Advanced book on Mathematics Olympiad

142 3 Real Analysis

3.2.6 Convex Functions........................................

A function is called convex if any segment with endpoints on its graph lies above the
graph itself. The picture you should have in mind is Figure 21. Formally, ifDis an
interval of the real axis, or more generally a convex subset of a vector space, then a
functionf:D→Ris called convex if

f (λx+( 1 −λ)y)≤λf (x)+( 1 −λ)f (y), for allx, y∈D, λ∈( 0 , 1 ).

Here we should remember that a setDis called convex if for anyx, y∈Dandλ∈( 0 , 1 )

the pointλx+( 1 −λ)yis also inD, which geometrically means thatDis an intersection

of half-spaces.

x λ y

λf(x)+(1− λ)f(y)

f( x+(1− λλ )y)

x+(1− λ)y

Figure 21

A functionfis called concave if−fis convex. Iffis both convex and concave,

thenfis linear, i.e.,f(x)=ax+bfor some constantsaandb.

Proposition.A twice-differentiable function on an interval is convex if and only if its

second derivative is nonnegative.

In general, a twice-differentiable function defined on a convex domain inRnis convex

if at any point its Hessian matrix is semipositive definite. This is a way of saying that

modulo a local change of coordinates, around each point the functionfis of the form

f(x 1 ,x 2 ,...,xn)=φ(x 1 ,x 2 ,...,xn)+x^21 +x 22 +···+xk^2 ,

wherek≤nandφ(x 1 ,x 2 ,...,xn)is linear.

As an application, we use convexity to prove Hölder’s inequality.

Hölder’s inequality.Ifx 1 ,x 2 ,...,xn,y 1 ,y 2 ,...,yn,p, andqare positive numbers with

1

p+

1

q=^1 ,then