Ostrowski’s Inequality in Complex Inner Product Spaces

S.S. Dragomir

Abstract. A version of Ostrowski’s inequality in complex inner product spaces is given. Applications for complex sequences and integrals are also provided.

1. Introduction

In 1951, A.M. Ostrowski [2, p. 289] proved the following result (see also [1, p. 92])

Theorem 1. Suppose that a, b and x are real n-tuples such that a ≠ 0 and

(1.1)  ∑_{i=1}^{n} a_{i}x_{i} = 0 and ∑_{i=1}^{n} b_{i}x_{i} = 1.

Then

(1.2)  ∑_{i=1}^{n} x_{i}^{2} ≥ \frac{∑_{i=1}^{n} a_{i}^{2}}{∑_{i=1}^{n} a_{i}^{2} ∑_{i=1}^{n} b_{i}^{2} − (∑_{i=1}^{n} a_{i}b_{i})^{2}}

with equality if and only if

(1.3)  x_{k} = \frac{b_{k} ∑_{i=1}^{n} a_{i}^{2} − a_{k} ∑_{i=1}^{n} a_{i}b_{i}}{∑_{i=1}^{n} a_{i}^{2} ∑_{i=1}^{n} b_{i}^{2} − (∑_{i=1}^{n} a_{i}b_{i})^{2}} ,

for k ∈ {1, ..., n} .

An integral version of this inequality was obtained by Pearce, Pečarić and Varošanec in 1998, [3].

H. Šikić and T. Šikić in 2001, [4], by the use of an argument based on orthogonal projection in inner product spaces have observed that Ostrowski’s inequality may be naturally stated in this abstract setting as follows:

Theorem 2. Let (H; ⟨., .⟩) be a real or complex inner product space and a, b ∈ H two linearly independent vectors. If x ∈ H is so that

(1.4)  ⟨x, a⟩ = 0 and ⟨x, b⟩ = 1,

then we has the inequality

(1.5)  ∥x∥^{2} ≥ \frac{∥a∥^{2}}{∥a∥^{2} ∥b∥^{2} − |⟨a, b⟩|^{2}} ,

2000 Mathematics Subject Classification. Primary 26D15, Secondary 46C05.

Key words and phrases. Ostrowski’s Inequality, Inner Product Spaces.

with equality if and only if

(1.6)  x = \frac{∥a∥^{2} b − \overline{⟨a, b⟩} · a}{∥a∥^{2} ∥b∥^{2} − |⟨a, b⟩|^{2}} .

In the present note, by the use of elementary arguments only and Schwarz’s inequality in inner product spaces, we show that Ostrowski’s inequality (1.5) holds true for a larger class of elements x ∈ H. The case of equality is analyzed. Applications for complex sequences and integrals are also provided.

2. The Results

The following theorem holds.

Theorem 3. Let (H; ⟨., .⟩) be a real or complex inner product space and a, b ∈ H two linearly independent vectors. If x ∈ H is so that

(2.1)  ⟨x, a⟩ = 0, and |⟨x, b⟩| = 1;

then one has the inequality

(2.2)  ∥x∥^{2} ≥ \frac{∥a∥^{2}}{∥a∥^{2} ∥b∥^{2} − |⟨a, b⟩|^{2}} .

The equality holds in (2.2) if and only if

(2.3)  x = μ (b − \frac{\overline{⟨a, b⟩}}{∥a∥^{2}} · a)

where μ ∈ 𝕂 (𝕂 = ℝ,ℂ) is so that

(2.4)  |μ| = \frac{∥a∥^{2}}{∥a∥^{2} ∥b∥^{2} − |⟨a, b⟩|^{2}} .

Proof. We use Schwarz’s inequality in the inner product space (H; ⟨., .⟩) , i.e.,

(2.5)  ∥u∥^{2} ∥v∥^{2} ≥ |⟨u, v⟩|^{2} ; u, v ∈ H

with equality iff there exists a scalar α ∈ 𝕂 so that u = αv.

If we apply (2.5) for

u = z − \frac{⟨z, c⟩}{∥c∥^{2}} · c, v = d − \frac{⟨d, c⟩}{∥c∥^{2}} · c

where c ≠ 0 and c, d, z ∈ H, we have

(2.6)  ∥z − \frac{⟨z, c⟩}{∥c∥^{2}} · c∥^{2} ∥d − \frac{⟨d, c⟩}{∥c∥^{2}} · c∥^{2} ≥ |⟨z − \frac{⟨z, c⟩}{∥c∥^{2}} · c, d − \frac{⟨d, c⟩}{∥c∥^{2}} · c⟩|^{2}

with equality iff there is a scalar β ∈ 𝕂 so that

(2.7)  z = \frac{⟨z, c⟩}{∥c∥^{2}} · c + β (d − \frac{⟨d, c⟩}{∥c∥^{2}} · c) .

Since simple calculation show that

∥z − \frac{⟨z, c⟩}{∥c∥^{2}} · c∥^{2} = \frac{∥z∥^{2} ∥c∥^{2} − |⟨z, c⟩|^{2}}{∥c∥^{2}} ,

∥d − \frac{⟨d, c⟩}{∥c∥^{2}} · c∥^{2} = \frac{∥d∥^{2} ∥c∥^{2} − |⟨d, c⟩|^{2}}{∥c∥^{2}} ,

and

⟨z − \frac{⟨z, c⟩}{∥c∥^{2}} · c, d − \frac{⟨d, c⟩}{∥c∥^{2}} · c⟩ = \frac{⟨z, d⟩ ∥c∥^{2} − ⟨z, c⟩ ⟨c, d⟩}{∥c∥^{2}},

then, by (2.6), we deduce

[∥z∥^{2} ∥c∥^{2} − |⟨z, c⟩|^{2}] [∥d∥^{2} ∥c∥^{2} − |⟨d, c⟩|^{2}] (2.8)

≥ |⟨z, d⟩ ∥c∥^{2} − ⟨z, c⟩ ⟨c, d⟩|^{2} ,

with equality if and only if there is a β ∈ 𝕂 so that (2.7) holds.

If a, x, b satisfy (2.1) then by (2.8) and (2.7) for the choices z = x, c = a and d = b we deduce the inequality (2.2) with equality iff there exists a μ ∈ 𝕂 so that

x = μ (b − \frac{\overline{⟨a, b⟩}}{∥a∥^{2}} · a)

and, by the second condition in (2.1) ,

(2.9)  |μ ⟨b − \frac{\overline{⟨a, b⟩}}{∥a∥^{2}} · a, b⟩| = 1.

Since (2.9) is clearly equivalent with (2.4), the theorem is completely proved.

3. Applications

The following particular cases hold.

1. If a, b, x ∈ ℓ^{2} (𝕂) , where ℓ^{2} (𝕂) := {x = (x_{i})_{i∈ℕ} , ∑_{i=1}^{∞} |x_{i}|^{2} < ∞} , with a, b linearly independent and

∑_{i=1}^{∞} x_{i}\overline{a_{i}} = 0 and |∑_{i=1}^{∞} x_{i}\overline{b_{i}}| = 1,

then one has the inequality

(3.1)  ∑_{i=1}^{∞} |x_{i}|^{2} ≥ \frac{∑_{i=1}^{∞} |a_{i}|^{2}}{∑_{i=1}^{∞} |a_{i}|^{2} ∑_{i=1}^{∞} |b_{i}|^{2} − |∑_{i=1}^{∞} a_{i}\overline{b_{i}}|^{2}}

with

(3.2)  x_{i} = μ [b_{i} − \frac{∑_{k=1}^{∞} \overline{a_{k}}b_{k}}{∑_{k=1}^{∞} |a_{k}|^{2}} · a_{i}] , i ∈ ℕ

and μ ∈ 𝕂 with the property

(3.3)  |μ| = \frac{∑_{i=1}^{∞} |a_{i}|^{2} ∑_{i=1}^{∞} |b_{i}|^{2} − |∑_{i=1}^{∞} a_{i}\overline{b_{i}}|^{2}}{∑_{i=1}^{∞} |a_{i}|^{2}} .

2. If f, g, h ∈ L^{2} (Ω, m), where Ω is a measurable space and L^{2} (Ω, m) := {f : Ω → 𝕂, ∫_{Ω} |f (x)|^{2} dm (x) < ∞} , with f, g are linearly independent and

∫_{Ω} h (x) \overline{f (x)}dm (x) = 0, |∫_{Ω} h (x) \overline{g (x)}dm (x)| = 1,

then one has the inequality (3.4)

∫_{Ω} |h (x)|^{2} dm (x) ≥ \frac{∫_{Ω} |f (x)|^{2} dm (x)}{∫_{Ω} |f (x)|^{2} dm (x) ∫_{Ω} |g (x)|^{2} dm (x) − |∫_{Ω} f (x) \overline{g (x)}dm (x)|^{2}}

with equality iff

(3.5)  h (x) = ν [g (x) − \frac{∫_{Ω} f\overline{(x)}g (x) dm (x)}{∫_{Ω} |f (x)|^{2} dm (x)} · f (x)]

for m − a.e.x ∈ Ω, and ν ∈ 𝕂 with

(3.6)  |ν| = \frac{∫_{Ω} |f (x)|^{2} dm (x)}{∫_{Ω} |f (x)|^{2} dm (x) ∫_{Ω} |g (x)|^{2} dm (x) − |∫_{Ω} f (x) \overline{g (x)}dm (x)|}.

References

[1] D.S. MITRINOVIĆ, J.E. PEČARIĆ and A.M. FINK, Classical and New Inequalities in Analysis, Kluwer, Dordrecht, 1993.

[2] A.M. OSTROWSKI, Vorlesungen über Differential und Integralrechnung II, Birkhäuser, Basel, 1951.

[3] C.E.M. PEARCE, J.E. PEČARIĆ and S. VAROŠANEC, An integral analogue of the Ostrowski inequality, J. Ineq. Appl., 2(1998), 275-283.

[4] H. ŠIKIĆ and T. ŠIKIĆ, A note on Ostrowski’s inequality, Math. Ineq. Appl., 4(2)(2001), 297-299.

School of Computer Science & Mathematics, Victoria University, Melbourne, Victoria, Australia

    E-mail address: sever@matilda.vu.edu.au
    URL: http://rgmia.vu.edu.au/SSDragomirWeb.html