ab}\le \frac{a+b}{2}$$
Applying (2) to (1) yields $(\text{?for?}c>0)$
$$\begin{array}{rl}{\int }_{E}fg\mathrm{d}x& ={\int }_E(\sqrt{c}f)(g/\sqrt{c})\mathrm{d}x\\ & \le \frac{c}{2}{\int }_E{f}^2\mathrm{d}x+\frac{1}{2c}{\int }_E{g}^2\mathrm{d}x\end{array}$$

LINK: https://math.stackexchange.com/questions/351905/cauchy-schwarz-inequality-for-integrals-for-any-two-functions-clarification

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