Mathematical inequalities are pervasive in economic theory, just as economic inequalities are pervasive in social life. The insistence that quantities (always) and prices (usually) be nonnegative, the constraint that expenditure not exceed wealth, the necessity in proving existence of competitive equilibrium that each agent’s resources have positive value, are so familiar that we scarcely think of them as requirements of inequality, though that is what they are.

Many of the basic results of economic theory (such as the non-positivity of the substitution effect) take the form of inequalities. These in turn often arise from the definiteness or semidefinitenes of certain matrices, such definiteness being again expressed by inequalities. Yet further along the chain of reasoning, those matrices usually derive such properties from their origin in the convexity or concavity of various functions. For real-valued functions, convexity is defined by Jensen’s Inequality (1906): The function f : X ⊂ Rn → R is convex if

$$ \forall {x}^1,{x}^2\in X,\kern0.5em \forall \alpha \in \left[0,1\right]f\left(\alpha {x}^1+\left(1-\alpha \right){x}^2\right)\times \le \alpha f\left({x}^1\right)+\left(1-\alpha \right)f\left({x}^2\right) $$
(1)

(A function g is concave if − g is convex).

There are close connections between convex functions and inequalities in general. Indeed, ‘The classical inequalities are … obtained by verifying that a certain function is convex and by calculating its transforms.’ (Young 1969, p. 112). To illustrate this general proposition by an important special case, consider the gauge J(·|C) of any set C ⊂ Rn, together with its polar transform J0(·|C), which is the gauge of the polar set C0 of C (see GAUGE FUNCTIONS). When C is convex and closed and contains the origin, J0(·|C) becomes the support function S(·|C) of C. A fundamental inequality of convexity for gauges and their polar transforms is Mahler’s Inequality (1939), which applied to the present situation reads:

$$ \forall x\in {R}^n,\kern1em \forall y\in {R}^n\kern1em \sum {x}_i{y}_i\le J\left(x|C\right)S\left(y|C\right) $$
(2)

Consider now Rn with its standard Euclidean norm \( \parallel x{\parallel}_2={\left(\sum {x}_i^2\right)}^{1/2} \), and suppose that C is the closed unit sphere \( Sc=\left\{x\in {R}^n:\parallel x{\parallel}_2\le 1\right\} \) of Rn. In this special case it happens that

$$ J\left(\cdot |{S}_z\right)={\left\Vert \cdot \right\Vert}_z=S\left(\cdot |{S}_c\right) $$
(3)

(see e.g. Rockafellar 1970, p. 130). So from (2) and (3),

$$ \forall x\in {R}^n,\kern1em \forall y\in {R}^n\kern1em \sum {x}_i{y}_i={\left(\sum {x}_i^2\right)}^{1/2}\kern1em {\left(\sum {y}_i^2\right)}^{1/2} $$
(4)

Since (4) is the famous Cauchy–Buniakowski–Schwarz Inequality, this illustrates Young’s general proposition above. Young (1969, pp. 112–113) gives further examples (with proofs) of the connections between convexity and the classical inequalities, such as that relating the arithmetic and geometric means, and Holder’s Inequality (1889):

$$ \forall x\in {R}^n,\kern1em \forall y\in {R}^n\kern1em \sum {x}_i{y}_i\le {\left(\sum {\left|{x}_i\right|}^p\right)}^{1/p}\kern1em {\left(\sum {\left|{y}_i\right|}^q\right)}^{1/q} $$
(5)

(where p > 0, q > 0, and p−1 + q−1 = 1), of which (4) is the special case p = 2 = q.

It is not surprising then that the classic work on inequalities, the delightful and indispensable book by Hardy et al. (1934), contains one of the earliest systematic treatments of convex functions in English. A later survey is Beckenbach and Bellman (1961).

See Also