General Norm Inequalities of Trapezoid Type for Fréchet Differentiable Functions in Banach Spaces

: In this paper we establish some error bounds in approximating the integral by general trapezoid type rules for Fréchet differentiable functions with values in Banach spaces.


Introduction
We recall some facts about differentiation of functions between normed vector spaces [1]. Let If there exists such an operator A, it is unique, so we write D f (x) = A and call it the Fréchet derivative of f at x.
A function f that is Fréchet differentiable for any point of O is said to be C 1 if the function O x → D f (x) ∈ B(E, F) is continuous, where B(E, F) is the space of all bounded linear operators defined on E with values in F. A function Fréchet differentiable at a point is continuous at that point. Fréchet differentiation is a linear operation. If f is Fréchet differentiable at x, it is also Gâteaux differentiable there, and ∇ x f (u) = D f (x)(u) for all u ∈ E.
We say that the function f : In [2] we established among others the following midpoint and trapezoid type inequalities for L-Lipschitzian functions f on an open and convex subset C in E and for all x, y ∈ C. The constant 1 4 is best possible in both inequalities (1) and (2). For Hermite-Hadamard's type inequalities for functions with scalar values, namely where f : [a, b] → R is convex on [a, b], see for instance [1,[3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] and the references therein. Some of the integrals used in these papers are taken in the sense of Riemann-Stieltjes while in the current paper we consider only vector valued functions with values in Banach spaces, in the first instance, and secondly, with values in Banach algebras where some applications for some fundamental functions like the exponential are also given.
In the recent paper [22] we obtained among others the following weighted version of the trapezoid inequality (2).
Moreover, we have the upper bounds , and B( f , p, x, y) ≤ for r, q > 1 with 1 Motivated by the above results, in this paper we establish some upper bounds for the quantity in the case that f : C ⊂ E → F is Fréchet differentiable on the open and convex subset C of the Banach space E with values into another Banach space F, x, y ∈ C, p : [0, 1] → C is a Lebesgue integrable function and γ ∈ C. Some particular cases of interest for different choices of γ are given. Applications for Banach algebras are also provided.

Some Identities of Interest
Consider a function f : C ⊂ E → F that is defined on the open and convex set C. We have the following properties for the auxiliary function where x, y ∈ C.

Lemma 1.
Assume that the function f : C ⊂ E → F is Fréchet differentiable on the open and convex set C. Then for all x, y ∈ C the auxiliary function ϕ (x,y) is differentiable on (0, 1) and Also and Proof. Let t ∈ (0, 1) and h = 0 small enough such that t + h ∈ (0, 1). Then Since f is Fréchet differentiable, hence by taking the limit over h → 0 in (11) we get which proves (8).
Additionally, we have since f is assumed to be Fréchet differentiable in x. This proves (9). The equality (10) follows in a similar way.
We have the following identity for the Riemann-Stieltjes integral: . Then for all x, y ∈ C and any γ, µ ∈ C, In particular, for µ = γ we have Proof. Using integration by parts rule for the Riemann-Stieltjes integral, we have for any s ∈ [0, 1].
If we add these two equalities, then we get for any s ∈ [0, 1], which proves the desired equality (12).

Remark 1.
From the equality (13) we have for s ∈ [0, 1] and γ = u(s) that and, in particular (13), then we get and, in particular The case of weighted integrals is as follows:  1]. Then for all x, y ∈ C and any γ, µ ∈ C, In particular, for µ = γ we have The proof follows by Lemma 2 applied for the function u : [0, 1] → C, u(t) = t 0 p(s)ds that is absolutely continuous on [0, 1] and therefore of bounded variation and

Remark 2. With the assumptions of Corollary 1 and by utilizing Remark 1 we get
and, in particular and, in particular

Main Results
We have:  1]. Then for all x, y ∈ C and any γ, µ ∈ C, In particular, for µ = γ we have Moreover, we have the upper bounds and for all x, y ∈ C and any γ, µ ∈ C. By taking the norm in (30), we get By using Hölder's inequality we have Therefore In particular, In particular, we have the trapezoid type inequalities

Remark 3. Since the Fréchet derivative satisfies the condition
for a ∈ C and b ∈ E, then for all x, y ∈ C we also have the chain of inequalities In particular, If α ∈ [0, 1], then for all x, y ∈ C we also have the chain of inequalities In particular, we have the trapezoid type inequalities

Some Examples for Banach Algebras
Let B be an algebra. An algebra norm on B is a map · : B→[0, ∞) such that (B, · ) is a normed space, and, further: for any a, b ∈ B. The normed algebra (B, · ) is a Banach algebra if · is a complete norm. We assume that the Banach algebra is unital, this means that B has an identity 1 and that 1 = 1.
Let B be a unital algebra. An element a ∈ B is invertible if there exists an element b ∈ B with ab = ba = 1. The element b is unique; it is called the inverse of a and written a −1 or 1 a . The set of invertible elements of B is denoted by InvB. If a, b ∈InvB then ab ∈InvB and (ab) Now, by the help of power series f (z) = ∑ ∞ n=0 α n z n we can naturally construct another power series which will have as coefficients the absolute values of the coefficients of the original series, namely, f a (z) := ∑ ∞ n=0 |α n |z n . It is obvious that this new power series will have the same radius of convergence as the original series. We also notice that if all coefficients α n ≥ 0, then f a = f .
Lemma 3. Let f (z) = ∑ ∞ n=0 α n z n be a function defined by power series with complex coefficients and convergent on the open disk D(0, R) ⊂ C, R > 0. For any x, y ∈ B with x , y < R we have We also have: for all t ∈ [0, 1].
Proof. Let u , v < R. Then there exists δ > 0 such that u + εv < R for all ε ∈ (−δ, δ) and by (40) we get and by taking ε → 0 we get, by the property of integral, that Now, if we take in (42) u = (1 − t)x + ty and v = y − x, then we get (41).
We have the following result: In particular, In particular, we have the trapezoid type inequalities The proof follows by Theorem 2 and Lemma 4. As some natural examples that are useful for applications, we can point out that if then the corresponding functions constructed by the use of the absolute values of the coefficients are Other important examples of functions as power series representations with nonnegative coefficients are: Γ(n + α)Γ(n + β)Γ(γ) n!Γ(α)Γ(β)Γ(n + γ) λ n , α, β, γ > 0, λ ∈ D(0, 1).
If we consider the exponential function f (x) = exp x, then for x, y ∈ B .
= max{ x , y }.     The interested reader may apply the above inequalities for the other functions listed in (47)-(49). The details are omitted.

Conclusions
In this paper we established some upper bounds for the quantity 1 0 p(s)ds − γ f (y) + γ f (x) − 1 0 p(t) f ((1 − t)x + ty)dt in the case that f : C ⊂ E → F is Fréchet differentiable on the open and convex subset C of the Banach space E with values into another Banach space F, x, y ∈ C, p : [0, 1] → C is a Lebesgue integrable function and γ ∈ C. Some particular cases of interest for different choices of γ are given. The symmetric case for the coefficients of f in the above inequality is analysed. Applications for Banach algebras are also provided.