On some Chebyshev type inequalities for the complex integral

Assume thatfandgare continuous onγ,γ⊂Cis a piecewisesmooth path parametrized byz(t), t∈[a, b]fromz(a) =utoz(b) =wwithw6=u, and thecomplex Chebyshev functionalis defined byDγ(f, g) :=1w−u∫γf(z)g(z)dz−1w−u∫γf(z)dz1w−u∫γg(z)dz.In this paper we establish some bounds for the magnitude of the functionalDγ(f, g)under Lipschitzian assumptions for the functionsfandg,and pro-vide a complex version for the well known Chebyshev inequality.

In this paper we establish some bounds for the magnitude of the functional D γ (f, g) under Lipschitzian assumptions for the functions f and g, and provide a complex version for the well known Chebyshev inequality. Keywords: Complex integral, Continuous functions, Holomorphic functions, Chebyshev inequality. MSC2010: 26D15, 26D10, 30A10, 30A86.

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 The constant 1 4 in (1) is sharp. 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 1 12 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 . 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 γ u,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.

[Revista Integración, temas de matemáticas
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, where f γ,∞ := sup z∈γ |f (z)| .
We also define the p-norm with p ≥ 1 by If p, q > 1 with 1 p + 1 q = 1, then, by Hölder's inequality, we have 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 D γ (f, g) under various assumptions for the functions f and g, and provide a complex version for the Chebyshev inequality (3).

Chebyshev type results
We start with the following identity of interest: Proof. For any z ∈ γ the integral γ (f (z) − f (w)) (g (z) − g (w)) dw exists and The function I (z) is also continuous on γ, then the integral γ I (z) dz exists and which proves the first equality in (6).
The rest follows in a similar manner and we omit the details.
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: [Revista Integración, temas de matemáticas We say that the function f : for any z, w ∈ G. If h (z) = z, we recapture the usual concept of L-Lipschitzian functions on G.
Proof. Taking the modulus in the first equality in (6), we get 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 [Revista Integración, temas de matemáticas and by (8) we get For g =f we have 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 Therefore, if f and g are L 1 , L 2 -Lipschitzian functions on [u, w] . and

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 for any t, s ∈ R and r > 0. In particular, for any t, s ∈ R.
If u = R exp (ia) and w = R exp (ib), then Since If γ = γ [a,b],R , then the circular complex Chebyshev functional is defined by We have the following result: