1. Introduction
For two Lebesgue integrable functions f, g : [a, b] → ℂ, in order to compare the integral mean of the product with the product of the integral means, we consider the Chebyshev functional defined by
In 1934, G. Grüss [17] showed that
provided m, M, n, N are real numbers with the property that
Another, however less known result, even though it was obtained by Chebyshev in 1882, [8], states that
provided that f’, g’ exist and are continuous on The constant cannot be improved in the general case.
The Chebyshev inequality (3) also holds if f, g: [a, b] → ℝ are assumed to be absolutely continuous and f’, g’ ∈ L ∞ [a,b], while
For other inequality of Grüss' type see [1]-[16] and [18]-[28].
In order to extend Grüss' inequality to complex integral we need the following preparations.
Suppose γ is a smooth path parametrized by z (t) , t ∈ [a, b] and f is a complex valued function which is continuous on γ. Put z (a) = u and z (b) = w with u, w ∈ ℂ. We define the integral of f on γu,w = γ as
We observe that the actual choice of parametrization of γ does not matter.
This definition immediately extends to paths that are piecewise smooth. Suppose γ is parametrized by z (t), t ∈ [a, b], which is differentiable on the intervals [a, c] and [c, b]; then, assuming that f is continuous on γ, we define
where v: = z (c). This can be extended for a finite number of intervals.
We also define the integral with respect to arc-length:
and the length of the curve γ is then
Let f and g be holomorphic in G, an open domain, and suppose γ ⊂ G is a piecewise smooth path from z (a) = u to z (b) = w. Then we have the integration by parts formula
We recall also the triangle inequality for the complex integral, namely,
We also define the p-norm with p≥1 by
For p = 1 we have
If p, q > 1 with then, by Hölder’s inequality, we have
Suppose γ ⊂ ℝ is a piecewise smooth path parametrized by z (t), t ∈ [a, b] from z (a) = u to z (b) = w with w ≠ u. If f and g are continuous on γ, we consider the complex Chebyshev functional defined by
In this paper we establish some bounds for the magnitude of the functional Dγ (f, g) under various assumptions for the functions f and g, and provide a complex version for the Chebyshev inequality (3).
2. Chebyshev type results
We start with the following identity of interest:
Lemma 2.1. Suppose γ ⊂ ℝ is a piecewise smooth path parametrized by z (t), t ∈ [a, b] from z (a) = u to z (b) = w with w ≠ u. If f and g are continuous on γ, then
Proof. For any z ∈ γ the integral exists and
The function I (z) is also continuous on γ, then the integral exists and
which proves the first equality in (6).
The rest follows in a similar manner and we omit the details. 0
Suppose γ ⊂ ℂ is a piecewise smooth path from z (a) = u to z (b) = w and h: γ → ℂ a continuous function on γ. Define the quantity:
We say that the function f: G ⊂ ℂ → ℂ is L-h-Lipschitzian on the subset G if
for any z, w ∈ G. If h (z) = z, we recapture the usual concept of L-Lipschitzian functions on G.
Theorem 2.2. Suppose γ ⊂ ℂ is a piecewise smooth path parametrized by z (t), t ∈ [a, b] from z (a) = u to z (b) = w with w ≠ u, h : γ → ℂ is continuous, f and g are L1, L2 -h-Lipschitzian functions on γ; then
Proof. Taking the modulus in the first equality in (6), we get
Now, observe that
Therefore, by (10) we get
and by (9) we get the desired result (8).
Further, for γ ⊂ ℂ a piecewise smooth path parametrized by z (t), and by taking h (z) = z in (7), we can consider the quantity
Corollary 2.3. Suppose γ ⊂ ℂ is a piecewise smooth path parametrized by z (t), t ∈ [a, b] from z (a) = u to z (b) = w with w ≠ u and f and g are L1, L2 -Lipschitzian functions on γ; then
Remark 2.4. Assume that f is L-h-Lipschitzian on γ. For g = f we have
and by (8) we get
and by (8) we get
If f is L-Lipschitzian on γ, then
And
If the path γ is a segment [u, w] connecting two distinct points u and w in ℂ, then we write as
Now, if f and g are L1, L2-Lipschitzian functions on [u, w] := {(1 - t) u + tw, t ∈ [0,1]} , then by (12) we have
where
Therefore,
if f and g are L 1 , L 2-Lipschitzian functions on [u, w].
If f is L-Lipschitzian on [u, w] , then
and
3. Examples for circular paths
Let [a, b] ⊆ [0, 2π] and the circular path γ[ a,b ],R centered in 0 and with radius R > 0:
If [a, b] = [0, π], then we get a half circle, while for [a, b] = [0, 2π] we get the full circle.
Since
for any t, s ∈ ℝ, then
for any t, s ∈ ℝ and r > 0. In particular,
for any t, s ∈ ℝ.
If u = R exp (¿a) and w = R exp(ib), then
Since
and
hence
If γ = γ [a,b],R then the circular complex Chebyshev functional is defined by
If γ = γ [a,b],R then
We have the following result:
Proposition 3.1. Let γ[a,b],R be a circular path centered in 0, with radius R > 0 and [a, b] ⊂ [0, 2π]. If f and g are L 1, L 2 -Lipschitzian functions on γ[a,b],R, then