Majorizing and Minorizing the Logarithm

The logarithm is concave. Consequently, for all positive $x$ and $y$ , we have the linear majorizer $\log x\leq\log y+\frac{1}{y}(x-y)=\log y+\frac{x}{y}-1.$ We can apply the same concavity to get a minorizer $\log\frac{1}{x}\leq\log\frac{1}{y}+y(\frac{1}{x}-\frac{1}{y}),$ which is $\log x\geq \log y-\frac{y}{x}+1.$ These majorizers and minorizers of the logarithm are illustrated in Figure 1 for $y=1$ and $y=5$ .

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Figure 1: Majorization and Minorization of Logarithm

More generally, we have $\log x\leq\log y+\frac{1}{p}\left\{\left(\frac{x}{y}\right)^p-1\right\}$ for all $p>0$ and $\log x\geq\log y+\frac{1}{p}\left\{\left(\frac{x}{y}\right)^p-1\right\}$ for all $p<0$ . See Figure 2, where $y=5$ and $p=\{-2,-1,1,2\}$ .

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Figure 2: Majorization and Minorization of Logarithm