Normal Density and Distribution

For a nice regular example we use the celebrated functions $$\begin{align*} \phi(x)& = \frac{1}{\sqrt{2\pi}}e^{-z^2/2},\\ \Phi(x)& = \int_{-\infty}^{x} \phi(z)\,dz. \end{align*}$$ Then $$\begin{align*} \Phi'(x)&=\phi(x),\\ \Phi''(x)& = \phi'(x) \;\;\, = \;\; -x\phi(x), \\ \Phi'''(x)&=\phi''(x) \;\, = \;\; -(1-x^{2})\phi(x),\\ \Phi''''(x)&=\phi'''(x)\; = \;\: -x(x^{2}-3)\phi(x). \end{align*}$$ To obtain quadratic majorizers we must bound the second derivatives. We can bound $\Phi''(x)$ by setting its derivative equal to zero. We have $\Phi'''(x)=0$ for $x=\pm 1$, and thus $|\Phi''(x)|\leq\phi(1)$. In the same way $\phi'''(x)=0$ for $x=0$ and $x=\pm\sqrt{3}$. At those values $|\phi''(x)|\leq 2\phi(\sqrt{3})$. More precisely, it follows that $$\begin{align*} 0&\leq\Phi'(x) \;\;\, = \; \phi(x)\; \leq \; \phi(0),\\ -\phi(1)&\leq\Phi''(x)\;\, = \; \phi'(x)\; \leq \; \phi(1),\\ -\phi(0)&\leq\Phi'''(x) \, = \; \phi''(x) \; \leq \; 2\phi(\sqrt{3}). \end{align*}$$ Thus we have the quadratic majorizers $$\Phi(x)\leq\Phi(y)+\phi(y)(x-y)+\frac12\phi(1)(x-y)^{2},$$ and $$\phi(x)\leq\phi(y)-y\phi(y)(x-y)+\phi(\sqrt{3})(x-y)^{2}.$$ The majorizers are illustrated for both $\Phi$ and $\phi$ at the points $y=0$ and $y=-3$ in Figures 1 and 2. The inequalities in this section may be useful in majorizing multivariate functions involving $\phi$ and $\Phi$. They are mainly intended, however, to illustrate construction of quadratic majorizers in the smooth case.}

The drawings are made by the code in normal.R.