A GENERALISATION OF KUREPA’S INEQUALITY

S.S. DRAGOMIR

Abstract. A generalisation of Kurepa’s inequality in inner product spaces that extends in its turn the de Bruijn refinement of the Cauchy-Buniakovsky-Schwarz inequality for sequences of real and complex numbers is given.

1. Introduction

In 1960, N.G. de Bruijn proved the following refinement of the celebrated Cauchy-Bunyakovsky-Schwarz (CBS) inequality, see [1] or [3, p. 48]:

Theorem 1. Let (a_{1}, …, a_{n}) be an n−tuple of real numbers and (z_{1}, …, z_{n}) a n−tuple of complex numbers. Then

(1.1)  |∑_{k=1}^{n} a_{k}z_{k}|^{2} ≤ \frac{1}{2} ∑_{k=1}^{n} a_{k}^{2} [∑_{k=1}^{n} |z_{k}|^{2} + |∑_{k=1}^{n} z_{k}^{2}|] (≤ ∑_{k=1}^{n} a_{k}^{2} ∑_{k=1}^{n} |z_{k}|^{2}) .

Equality holds in (1.1) if and only if, for k ∈ {1, …, n} , a_{k} = Re (λz_{k}), where λ is a complex number such that λ^{2} ∑_{k=1}^{n} z_{k}^{2} is a nonnegative real number.

For various results in connection with the discrete (CBS) −inequality see the recent book [3].

In 1966, in an effort to extend this result to inner products, S. Kurepa considered the following setting (see [2]):

Let H be a real inner product space endowed with the scalar product ⟨·, ·⟩ and the norm generated by the scalar product ∥·∥ = \sqrt{⟨·, ·⟩}. The complexification H_{ℂ} of H is defined as a complex linear space H × H of all ordered pairs (x, y) (x, y ∈ H) endowed with the operations:

(1.2)  (x, y) + (x′, y′) := (x + x′, y + y′) ,
(1.3)  (σ + iτ) · (x, y) := (σx − τy, τx + σy) ,

where x, y, x′, y′ ∈ H and σ, τ ∈ ℝ.

If z = (x, y) ∈ H_{ℂ} and z′ ∈ (x′, y′) ∈ H_{ℂ} then we can define on H_{ℂ} the following complex scalar product

(1.4)  ⟨z, z′⟩_{ℂ} := ⟨x, x′⟩ + ⟨y, y′⟩ + i ⟨y, x′⟩ − i ⟨x, y′⟩

generating the norm

∥z∥_{ℂ} = (∥x∥^{2} + ∥y∥^{2})^{1/2} , z = (x, y) ∈ H_{ℂ}.

Date: 27 October, 2004.

2000 Mathematics Subject Classification. 46C05, 46C10, 26D15, 26D10.

Key words and phrases. Kurepa’s inequality, Schwarz’s inequality, de Bruijn inequality. Inner products, Discrete inequalities, Integral inequalities.

With this inner product (H_{ℂ}; ⟨·, ·⟩_{ℂ}) becomes a complex inner product space. In this space we can also define the conjugate vector z̄ of z = (x, y) by

(1.5)  z̄:= (x, −y) .

Similarly with the scalar case we denote

(1.6)  Re z := (x, 0) ,  Im z := (0, y) .

Formally, we can write z = x + iy = Re z + i Im z and z̄ = x − iy = Re z − i Im z.

We can now state the result due to Kurepa that provides a generalisation of the de Bruijn inequality.

Theorem 2. Let (H; ⟨·, ·⟩) be a real inner product and (H_{ℂ}; ⟨·, ·⟩) its complexification. For any a ∈ H and z ∈ H_{ℂ} we have the inequality

(1.7)  |⟨z, a⟩_{ℂ}|^{2} ≤ ∥a∥^{2} [\frac{1}{2} ∥z∥_{ℂ}^{2} + \frac{1}{2} |⟨z, z̄⟩_{ℂ}|] (≤ ∥a∥^{2} ∥z∥_{ℂ}^{2}) .

If we take some particular examples of inner products, then we may state some interesting refinements of the (CBS) −inequality.

Corollary 1. Let (S, Σ, μ) be a positive measure space and f, g ∈ L_{2} (S, Σ, μ) . If f is a real function and g a complex function on S, then

|∫_{S} f (s) g (s) dμ (s)|^{2} (1.8)

≤ ∫_{S} f^{2} (s) dμ (s) [\frac{1}{2} ∫_{S} |g (s)|^{2} dμ (s) + \frac{1}{2} |∫_{S} g^{2} (s) dμ (s)|]

(≤ ∫_{S} f^{2} (s) dμ (s) ∫_{S} |g (s)|^{2} dμ (s)) .

Corollary 2. If (a_{1}, …, a_{n}) is an n−tuple of real numbers, (z_{1}, …, z_{n}) an n−tuple of complex numbers and (A_{ij}) a positive definite n × n real matrix, then:

| ∑_{i,j=1}^{n} A_{ij}a_{i}z_{j}|^{2} ≤ ( ∑_{i,j=1}^{n} A_{ij}a_{i}a_{j}) [\frac{1}{2} ∑_{i,j=1}^{n} A_{ij}z_{i}\overline{z_{j}} + \frac{1}{2} | ∑_{i,j=1}^{n} A_{ij}z_{i}\overline{z_{j}}|] (1.9)

(≤ ∑_{i,j=1}^{n} A_{ij}a_{i}a_{j} ∑_{i,j=1}^{n} A_{ij}z_{i}\overline{z_{j}}) .

In this paper, a generalisation of Kurepa’s inequality for the case of three vectors in real or complex inner product spaces is given.

Applications for discrete and integral inequalities extending the de Bruijn result (1.7) are also pointed out.

2. A Generalisation of Kurepa’s Inequality

The following lemma is of interest.

Lemma 1. Let f : [0, 2π] → ℝ given by

(2.1)  f (α) = λ sin^{2} α + 2β sin α cos α + γ cos^{2} α,

where λ, β, γ ∈ ℝ. Then

(2.2)  sup_{α∈[0,2π]} f (α) = \frac{1}{2} (λ + γ) + \frac{1}{2} [(γ − λ)^{2} + 4β^{2}]^{\frac{1}{2}} .

Proof. Since

sin^{2} α = \frac{1 − cos 2α}{2},  cos^{2} α = \frac{1 + cos 2α}{2},  2 sin α cos α = sin 2α,

hence f may be written as

(2.3)  f (α) = \frac{1}{2} (λ + γ) + \frac{1}{2} (γ − λ) cos 2α + β sin 2α.

If β = 0, then (2.3) becomes

f (α) = \frac{1}{2} (λ + γ) + \frac{1}{2} (γ − λ) cos 2α.

Obviously, in this case

sup_{α∈[0,2π]} f (α) = \frac{1}{2} (λ + γ) + \frac{1}{2} |γ − λ| = max (γ, λ) .

If β ≠ 0, then (2.3) becomes

f (α) = \frac{1}{2} (λ + γ) + β [sin 2α + \frac{(γ − λ)}{β} cos 2α] .

Let φ ∈ (−\frac{π}{2} , \frac{π}{2} ) for which tan φ = \frac{γ−λ}{2β}. Then f can be written as

f (α) = \frac{1}{2} (λ + γ) + \frac{β}{cos φ} sin (2α + φ) .

For this function, obviously

(2.4)  sup_{α∈[0,2π]} f (α) = \frac{1}{2} (λ + γ) + \begin{matrix}| & β & | \\ | & cos & φ & | \end{matrix}.

Since

\begin{matrix}sin & 2 & φ \\ cos & 2 & φ \end{matrix} = \frac{(γ − λ)^{2}}{4β^{2}} ,

hence

\frac{1}{|cos φ|} = \frac{[(γ − λ)^{2} + 4β^{2}]^{\frac{1}{2}}}{2 |β|} ,

and from (2.4) we deduce the desired result (2.2).

We recall now, a functional (·, ·) : X × X → 𝕂, (𝕂 = ℝ, ℂ) is a nonnegative Hermitian form if

(h) (x, x) ≥ 0 for all x ∈ X; (hh) (αx + βy, z) = α (x, z) + β (y, z) for all α, β ∈ 𝕂 and x, y, z ∈ X; (hhh) \overline{(y, x)} = (x, y) for all x, y ∈ X. Such a functional (·, ·) generates the semi-norm ∥·∥ by

∥x∥ := (x, x)^{\frac{1}{2}} , x ∈ X.

The following result holds.

Theorem 3. Let X be a complex space and (·, ·) a nonnegative Hermitian form defined on X. If x, y, z ∈ X are such that

(2.5)  Im (x, z) = Im (y, z) = 0,

then we have the inequality:

Re^{2} (x, z) + Re^{2} (y, z) (2.6)

= |(x + iy, z)|^{2}

≤ \frac{1}{2} \{∥x∥^{2} + ∥y∥^{2} + [(∥x∥^{2} − ∥y∥^{2})^{2} + 4 Re^{2} (x, y)]^{\frac{1}{2}} \} ∥z∥^{2}

≤ (∥x∥^{2} + ∥y∥^{2}) ∥z∥^{2} .

Proof. Obviously, by (2.5), we have

(x + iy, z) = Re (x, z) + i Re (y, z)

and the first part of (2.6) holds true.

Now, let φ ∈ [0, 2π] be such that

(x + iy, z) = e^{iφ} |(x + iy, z)| .

Then

|(x + iy, z)| = e^{−iφ} (x + iy, z) = (e^{−iφ} (x + iy) , z) .

Utilising the above inequality, we can write:

|(x + iy, z)| = Re (e^{−iφ} (x + iy) , z)
= Re ((cos φ − i sin φ) (x + iy) , z)
= Re (cos φ · x + sin φ · y − i sin φ · x + i cos φ · y, z)
= Re (cos φ · x + sin φ · y, z) + Im (sin φ · x − cos φ · y, z)
= Re (cos φ · x + sin φ · y, z) + sin φ Im (x, z) − cos φ Im (y, z)
= Re (cos φ · x + sin φ · y, z) ,

and for the last equality, we have used the assumption (2.5).

Taking the square and using the Schwarz inequality for the nonnegative Hermitian form (·, ·) , we have

(2.7)  |(x + iy, z)|^{2} = [Re (cos φ · x + sin φ · y, z)]^{2}
≤ ∥cos φ · x + sin φ · y∥^{2} ∥z∥^{2} .

On making use of Lemma 1, we have

sup_{φ∈[0,2π]} ∥cos φ · x + sin φ · y∥^{2}

= sup_{φ∈[0,2π]} [∥x∥^{2} cos^{2} φ + 2 sin φ cos φ Re ⟨x, y⟩ + ∥y∥^{2} sin^{2} φ]

= \frac{1}{2} \{∥x∥^{2} + ∥y∥^{2} + [(∥x∥^{2} − ∥y∥^{2})^{2} + 4 Re^{2} (x, y)]^{\frac{1}{2}} \}

and the first inequality in (2.6) is also proved.

Observe that

(∥x∥^{2} − ∥y∥^{2})^{2} + 4 Re^{2} (x, y) = (∥x∥^{2} − ∥y∥^{2})^{2} + 4 [∥x∥^{2} ∥y∥^{2} − Re^{2} (x, y)]
≤ (∥x∥^{2} − ∥y∥^{2})^{2} ,

and the last part of (2.6) is proved.

Remark 1. Observe that if X is a real space and (·, ·) a nonnegative Hermitian form on X, then for any x, y, z ∈ X one has:

(x, z)^{2} + (y, z)^{2} (2.8)

≤ \frac{1}{2} \{∥x∥^{2} + ∥y∥^{2} + [(∥x∥^{2} − ∥y∥^{2})^{2} + 4 Re^{2} (x, y)]^{\frac{1}{2}} \} ∥z∥^{2}

≤ (∥x∥^{2} + ∥y∥^{2}) ∥z∥^{2} .

Remark 2. If X is a real space, (·, ·) a nonnegative Hermitian form on X, X_{ℂ} its complexification and (·, ·)_{ℂ} the corresponding nonnegative Hermitian form on X, then for x, y ∈ X and w := x + iy ∈ X_{ℂ} and for e ∈ X we have

Im (x, e)_{ℂ} = Im (y, e)_{ℂ} = 0,
∥w∥_{ℂ}^{2} = ∥x∥^{2} + ∥y∥^{2} ,
|(w, w̄)_{ℂ}| = (∥x∥^{2} − ∥y∥^{2})^{2} + 4 (x, y)^{2} ,

where w̄ := x − iy ∈ X_{ℂ}.

Applying Theorem 3 for the complex space X_{ℂ} and complex nonnegative Hermitian form (·, ·)_{ℂ} , we deduce

(2.9)  |(w, e)_{ℂ}|^{2} ≤ \frac{1}{2} ∥e∥^{2} [∥w∥_{ℂ}^{2} + |(w, w̄)_{ℂ}|] ≤ ∥e∥^{2} ∥w∥_{ℂ}^{2} ,

which is a Kurepa type inequality for the Hermitian form (·, ·)_{ℂ} .

Corollary 3. Let x, y, z be as in Theorem 3. In addition, if Re (x, y) = 0, then

[Re^{2} (x, z) + Re^{2} (y, z)]^{\frac{1}{2}} ≤ ∥z∥ max \{∥x∥ , ∥y∥\} .

Remark 3. If X is a real space and (·, ·) a real nonnegative Hermitian form on X, then for any x, y, z ∈ X with (x, y) = 0 we have

[(x, z)^{2} + (y, z)^{2}]^{\frac{1}{2}} ≤ ∥z∥ max \{∥x∥ , ∥y∥\} .

3. A Related Result

Utilising Lemma 1, we may state and prove the following result as well.

Theorem 4. Let X be a real or complex linear space and (·, ·) a nonnegative Hermitian form on X. Then we have the inequalities

\frac{1}{2} \{|(v, t)|^{2} + |(w, t)|^{2} (3.1)

+ [(|(v, t)|^{2} − |(w, t)|^{2})^{2} + 4 (Re (v, t) Re (w, t) + Im (v, t) Im (w, t))^{2}]}

≤ \frac{1}{2} ∥t∥^{2} \{∥v∥^{2} + ∥w∥^{2} + [(∥v∥^{2} − ∥w∥^{2})^{2} + 4 Re^{2} (v, w)]^{\frac{1}{2}} \}

≤ (∥v∥^{2} + ∥w∥^{2}) ∥t∥^{2}

for all v, w, t ∈ X.

Proof. Observe that, by Schwarz’s inequality for the nonnegative Hermitian form (·, ·), we have

(3.2)  |(cos φ · v + sin φ · w, z)|^{2} ≤ ∥cos φ · v + sin φ · w∥^{2} ∥z∥^{2}

for any φ ∈ [0, 2π] .

Since

I (φ) := ∥cos φ · v + sin φ · w∥^{2}
= ∥x∥^{2} cos^{2} φ + 2 sin φ cos φ Re (v, w) + ∥w∥^{2} sin^{2} φ

hence, as in Theorem 3,

sup_{φ∈[0,2π]} I (φ) = \frac{1}{2} \{∥v∥^{2} + ∥w∥^{2} + [(∥v∥^{2} − ∥w∥^{2})^{2} + 4 Re^{2} (v, w)]^{\frac{1}{2}} \} .

Also, denoting

J (φ) := |(v, z) cos φ + (w, z) sin φ|^{2}
= |(v, z)|^{2} cos^{2} φ + 2 sin φ cos φ Re [(v, z) \overline{(w, z)}] + |(w, z)|^{2} sin^{2} φ,

then, on applying Lemma 1, we deduce that

sup_{φ∈[0,2π]} J (φ)

= \frac{1}{2} \{|(v, t)|^{2} + |(w, t)|^{2} + [(|(v, t)|^{2} − |(w, t)|^{2})^{2} + 4 Re^{2} [(v, t) \overline{(w, t)}]]^{\frac{1}{2}} \}

and, since

Re [(v, t) \overline{(w, t)}] = Re (v, t) Re (w, t) + Im (v, t) Im (w, t) ,

hence, on taking the supremum in the inequality (3.2), we deduce the desired inequality (3.1).

Remark 4. In the real case, (3.1) provides the same inequality we obtained in (2.8).

In the complex case, if we assume that v, w, t ∈ X are such that

Re (v, t) Re (w, t) = − Im (v, t) Im (w, t) ,

then (3.1) becomes:

(3.3) max {|(v, t)|^{2} , |(w, t)|^{2}}

≤ \frac{1}{2} ∥t∥^{2} \{∥v∥^{2} + ∥w∥^{2} + [(∥v∥^{2} − ∥w∥^{2})^{2} + 4 Re^{2} (v, w)]^{\frac{1}{2}} \} .

4. Applications to Discrete Inequalities

Assume that (K; ⟨·, ·⟩) is a Hilbert space over the real or complex number field 𝕂. Assume also that p_{k} ≥ 0, k ∈ ℕ with ∑_{k=1}^{∞} p_{k} = 1 and define

ℓ_{p}^{2} (K) := {x := (x_{k})_{k∈ℕ} |x_{k} ∈ K, k ∈ ℕ and ∑_{k=1}^{∞} p_{k} ∥x_{k}∥^{2} < ∞} .

It is well known that ℓ_{p}^{2} (K) endowed with the inner product ⟨·, ·⟩_{p} defined by

⟨x, y⟩_{p} := ∑_{k=1}^{∞} p_{k} ⟨x_{k}, y_{k}⟩

and generating the norm

∥x∥_{p} := (∑_{k=1}^{∞} p_{k} ∥x_{k}∥^{2})^{\frac{1}{2}}

is a Hilbert space over 𝕂.

Utilising Theorem 3, one may state the following proposition.

Proposition 1. Let x = (x_{k})_{k∈ℕ} , y = (y_{k})_{k∈ℕ} and z = (z_{k})_{k∈ℕ} ∈ ℓ_{p}^{2} (K) such that

∑_{k=1}^{∞} p_{k} Im ⟨x_{k}, z_{k}⟩ = ∑_{k=1}^{∞} p_{k} Re ⟨y_{k}, z_{k}⟩ = 0.

Then we have the inequalities:

(∑_{k=1}^{∞} p_{k} Re ⟨x_{k}, y_{k}⟩)^{2} + (∑_{k=1}^{∞} p_{k} Re ⟨y_{k}, z_{k}⟩)^{2} (4.1)

≤ \frac{1}{2} \{∑_{k=1}^{∞} p_{k} (∥x_{k}∥^{2} + ∥y_{k}∥^{2}) + [(∑_{k=1}^{∞} p_{k} (∥x_{k}∥^{2} − ∥y_{k}∥^{2}))^{2}

+ 4 [∑_{k=1}^{∞} p_{k} Re ⟨x_{k}, y_{k}⟩]^{2}]^{\frac{1}{2}} \} ∑_{k=1}^{∞} p_{k} ∥z_{k}∥^{2}

≤ ∑_{k=1}^{∞} p_{k} (∥x_{k}∥^{2} + ∥y_{k}∥^{2}) ∑_{k=1}^{∞} p_{k} ∥z_{k}∥^{2} .

In particular, if ∑_{k=1}^{∞} p_{k} Re ⟨y_{k}, z_{k}⟩ = 0, then

(4.2) [(∑_{k=1}^{∞} p_{k} Re ⟨x_{k}, y_{k}⟩)^{2} + (∑_{k=1}^{∞} p_{k} Re ⟨y_{k}, z_{k}⟩)^{2}]^{\frac{1}{2}}

≤ max \{(∑_{k=1}^{∞} p_{k} ∥x_{k}∥^{2})^{\frac{1}{2}} , (∑_{k=1}^{∞} p_{k} ∥y_{k}∥^{2})^{\frac{1}{2}} \} · (∑_{k=1}^{∞} p_{k} ∥z_{k}∥^{2})^{\frac{1}{2}} .

Remark 5. For the case of complex numbers, if (x_{k})_{k∈ℕ} , (y_{k})_{k∈ℕ} , (z_{k})_{k∈ℕ} ∈ ℓ_{p}^{2} (ℂ) are such that

(4.3)  ∑_{k=1}^{∞} p_{k} Re x_{k} · Im z_{k} = ∑_{k=1}^{∞} p_{k} Re z_{k} · Im x_{k}

and

∑_{k=1}^{∞} p_{k} Re y_{k} · Im z_{k} = ∑_{k=1}^{∞} p_{k} Re z_{k} · Im y_{k},

then we have the inequalities

[∑_{k=1}^{∞} p_{k} (Re x_{k} · Re z_{k} + Im x_{k} · Im z_{k})]^{2} (4.4)

+ [∑_{k=1}^{∞} p_{k} (Re y_{k} · Re z_{k} + Im y_{k} · Im z_{k})]^{2}

≤ \frac{1}{2} \{∑_{k=1}^{∞} p_{k} (Re^{2} x_{k} + Im^{2} x_{k} + Re^{2} y_{k} + Im^{2} y_{k})

+ (∑_{k=1}^{∞} p_{k} (Re^{2} x_{k} + Im^{2} x_{k} − Re^{2} y_{k} − Im^{2} y_{k}))^{2}

+4 [∑_{k=1}^{∞} p_{k} (Re x_{k} · Re y_{k} + Im x_{k} · Im y_{k})]^{2}}

× ∑_{k=1}^{∞} p_{k} (Re^{2} z_{k} + Im^{2} z_{k})

≤ ∑_{k=1}^{∞} p_{k} (Re^{2} x_{k} + Im^{2} x_{k} + Re^{2} y_{k} + Im^{2} y_{k}) ∑_{k=1}^{∞} p_{k} (Re^{2} z_{k} + Im^{2} z_{k}) .

Observe that if above, in particular, we assume that Re x_{k} = a_{k}, Im x_{k} = 0, Re y_{k} = b_{k}, Im y_{k} = 0, Re z_{k} = c_{k}, Im z_{k} = 0 where a_{k}, b_{k}, c_{k} ∈ ℝ, k = {1, …, n} , then (4.3) is satisfied, and, from (4.4), we deduce

(∑_{k=1}^{∞} p_{k}a_{k}c_{k})^{2} + (∑_{k=1}^{∞} p_{k}b_{k}c_{k})^{2}

≤ \frac{1}{2} \{∑_{k=1}^{∞} p_{k} (a_{k}^{2} + b_{k}^{2}) + [(∑_{k=1}^{∞} p_{k} (a_{k}^{2} − b_{k}^{2}))^{2} + 4 (∑_{k=1}^{∞} p_{k}a_{k}b_{k})^{2}]^{\frac{1}{2}} \} ∑_{k=1}^{∞} p_{k}c_{k}^{2}

≤ ∑_{k=1}^{∞} p_{k} (a_{k}^{2} + b_{k}^{2}) ∑_{k=1}^{∞} p_{k}c_{k}^{2},

which is the de Bruijn inequality for the complex numbers w_{k} = a_{k} + ib_{k} and real numbers c_{k}, k ∈ ℕ, i.e.,

|∑_{k=1}^{∞} p_{k}c_{k}w_{k}|^{2} ≤ \frac{1}{2} ∑_{k=1}^{∞} p_{k}c_{k}^{2} [∑_{k=1}^{∞} p_{k} |w_{k}|^{2} + |∑_{k=1}^{∞} p_{k}w_{k}^{2}|^{2}]

≤ ∑_{k=1}^{∞} p_{k}c_{k}^{2} ∑_{k=1}^{∞} p_{k} |w_{k}|^{2} ,

provided (c_{k})_{k∈ℕ} , (w_{k})_{k∈ℕ} ∈ ℓ_{p}^{2} (ℂ) .

Remark 6. We note that the condition (4.3) may hold in other cases, for instance, if:

\frac{Re x_{k}}{Im x_{k}} = \frac{Re z_{k}}{Im z_{k}} = \frac{Re y_{k}}{Im y_{k}} ,

for each k ∈ ℕ, provided Im x_{k}, Im z_{k} and Im y_{k} are not zero for k ∈ ℕ.

5. Applications to Integral Inequalities

Assume that (K; ⟨·, ·⟩) is a Hilbert space over the real or complex number field 𝕂. If ρ : [a, b] ⊂ ℝ → [0, ∞) is a Lebesgue integrable function with ∫_{a}^{b} ρ (t) dt = 1, then we may consider the space L_{ρ}^{2} ([a, b] ; K) of all functions f : [a, b] → K, that are Bochner measurable and ∫_{a}^{b} ρ (t) ∥f (t)∥^{2} dt < ∞. It is well known that L_{ρ}^{2} ([a, b] ; K) endowed with the inner product ⟨·, ·⟩_{ρ} defined by

⟨f, g⟩_{ρ} := ∫_{a}^{b} ρ (t) ⟨f (t) , g (t)⟩ dt

and generating the norm

∥f∥_{ρ} := (∫_{a}^{b} ρ (t) ∥f (t)∥^{2} dt)^{\frac{1}{2}}

is a Hilbert space over 𝕂.

Applying Theorem 3 for L_{ρ}^{2} ([a, b] ; K), we may state the following proposition.

Proposition 2. Let f, g, h ∈ L_{ρ}^{2} ([a, b] ; K) such that

∫_{a}^{b} ρ (t) Im ⟨f (t) , h (t)⟩ dt = ∫_{a}^{b} ρ (t) Im ⟨g (t) , h (t)⟩ dt = 0.

Then we have the inequalities:

(∫_{a}^{b} ρ (t) Re ⟨f (t) , h (t)⟩ dt)^{2} + (∫_{a}^{b} ρ (t) Re ⟨g (t) , h (t)⟩ dt)^{2}

≤ \frac{1}{2} \{∫_{a}^{b} ρ (t) (∥f (t)∥^{2} + ∥g (t)∥^{2}) dt

+ [(∫_{a}^{b} ρ (t) (∥f (t)∥^{2} − ∥g (t)∥^{2}) dt)^{2} + 4 (∫_{a}^{b} ρ (t) Re ⟨f (t) , g (t)⟩ dt)^{2}]^{\frac{1}{2}} \}

× ∫_{a}^{b} ρ (t) ∥h (t)∥^{2} dt

≤ ∫_{a}^{b} ρ (t) (∥f (t)∥^{2} + ∥g (t)∥^{2}) dt · ∫_{a}^{b} ρ (t) ∥h (t)∥^{2} dt.

In particular, if

∫_{a}^{b} ρ (t) Re ⟨f (t) , g (t)⟩ dt = 0

then

[(∫_{a}^{b} ρ (t) Re ⟨f (t) , h (t)⟩ dt)^{2} + (∫_{a}^{b} ρ (t) Re ⟨g (t) , h (t)⟩ dt)^{2}]^{\frac{1}{2}}

≤ max \{(∫_{a}^{b} ρ (t) ∥f (t)∥^{2} dt)^{\frac{1}{2}} , (∫_{a}^{b} ρ (t) ∥g (t)∥^{2} dt)^{\frac{1}{2}} \} (∫_{a}^{b} ρ (t) ∥h (t)∥^{2} dt)^{\frac{1}{2}} .

References

[1] N.G. de BRUIJN, Problem 12, Wisk. Opgaven, 21 (1960), 12-14.

[2] S. KUREPA, On the Buniakowsky-Cauchy-Schwarz inequality, Glasnick Mathematički, 1(21)(2) (1966), 147-158.

[3] S.S. DRAGOMIR, Discrete Inequalities of the Cauchy-Bunyakovsky-Schwarz Type, Nova Science Publishers, NY, 2004.

School of Computer Science and Mathematics, Victoria University of Technology, PO Box 14428, MCMC 8001, VIC, Australia.

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