Some Bounds for the Complex Čebyšev Functional of Functions of Bounded Variation

: In this paper, we provide several bounds for the modulus of the complex ˇCebyšev functional . Applications to the trapezoid and mid-point inequalities, that are symmetric inequalities, are also provided.


Introduction
For Lebesgue integrable functions f , g : [a, b] → C we consider the complex Cebyšev functional For two integrable real-valued functions f , g : [a, b] → R, in order to compare the integral mean of the product with the product of the integral means, in 1934, G. Grüss [1] showed that provided m, M, n, N are real numbers with the property that The constant 1 4 is most possible in (1) in the sense that it cannot be replaced by a smaller one. For other results, see [2][3][4][5][6][7].
To extend this symmetric inequality for complex-valued functions we need the following preparations.
We observe that for any z ∈ C we have the equivalence This follows from the equality that holds for any z ∈ C. The equality (3) is thus a simple consequence of this fact. For any φ, Φ ∈ C, φ = Φ,we also have that +(Im Φ − Im g(t))(Im g(t) − Im φ) ≥ 0 for a.e. t ∈ [a, b]}.
One can easily observe thatS [a,b] (φ, Φ) is closed, convex and This fact provides also numerous example of complex functions belonging to the class In [4] we obtained the following complex version of Grüss' inequality: provided f ∈∆ [a,b] (φ, Φ) and g ∈∆ [a,b] (ψ, Ψ), where g denotes the complex conjugate function of g.
We denote the variance of the complex-valued function f : [a, b] → C by D( f ) and defined as wheref denotes the complex conjugate function of f .
If we apply the inequality (7) for g = f , then we obtain We observe that, if g ∈∆ [a,b] (ψ, Ψ), then g(t) − ψ+Ψ [a,b] ψ, Ψ and by (7), for g instead of g we also have We can also consider the following quantity associated with a complex-valued function f : [a, b] → C, For an integrable function f : [a, b] → C, consider the mean deviation of f defined by The following result holds (see [10] or the more extensive preprint version [11]).
where b a ( f ) denotes the total variation of f on the interval [a, b]. The constant 1 2 is best possible in (11).
The constant 1 4 is best possible in (12). We also have and the constant 1 2 is best possible in (13).
Using the above results we can state, for a function of bounded variation f : [a, b] → C, that In the recent paper [12] we obtained the following result that extends to complex functions the inequalities obtained in [13]: and p, q > 1 with 1 An important corollary of this result is: Then In particular, we have This generalizes the following result obtained by Cheng and Sun [14] by a more complicated technique provided m ≤ g ≤ M for a.e. x ∈ [a, b]. The constant 1 2 is best in (18) as shown by Cerone and Dragomir in [15] where a general version for Lebesgue integral and measurable spaces was also given.
Motivated by the above results, in this paper we establish other bounds for the absolute value of theČebyšev functional when the complex-valued functions are of bounded variation. Applications to the trapezoid and mid-point inequalities are also provided.

Main Results
We have the following inequality for the complexČebyšev functional that extends naturally the real case: and Proof. As in the real case, we have Korkine's identity that can be proved directly by doing the calculations in the right hand side. By the properties of modulus, we have Using the properties of the integral versus the modulus, we also have Using the Cauchy-Bunyakovsky-Schwarz integral inequality, we have and a similar equality for g, hence we obtain from (21) and (22) It is know that the CVF is monotonic nondecreasing on [a, b] and is continuous in a point c ∈ [a, b] if and only if the generating function f is continuous in that point. If f is Lipschitzian with the constant L > 0, i.e., then V f is also Lipschitzian with the same constant.
(i) If f is of bounded variation and g is Lebesgue integrable on [a, b], then (ii) If f and g are of bounded variation on [a, b], then (iii) If f is Lipschitzian with the constant L > 0 and g is of bounded variation on [a, b], then where is the identity mapping of the interval .
Using the Cauchy-Bunyakovsky-Schwarz integral inequality, we have Using the inequality (21) and (26) we obtain the desired result (23).
(ii) If f and g are of bounded variation on [a, b], then for any t, s ∈ [a, b] we have since V f (·) and V g (·) are monotonic nondecreasing on [a, b]. Then Using the inequality (21) and (27) Using the inequality (21) and (28) we obtain the desired result (25).
In 1970, A. M. Ostrowski [16] proved among other things the following result, which is somehow a mixture of theČebyšev and Grüss results provided f is Lebesgue integrable on [a, b] and satisfying (2) while g : [a, b] → R is absolutely continuous and g ∈ L ∞ [a, b]. Here the constant 1 8 is also sharp. The following lemma for real-valued functions holds [17].
Then we have the inequality The constants 1 2 and 1 8 are sharp in the sense that they cannot be replaced by smaller constants.
When one function is complex-valued, we can state the following refinement and extension of Ostrowski's inequality (29). This extends the corresponding result from [17] in which both functions are real-valued.
The constants 1 2 and 1 8 are sharp in the above sense.
Proof. Integrating by parts, we have Taking the modulus, we have The sharpness of the constants follows from the real-valued case outlined in [17].

Some Examples
Assume that the function f : for any g Lebesgue integrable function on [a, b].
If g ∈∆ [a,b] (ψ, Ψ) for some distinct complex numbers ψ, Ψ, then by (34) we have If f and g are of bounded variation on [a, b], then V f and V g are monotonic nondecreasing on [a, b] and by (12) we have Using the inequality (24) we then have If f and g are of bounded variation on [a, b], then by (15) we have Using the inequality (24) we then have From (39) we obtain in particular Then we have by (18) that Therefore, if f is Lipschitzian with the constant L > 0 and g is of bounded variation on [a, b], then by (25) and (41) we have Therefore, if f is Lipschitzian with the constant L > 0 and g is of bounded variation on [a, b], then by (25) and (43) we have This is an improvement of the inequality (42)  Assume that g : [a, b] → C is absolutely continuous with g ∈ L ∞ [a, b] and f : [a, b] → C is of bounded variation. From (33) for V f , V g we have Using (24) we then have

Applications to Trapezoid Inequality
Let h : [a, b] → C be an absolutely continuous function on [a, b]. Then we have the trapezoid equality for any δ ∈ C. This is obvious integrating by parts in the right hand side of the equality. Using the inequality (12) for f = h − δ and g = − a+b 2 we obtain Since then by (47) and (48) we obtain provided that h is of bounded variation and δ ∈ C.
for some complex numbers ψ, Ψ ∈ C, then by (49) we obtain provided h satisfies the condition (50). We observe that, a sufficient condition for the condition (50) to hold is that h is twice differentiable on (a, b) and h ∈∆ [a,b] (ψ, Ψ). Using the inequality (12) for f = − a+b 2 and g = h − δ we obtain then by (52) and (47) we obtain Using (47) we then obtain If we use the inequality (44) for f = h and g = − a+b 2 we obtain provided h is Lipschitzian with the constant K > 0. If we use (47) and (56), we obtain provided h is Lipschitzian with the constant K > 0.

Applications to Mid-Point Inequality
Let h : [a, b] → C be an absolutely continuous function on [a, b]. Then we have the mid-point equality for any δ ∈ C, where the kernel p : [a, b] → R is given by This is obvious integrating by parts in the right hand side of the equality. Using the inequality (12) for f = h − δ and g = p we obtain Since then by (57) and (59) we obtain provided that h is of bounded variation and δ ∈ C.
If h satisfies the condition (50) for some complex numbers ψ, Ψ ∈ C, then by (60) we obtain provided h satisfies the condition (50). We observe that, a sufficient condition for the condition (50) to hold is that h is twice differentiable on (a, b) and h ∈∆ [a,b] (ψ, Ψ).
Using the inequality (12) for f = p and g = h − δ we obtain then by (57) and (62) we obtain provided h : [a, b] → C is an absolutely continuous function on [a, b].
If we use the inequality (16) for f = h with h ∈∆ [a,b] (φ, Φ) for some distinct complex numbers φ, Φ and g = p, then By (57) and (64) we obtain  (p) = t − a, therefore, for any t ∈ [a, b] we have V p (t) = t − a. We observe that, this is an example of a function of bounded variation for the CVF is differentiable for every t ∈ [a, b]. Now, if we take g = p and f = h − δ and use the inequality (24), then we obtain Using the inequality (33) we also have |C(V h −δ , − a)| (66) Then by (57) and (66) we obtain for any δ ∈ C, where h : [a, b] → C is an absolutely continuous function on [a, b] and the derivative is of bounded variation. This is a refinement of (60).

Conclusions
In this paper, we provide several bounds for the modulus of the complexČebyšev functional

Conflicts of Interest:
The author declares no conflict of interest.