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 Aug6 awarded Tumbleweed Jul30 asked A different proof of the distributive law for gcd and lcm Jul9 answered Dimensions of symmetric and skew-symmetric matrices Sep4 accepted Monotonicity of $\ell_{p}$ norm and Holder's inequality Sep4 comment Monotonicity of $\ell_{p}$ norm and Holder's inequality Nice. I would only add that the second inequality is equivalent to $\sum |z_{i}|^{k} \leq (\sum |z_{i}|)^{k}$, which is obvious because the difference between the right and left sides of the inequality is the sum of all the cross terms in $(\sum |z_{i}|)^{k}$, all of which are nonnegative. Sep4 asked Monotonicity of $\ell_{p}$ norm and Holder's inequality Jul12 accepted Logarithm inequality for vectors Jul12 awarded Yearling Jul10 revised Logarithm inequality for vectors corrected missing absolute value Jul10 comment Logarithm inequality for vectors That works, but I don't think it's that easy to show the starting inequality. However, I was able to do it by differentiating $f(x) = \log(1+x^{2}) - x^{2} + \frac{x^{4}}{2 (1-x^{2})}$ to show that $f(x)$ is increasing on $(0,1)$. The desired inequality follows. ZachL's comment inspired my answer below, which I think is easier. Jul10 answered Logarithm inequality for vectors Jul10 answered Monty Hall/Bayes' Theorem Jul10 awarded Critic Jul10 comment Logarithm inequality for vectors I don't think that this works. I want my inequality to hold for all $d \in \mathbf{R}^{n}$ with $||d||_{\infty} < 1$, but the inequality $\log(1+x) \geq x - \frac{1}{2} x^{2}$ only holds for $x \geq 0$. For example, suppose $n=1$ and $d = -0.9$. Then, we have that $\sum_{i=1}^{n} \log(1+d_{i}) = \log(0.1) \approx -2.3 < \mathbf{1}^{T} d - \frac{1}{2} ||d||_{2}^{2} = -0.9 - 0.5 * (-0.9)^{2} \approx -1.3$. Jul10 asked Logarithm inequality for vectors Apr23 accepted Maximizing a convex function Apr22 asked Maximizing a convex function Dec11 awarded Scholar Dec11 accepted Regularity Conditions for Constrained Optimization Dec10 comment Variance Formula Looks correct to me.