A “DOUBLE” CAUCHY–SCHWARZ TYPE INEQUALITY

Manolis Magiropoulos and Dimitri Karayannakis

Abstract. A “double” version of the C–S inequality in any complex pre– Hilbert space is given, along with some numerical applications.

A substantial part of the mathematical folklore in the frame of inner–product spaces involves applications and/or extensions of the Cauchy–Schwarz (C–S) inequality

|⟨e, f⟩| ≤ ∥e∥ ∥f∥,

where by ∥ · ∥ we indicate the norm induced by the inner–product ⟨, ⟩.

For the case of a (real) Hilbert space M. Lambrou (Univ. of Crete) indicated to the second author, by personal communication, the following “double” version of the C–S inequality.

|⟨e, f⟩| |⟨e, g⟩| ≤ \frac{1}{2}\{∥f∥ ∥g∥ + |⟨f, g⟩|\}∥e∥^{2}.  (∗)

(Note that if f, g are considered to be linearly dependent we simply obtain the C–S inequality.)

The fact that the R.H.S. of (∗) provides a better bound than the “natural” ∥e∥^{2} ∥f∥ ∥g∥ is evident by use of the C–S inequality itself. That (∗) gives, in certain cases, a much better bound for |⟨e, f⟩||⟨e, g⟩| becomes clear from the following example.

Let f = sin x, g = cos x, e = 1/x be considered as members of the classical Hilbert space L^{2}[α, β], α > 0. Then

|⟨e, f⟩| |⟨e, g⟩| ≤ (∫_{α}^{β} \frac{dx}{x^{2}})(∫_{α}^{β} sin^{2} xdx)^{1/2}(∫_{α}^{β} cos^{2} xdx)^{1/2}.

On the other hand (∗) provides the bound

\frac{1}{2}(∫_{α}^{β} \frac{dx}{x^{2}})[(∫_{α}^{β} sin^{2} xdx)^{1/2}(∫_{α}^{β} cos^{2} xdx)^{1/2} + | ∫_{α}^{β} sin x cos xdx|].

Subtracting the (∗)–bound from the C–S–bound one has

\frac{1}{8}(\frac{1}{α} − \frac{1}{β} )[(β − α + sin 2α − sin 2β)^{1/2}(β − α + sin 2β − sin 2α)^{1/2}

− | cos 2β − cos 2α|],

which increases to +∞ for a (number of) suitable limit behavior of α or β.

In the present work we present an elementary proof of (∗) for any pre–Hilbert space over the complex field (which naturally also covers the real case), along with a few applications of (∗).

\underline{Theorem 1}. Let e, f, g be elements of a complex pre–Hilbert space (H, ⟨, ⟩); then

2|⟨e, f⟩| |⟨e, g⟩| ≤ {∥f∥ ∥g∥ + |⟨f, g⟩|}∥e∥^{2}.

\underline{Proof}. Based on a previous remark, let f, g be linearly independent, and let e = λf + μg. W.L.O.G. we may also assume that ∥e∥ = ∥f∥ = ∥g∥ = 1. If f ⊥ g the L.H.S. of (∗) becomes 2|λ| |μ| whereas the R.H.S. becomes |λ|^{2} + |μ|^{2} and we are done. If f and g are not orthogonal to each other, by the Gram– Schmidt construction we obtain e = kf + sh with h = (g − cf)(1 − |c|^{2})^{−1/2} with c = ⟨g, f⟩ ≠ 0, 1. Then h ⊥ f and ∥h∥ = 1. It is easily seen that we may consider c > 0 since, otherwise, by switching from f to (cf)/|c| we find ourselves in an equivalent position. Then |κ|^{2} + |s|^{2} = 1 with κ = ⟨e, f⟩ and s = ⟨e, h⟩. In a similar manner we may assume that κ ≥ 0. Then

|⟨e, f⟩| |⟨e, g⟩| = κ|κc + s(1 − c^{2})^{1/2}|.

But

\begin{aligned}|κc + s(1 − c^{2})^{1/2}|^{2} = κ^{2}c^{2} + 2Re(κc(1 − c^{2})^{1/2}s) & + |s|^{2}(1 − c^{2}) \\ & ≤ (κc + |s|(1 − c^{2})^{1/2})^{2}. \end{aligned}

Thus,

|⟨e, f⟩| |⟨e, g⟩| ≤ κ^{2}c + κ(1 − κ^{2})^{1/2}(1 − c^{2})^{1/2}.  (∗∗)

Using the first derivative criterion, etc. for local extremes, it can be easily seen that the R.H.S. of (∗∗) is bounded by (1 + c)/2 and we are done.

It remains now to prove (∗) for the case e ∉ sp{f, g}. Then, e = e_{1} + e_{2} with e_{1} ∈ sp{f, g} and e_{2} ⊥ sp{f, g}. The L.H.S. of (∗) becomes

2|⟨e, f⟩| |⟨e, g⟩| ≤ {∥f∥ ∥g∥ + |⟨f, g⟩|}∥e_{1}∥^{2},

because of the first part. Since ∥e_{1}∥ ≤ ∥e∥ we obtain the announced result. Q.E.D. In case (H, ⟨·⟩) is Hilbert, we can generalize (∗) as follows.

\underline{Theorem 2}. For any projection P and any vectors f, g

2|⟨P f, g⟩| ≡ 2|⟨P f, P g⟩| ≤ ∥f∥ ∥g∥ + |⟨f, g⟩|.

\underline{Proof}. Let Q = I − P. Then

∥f∥^{2} = ∥P f∥^{2} + ∥Qf∥^{2}, ∥g∥^{2} = ∥P g∥^{2} + ∥Qg∥^{2}.

Since

⟨f, g⟩ = ⟨P f, P g⟩ + ⟨Qf, Qg⟩,

by the classical Schwarz inequality we have

|⟨f, g⟩| ≥ |⟨P f, P g⟩| − ∥Qf∥ · ∥Qg∥.

Therefore, for the assertion, it suffices to prove that

|⟨P f, P g⟩| + ∥Qf∥ · ∥Qg∥ ≤ \sqrt{(∥P f∥^{2} + ∥Qf∥^{2})(∥P g∥^{2} + ∥Qg∥^{2})}.

Then, using the classical Schwarz inequality once more, it suffices to prove

∥P f∥ · ∥P g∥ + ∥Qf∥ · ∥Qg∥ ≤ \sqrt{(∥P f∥^{2} + ∥Qf∥^{2})(∥P g∥^{2} + ∥Qg∥^{2})}

which is nothing but the classical Cauchy inequality.

\underline{Remark}. The result of (∗) in the case where (H, ⟨·⟩) of Theorem 1 is considered complete corresponds to the case where rank(P) = 1.

We turn now to a couple of applications of Theorem 1 starting with a complex Hilbert space (H, ⟨, ⟩).

(i) Notice first that if f and g in H are mutually orthogonal then (∗) reduces to

2|⟨e, f⟩| |⟨e, g⟩| ≤ ∥e∥^{2} ∥f∥ ∥g∥  (∗ ∗ ∗)

If now e ∈ H and {f_{i}}, i ∈ I, an orthogonal family in H, we can consider finite products of the Fourier coefficients of e w.r.t. {f_{i}} namely

∏_{j∈J} |⟨e, f_{j}⟩|,

where J ⊂ I a finite set with at least two elements (in which case, naturally, we also impose dim H ≥ 2). It is evident that the C–S inequality would have provided the crude (upper) bound ∥e∥^{n}, where n is the cardinality of J.

In view of (∗ ∗ ∗) though, we obtain the following far better bound, namely:

∏_{j∈J} |⟨e, f_{j}⟩| ≤ \{ \begin{matrix}2^{−n/2}∥e∥^{n}, & if n is even; \\ 2^{−(n−1)/2}∥e∥^{n}, & if n is odd. \end{matrix}

(ii) For another application of (∗) let us employ the pre–Hilbert space L^{1}(0, +∞) and consider its (linearly independent) elements

\frac{sin x}{x}, \frac{cos 2x}{1 + x^{2}} and \frac{1}{1 + x^{2}}

in the roles of e, f and g, respectively. The direct calculation of I = ⟨e, f⟩ is a rather painful experience within the techniques of contour integration, or even by tracing it in suitable tables.

On the other hand by elementary contour integration and/or by reference to [1], we have

\begin{matrix}⟨e, g⟩ = \frac{π}{2} (1 & − e^{−1}), ∥f∥ = [\frac{π}{8} (1 + 5e^{−4})]^{1/2}, ∥g∥ = \frac{π^{1/2}}{2} , \\ ⟨f, g⟩ = \frac{3}{4}πe^{−2}, and ∥e∥^{2} = \frac{π}{2} . \end{matrix}

Thanks to (∗) we obtain the following (strict) upper bound

I < \frac{π}{8(1 − e^{−1})}[(\frac{1}{2} + \frac{5}{2}e^{−4})^{1/2} + 3e^{−2}] < 0.7112.

\underline{Reference}

1. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, ed. A. Jeffrey, Academic Press, 1980.

Manolis Magiropoulos Technological & Educational Institute of Crete Heraklion, Greece email: mageir@stef.teiher.gr

Dimitri Karayannakis Technological & Educational Institute of Crete Heraklion, Greece email: dkar@stef.teiher.gr