Reversed Cauchy-Schwarz inequality

This post will state a couple forms of the Cauchy-Schwarz inequality and then present the lesser-known reverse of the Cauchy-Schwarz inequality due to Polya and Szego.

Cauchy-Schwarz inequality

The summation form of the Cauchy-Schwarz inequality says that

for sequences of real numbers x_n and y_n.

The integral form of the Cauchy-Schwarz inequality says that

for any two real-valued functions f and g over a measure space (E, ) provided the integrals above are defined.

You can derive the sum form from the integral form by letting your measure space be the integers with counting measure. You can derive the integral form by applying the sum form to the integrals of simple functions and taking limits.

Flipping Cauchy-Schwarz

The Cauchy-Schwarz inequality is well known [1]. There are reversed versions of the Cauchy-Schwarz inequality that not as well known. The most basic such reversed inequality was proved by Polya and Szego in 1925 and many variations on the theme have been proved ever sense.

Polya and Szego's inequality says

for some constant C provided f and g are bounded above and below. The constant C does not depend on the functions per se but on their upper and lower bounds. Specifically, assume

Then

where

Sometimes you'll see C written in the equivalent form

This way of writing C makes it clear that the constant only depends on m and M via their ratio.

Note that if f and g are constant, then the inequality is exact. So the constant C is best possible without further assumptions.

The corresponding sum form follows immediately by using counting measure on the integers. Or in more elementary terms, by integrating step functions that have width 1.

Sum example

Let x = (2, 3, 5) and y = (9, 8, 7).

The sum of the squares in x is 38 and the sum of the squares in y is 194. The inner product of x and y is 18+24+35 = 77.

The product of the lower bounds on x and y is m = 14. The product of the upper bounds is M = 45. The constant C = 59^2/(4*14*45) = 1.38.

The left side of the Polya and Szego inequality is 38*194 = 7372. The right side is 1.38*77^2= 8182.02, and so the inequality holds.

Integral example

Let f(x) = 3 + cos(x) and let g(x) = 2 + sin(x). Let E be the interval [0, 2].

The following Mathematica code shows that the left side of the Polya and Szego inequality is 171^2 and the right side is 294 ^2.

The function f is bound below by 2 and above by 4. The function g is bound below by 1 and above by 3. So m = 2 and M = 12.

 In[1]:= f[x_] := 3 + Cos[x] In[2]:= g[x_] := 2 + Sin[x] In[3]:= Integrate[f[x]^2, {x, 0, 2 Pi}] Integrate[g[x]^2, {x, 0, 2 Pi}] Out[3]= 171 ^2 In[4]:= {m, M} = {2, 12}; In[5]:= c = (m + M)^2/(4 m M); In[6]:= c Integrate[f[x] g[x], {x, 0, 2 Pi}]^2 Out[6]= 294 ^2

Related posts

• Three books on inequalities
• Sum and mean inequalities move in opposite directions

[1] The classic book on inequalities by Hardy, Littlewood, and Polya mentions the Polya-Szego inequality on page 62, under Miscellaneous theorems and examples." Maybe Polya was being inappropriately humble, but it's odd that his inequality isn't more prominent in his book.

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