This paper introduces a novel type of convex function known as the refined modified
Fractional calculus is a branch of mathematics that deals with the concepts of fractional differentiation and integration. The history of fractional calculus dates back to the seventeenth century, when G. W. Leibniz and Marquis de l’Hospital initiated a discussion on semi-derivatives [1]. Since then, fractional calculus has been an active area of research that has found applications in many fields of science and engineering, including control theory, signal processing, and fluid mechanics.
The development of fractional calculus owes much to the contributions of Riemann and Liouville. In 1854, Riemann introduced the first integral operator, known as the Riemann-Liouville fractional integral operator [2]. This operator provides a natural extension of the classical concept of integration to non-integer orders. The Riemann-Liouville fractional derivative, which is obtained by applying the fractional integral operator to a function, was introduced by Liouville in 1832 [1]. The Caputo fractional derivative formula, which is a refinement of the Riemann-Liouville fractional derivative formula, was later introduced by Caputo [3]. The Caputo fractional derivative is now widely used in various fields of science and engineering, including fractional control theory and fractional differential equations.
In addition to the classical definitions of fractional derivatives and fractional integrals, numerous other definitions have been proposed by various authors. For example, the Grünwald-Letnikov fractional derivative, the Riesz fractional derivative, and the Marchaud fractional derivative are among the most widely used definitions. These different definitions have their own advantages and disadvantages and find different applications in different fields. The study of fractional derivatives and fractional integrals is an active area of research, and many open problems remain to be solved in this field [2, 4].
The Caputo fractional derivative has a rich history dating back to the seventeenth century when G. W. Leibniz and Marquis de l’Hospital discussed semi-derivatives [3]. It is an essential tool in the field of fractional calculus, which is based on fractional differentiation and integration. The Riemann-Liouville fractional integral operator is the first integral operator in the field of fractional calculus, and the Riemann-Liouville fractional derivative was obtained using this operator. Caputo later improved the Riemann-Liouville fractional derivative formula, resulting in the well-known formula for Caputo fractional derivatives [3].
Definition 1. [3] Let
In particular we have
The Caputo
Definition 2. [5] Let
Definition 3. [6] The beta function of two variable u and v
is defined as follows:
Convex functions have been a fundamental concept in mathematics for over a century, and have found numerous applications in fields ranging from optimization and statistics to economics and physics. A convex function is defined as a function whose graph lies above its secant lines, and has several important properties such as being differentiable almost everywhere, and having a unique minimum.
Classical convexity is the most commonly studied version of convexity, and requires a function to have a non-negative second derivative. A function that is twice differentiable and has a positive second derivative at every point is called strictly convex. Convex functions have a wide range of applications in optimization, as they guarantee the existence of a unique global minimum. Examples of classical convex functions include the exponential function, the logarithm function, and the quadratic function[7, 8].
Another important version of convexity is quasi-convexity, which requires that a function’s sublevel sets are convex. A function is quasi-convex if the set of points below its level set is convex. Quasi-convexity is a weaker form of convexity that is still useful in optimization and other areas of mathematics. Examples of quasi-convex functions include the absolute value function and the maximum function [7, 9].
Another version of convexity is known as strongly convexity, which requires that a function’s second derivative is bounded below by a positive constant. Strong convexity is a more stringent condition than classical convexity and has applications in optimization algorithms, such as gradient descent. Examples of strongly convex functions include the quadratic function with a positive-definite Hessian matrix [7, 10].
Other versions of convexity include p-convexity, which requires a function to be a convex combination of its pth powers, and generalized convexity, which is defined by a broad class of convex functions including quasi-convex, pseudo-convex, and logarithmically convex functions. P-convexity has applications in mathematical economics, as it provides a way to model preferences of consumers with respect to income distributions [11,12,13]). Generalized convexity has applications in optimization, game theory, and statistics [14].
In conclusion, the study of convex functions and their various versions has been an active area of research for many years, and continues to be an important topic in mathematics and other fields. The different types of convexity provide a rich framework for understanding the behavior of functions in different settings, and their study has led to important insights and applications in many areas of science and engineering.
Inequality theory is a fundamental branch of mathematics that deals with the study of inequalities and their applications. Inequalities play a crucial role in many areas of mathematics, such as analysis, geometry, number theory, and optimization, as well as in other fields such as physics and economics. One of the most important inequalities in mathematics is the Hadamard inequality, which provides a bound on the determinant of a positive definite matrix in terms of its diagonal entries. The inequality is named after Jacques Hadamard, who first proved it in 1893. The Hadamard inequality has numerous applications in matrix theory, optimization, and statistics, and has been extended to many other settings, including complex matrices and operator theory [15, 16, 17].
The Hadamard inequality can also be generalized to functions, where it provides a bound on the product of the values of a convex function over an interval in terms of its integral over that interval. In this context, the inequality is known as the Hadamard-Fischer inequality, and has important applications in probability theory, functional analysis, and optimization [18, 19].
In recent years, there has been growing interest in the study of fractional calculus and its applications in inequality theory. Fractional calculus provides a way to extend the classical calculus to non-integer orders, and has been used to develop new inequalities and bounds for various mathematical objects, such as functions, integrals, and differential equations. The study of fractional calculus and its applications in inequality theory has led to new insights and results in many areas of mathematics and physics [19, 20]. In conclusion, the study of inequality theory is an important and active area of research in mathematics and other fields. The Hadamard inequality, in particular, has been a fundamental result in matrix theory and has numerous applications in optimization and statistics. The generalization of the Hadamard inequality to functions and its extensions to fractional calculus have further enriched the field, and continue to be an active area of research.
The Hadamard inequality for convex function is stated in undermentioned theorem:
Theorem 1. [21] If
The Hadamard inequality is a fundamental inequality that is a
straightforward consequence of convexity for traditional convex
functions, and has been extensively studied in the literature. However,
the extension of the Hadamard inequality to the Caputo
This paper presents a study of Caputo fractional derivatives for a
generalized convex function, with the goal of exploring new notions and
concepts motivated by analytic representation of convex functions. The
study will focus on the Hadamard inequality for Caputo
This paper is structured as follows; In Section 2, we provide the
necessary definitions and background information that will be used
throughout the paper. In Section 3, we introduce two versions of the Hadamard
inequality for refined modified
This section provides the necessary definitions and background
information that will be used throughout the paper. We begin by
introducing the concept of convexity and its various definitions, which
are central to the study of refined modified
Definition 4. [22] A function
Definition 5. [23] A function
[24] introduced the
concept of modified
Definition 6. [24] Let
Definition 7 (Modified
Definition 1 (Refined Modified
In both cases, the functions
In this section, we present two versions of Hadamard inequality for
refined modified
Theorem 2. Let
Proof. Since
Corollary 1. By setting
Corollary 2. By setting
Corollary 3. By setting
Corollary 4. If we choose “ h” is identity
function in (
Corollary 5. If “ h” is identity function and
set
Corollary 6. If “ h” is identity function and
set
Corollary 7. If “ h” is identity function and
set
Theorem 3. Let
Proof. By putting
Corollary 8. By setting
Corollary 9. By setting
Corollary 10. By setting
Corollary 11. If we choose “ h” is identity
function in (24), the following Caputo
Corollary 12. If we choose “ h” is identity
function
Corollary 13. If we choose “ h” is identity
function and
Corollary 14. If we choose “ h” is identity
function and
Theorem 4. Let
Proof. Since
Corollary 15. By setting
Corollary 16. By setting
Corollary 17. By setting
In order to establish error estimates of Hadamard-type inequalities, we first introduce a supporting lemmas that utilizes two integral identities. These lemmas plays a crucial role in our analysis and provides a key step towards our main result. Overall, these supporting lemmas serves as an important foundation for our subsequent analysis of Hadamard-type inequalities.
Lemma 1. [25] Let
Theorem 5. Let
Proof. From Lemma 1 and by using the
property of modulus, we get
Corollary 18. By setting
Corollary 19. By setting
Corollary 20. By setting
Lemma 2. [25] Let
Theorem 6. Let
Proof. Using Lemma 2 and refined
modified (h,m)-convexity of
Corollary 21. By setting
Corollary 22. By setting
Corollary 23. By setting
In conclusion, this paper has presented new refinements and special
cases of the Hadamard inequality for modified
The authors declare no conflict of interest.
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