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Article

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

by
Silvestru Sever Dragomir
1,2
1
Mathematics, College of Engineering & Science, Victoria University, P.O. Box 14428, Melbourne 8001, Australia
2
DST-NRF Centre of Excellence in the Mathematical and Statistical Sciences, School of Computer Science & Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa
Symmetry 2021, 13(7), 1288; https://doi.org/10.3390/sym13071288
Submission received: 22 June 2021 / Revised: 11 July 2021 / Accepted: 12 July 2021 / Published: 17 July 2021
(This article belongs to the Special Issue New Directions in Theory of Approximation and Related Problems)

## Abstract

:
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.

## 1. Introduction

We recall some facts about differentiation of functions between normed vector spaces [1].
Let O be an open subset of a normed vector space E, f a real-valued function defined on $O ,$$a ∈ O$ and u a nonzero element of E. The function $f u$ given by is defined on an open interval containing $0 .$ If the derivative $d f u d t 0$ is defined, i.e., if the limit
exists, then we denote this derivative by $∇ a f u .$ It is called the Gâteaux derivative (directional derivative) of f at a in the direction $u .$ If $∇ a f u$ is defined and $λ ∈ R \ 0$, then is defined and The function f is Gâteaux differentiable at a if $∇ a f u$ exists for all directions $u .$
Let E and F be normed vector spaces, and O be an open subset of E. A function $f : O → F$ is called Fréchet differentiable at $x ∈ O$ if there exists a bounded linear operator $A : E → F$ such that
If there exists such an operator A, it is unique, so we write $D f ( x ) = A$ and call it the Fréchet derivative of f at x.
A function f that is Fréchet differentiable for any point of O is said to be $C 1$ if the function is continuous, where is the space of all bounded linear operators defined on E with values in F. A function Fréchet differentiable at a point is continuous at that point. Fréchet differentiation is a linear operation. If f is Fréchet differentiable at x, it is also Gâteaux differentiable there, and $∇ x f u = D f ( x ) u$ for all $u ∈ E .$
We say that the function $f : O ⊂ E → F$ is L-Lipschitzian on O with the constant $L > 0$ if
In [2] we established among others the following midpoint and trapezoid type inequalities for L-Lipschitzian functions f on an open and convex subset C in E
and
for all $x ,$$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 is convex on 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).
Theorem 1.
Let f$: C ⊂ E → F$ be a function of class $C 1$ on the open and convex subset C of the Banach space E with values into another Banach space F and a Lebesgue integrable and symmetric function on namely for all Then for all $x ,$$y ∈ C$
Moreover, we have the upper bounds
and
for $r ,$$q > 1$ with $1 r + 1 q = 1 .$
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$, 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.

## 2. 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 is differentiable on and
Also
and
Proof.
Let and $h ≠ 0$ small enough such that . Then
Since f is Fréchet differentiable, hence by taking the limit over $h → 0$ in (11) we get
which proves (8).
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:
Lemma 2.
Assume that the function f$: C ⊂ E → F$ is Fréchet differentiable on the open and convex set $C .$ Let be of bounded variation on and . 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
and
for any
If we add these two equalities, then we get
for any which proves the desired equality (12). □
Remark 1.
From the equality (13) we have for and $γ = u s$ that
and, in particular
Additionally, if is such that $u m = u 0 + u 1 2 ,$ then from (14) we get
Now, if we take in (13), then we get
and, in particular
The case of weighted integrals is as follows:
Corollary 1.
Assume that the function f$: C ⊂ E → F$ is Fréchet differentiable on the open and convex set $C .$ Let be Lebesgue integrable on and . 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 , $u t = ∫ 0 t p s d s$ that is absolutely continuous on and therefore of bounded variation and
Remark 2.
With the assumptions of Corollary 1 and by utilizing Remark 1 we get
and, in particular
Additionally, if is such that $∫ 0 m p τ d τ = 1 2 ∫ 0 1 p τ d τ ,$ then
Now, for we get
and, in particular

## 3. Main Results

We have:
Theorem 2.
Assume that the function f$: C ⊂ E → F$ is Fréchet differentiable on the open and convex set $C .$ Let be Lebesgue integrable on and . Then for all $x ,$$y ∈ C$ and any $γ ,$ $μ ∈ C$,
In particular, for $μ = γ$ we have
Moreover, we have the upper bounds
and
Proof.
We have by (19) that
for all $x ,$$y ∈ C$ and any $γ ,$$μ ∈ C$.
By taking the norm in (30), we get
By using Hölder’s inequality we have
and
Therefore
which proves the inequality (27). □
Corollary 2.
Assume that the function f$: C ⊂ E → F$ is Fréchet differentiable on the open and convex set $C .$ Let be Lebesgue integrable on and . Then for all $x ,$$y ∈ C$ we have
In particular,
The proof follows by Theorem 2 on choosing $γ = ∫ 0 s p τ d τ ,$
Corollary 3.
Assume that the function f$: C ⊂ E → F$ is Fréchet differentiable on the open and convex set $C .$ Let be Lebesgue integrable on . Then for all $x ,$$y ∈ C$
for all
In particular, we have the trapezoid type inequalities
Remark 3.
Since the Fréchet derivative satisfies the condition
for $a ∈ C$ and $b ∈ E ,$ then for all $x ,$$y ∈ C$ we also have the chain of inequalities
In particular,
If then for all $x ,$$y ∈ C$ we also have the chain of inequalities
In particular, we have the trapezoid type inequalities

## 4. Some Examples for Banach Algebras

Let $B$ be an algebra. An algebra norm on $B$ is a map $· : B → [ 0 , ∞ )$ such that is a normed space, and, further:
for any $a ,$$b ∈ B .$ The normed algebra is a Banach algebra if $·$ is a complete norm.
We assume that the Banach algebra is unital, this means that $B$ has an identity 1 and that $1 = 1 .$
Let $B$ be a unital algebra. An element $a ∈ B$ is invertible if there exists an element $b ∈ B$ with $a b = b a = 1 .$ The element b is unique; it is called the inverse of a and written $a − 1$ or $1 a .$ The set of invertible elements of $B$ is denoted by Inv$B$. If $a ,$$b ∈$Inv$B$ then $a b ∈$Inv$B$ and
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, 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 $f a = f .$
The following result holds [23].
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 , $R > 0 .$ For any $x ,$ $y ∈ B$ with $x ,$$y < R$ we have
We also have:
Lemma 4.
Let $f z = ∑ n = 0 ∞ α n z n$ be a function defined by power series with complex coefficients and convergent on the open disk , $R > 0 .$ For any $x ,$ $y ∈ B$ with $x ,$$y < R$ we have
for all
Proof.
Let $u ,$$v < R .$ Then there exists $δ > 0$ such that for all and by (40) we get
namely
and by taking $ε → 0$ we get, by the property of integral, that
Now, if we take in (42) and $v = y − x ,$ then we get (41). □
We have the following result:
Theorem 3.
Let $f z = ∑ n = 0 ∞ α n z n$ be a function defined by power series with complex coefficients and convergent on the open disk , $R > 0 .$ Also, let be a Lebesgue integrable function on For any $x ,$$y ∈ B$ with $x ,$$y < R$ we have
In particular,
Also,
for all
In particular, we have the trapezoid type inequalities
The proof follows by Theorem 2 and Lemma 4.
As some natural examples that are useful for applications, we can point out that if
then the corresponding functions constructed by the use of the absolute values of the coefficients are
Other important examples of functions as power series representations with non-negative coefficients are:
$exp λ = ∑ n = 0 ∞ 1 n ! λ n λ ∈ C ,$
If we consider the exponential function $f x = exp x ,$ then for $x ,$$y ∈ B$
Also
Moreover, for $x ,$$y ∈ B$
By making use of Theorem 3 and (50)–(52), we have for , a Lebesgue integrable function on and any $x ,$$y ∈ B$ and that
In particular,
Also,
for all
In particular, we have the trapezoid type inequalities
The interested reader may apply the above inequalities for the other functions listed in (47)–(49). The details are omitted.

## 5. Conclusions

In this paper we established 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$, 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.

## Funding

This research received no external funding.

Not applicable.

Not applicable.

Not applicable.

## Conflicts of Interest

The author declares no conflict of interest.

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Dragomir, S.S. General Norm Inequalities of Trapezoid Type for Fréchet Differentiable Functions in Banach Spaces. Symmetry 2021, 13, 1288. https://doi.org/10.3390/sym13071288

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Dragomir SS. General Norm Inequalities of Trapezoid Type for Fréchet Differentiable Functions in Banach Spaces. Symmetry. 2021; 13(7):1288. https://doi.org/10.3390/sym13071288

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Dragomir, Silvestru Sever. 2021. "General Norm Inequalities of Trapezoid Type for Fréchet Differentiable Functions in Banach Spaces" Symmetry 13, no. 7: 1288. https://doi.org/10.3390/sym13071288

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