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In this paper, we establish several relations between convex order, variance order, and comonotonicity. In the first part, we extend Cheung (2008b) to show that when the marginal distributions are fixed, a sum with maximal variance is in fact a comonotonic sum. Thus the convex upper bound is achieved if and only if the marginal variables are comonotonic. Next, we study the situation where besides the marginal distributions; the variance of the sum is also fixed. Intuitively one expects that adding this information may lead to a bound that is sharper than the comonotonic upper bound. However, we show that such upper bound does not even exist. Nevertheless, we can still identify a special dependence structure known as upper comonotonicity, in which case the sum behaves like a convex largest sum in the upper tail. Finally, we investigate when the convex order is equivalent to the weaker variance order …