Strongly Reciprocally -Convex Functions and Some Inequalities
In this paper, we generalize the concept of strong and reciprocal convexity. Some basic properties and results will be presented for the new class of strongly reciprocally -convex functions. Furthermore, we will discuss the Hermite–Hadamard-type, Jensen-type, and Fejér-type inequalities for the strongly reciprocally -convex functions.
The importance of convex functions and convex sets cannot be ignored, especially in nonlinear programing [1–5] and optimization theory , see, for instance, [7–14]. Generalization in the convexity is always appreciable. Also, many generalizations and extensions have been made in the theory of inequalities as well as in convexity. Several inequalities have been studied and established for the convexity of functions, and many generalizations, applications, and refinements take place, see [7, 9, 13, 15–18], for further study.
In the theory of inequalities, the famous inequality, Hermite–Hadamard inequality was established by Jaques Hadamard . If is a convex function, thenholds for all with .
In , Lipot Fejér established the weighted version of the Hermite–Hadamard inequality.
If is a convex function, then the inequalityholds for all with and is integrable, nonnegative, and symmetric about .
Mathematically, Jensen-type inequality is stated as if is a convex function defined on , thenholds for all , and with .
This inequality has applications in probability and statistics.
The article is organized as follows: Section 2 is devoted to preliminaries and basic results, whereas in the last section, we will develop the main results for strongly reciprocally -convex functions.
This section concerns preliminaries and basic results for the strongly reciprocally -convex functions.
Definition 1 (-convex set; see ). An interval is called the -convex set if for all and , where or , , , and .
Definition 2 (-convex function; see ). A function is called -convex function iffor all and , where is the -convex set.
Definition 3 (strongly convex function; see ). Let be a positive number. A function is called a strongly convex function iffor all and .
Definition 4 (strongly -convex function; see ). Let be a positive number. A function is called strongly -convex function iffor all and .
Definition 5 (harmonic convex function; see ). Let be an interval. A function is harmonic convex iffor all and .
Definition 6 (harmonic -convex function; see ). A function is called a harmonic -convex function iffor all and .
Definition 7. (strongly reciprocally convex function; see ). Let and . A function is said to be strongly reciprocally convex with modulus on if the inequalityholds for all and .
Now, we are ready to introduce a new class of convexity named as strongly reciprocally -convex function.
Definition 8 (strongly reciprocally -convex function). A function is called strongly reciprocally -convex with modulus on if the inequalityholds, for all and .
Remark 1. (1)If we insert in inequality (10), then we retrace the strong and reciprocal convexity (2)If we insert in inequality (10), then we retrace the harmonic -convexity (3)If we insert and in inequality (10), then we retrace the harmonic convexity The following proposition expresses the algebraic property of strongly reciprocally -convex functions.
Proposition 1. Let be two strongly reciprocally -convex functions; then, the following statements hold:(i) is strongly reciprocally -convex(ii)For any , is strongly reciprocally -convex corresponding to
Proof. (i)Choose ; then, by the definition of and , we obtain where .(ii)Let ; then, by definition, we obtainwhere and .
The next lemma establishes the connection between the strong and reciprocal -convexity and harmonic -convexity.
Lemma 1. Let be a function; is strongly reciprocally -convex iff the function , defined by , is harmonically -convex.
Proof. Let be strongly reciprocally -convex; then, we haveThis shows that is a harmonic -convex function.
Conversely, if is harmonically -convex, thenThis implies that is a strongly reciprocally -convex function for all and .
3. Main Results
In this section, Hermite–Hadamard-, Fejér-, and Jensen-type inequalities are investigated. The next theorem gives the generalization of the Hermite–Hadamard inequality for strongly reciprocally -convex functions.
Theorem 1. (Hermite–Hadamard-type inequality). Let be an interval on the real line. If is a strongly reciprocally -convex function with modulus and , thenfor all with .
Proof. We start by the definition; set in inequality (10), and we haveLet and , and by integrating w.r.t over [0, 1], the above inequality yieldsand then inequality (17) is reduced towhich is the left side of the inequality.
For the right side of inequality (15), set and in (10); we haveIntegrating w.r.t over [0, 1], the above inequality yieldsSincethen we obtainFrom (18) and (22), we get (15).
Remark 2. (1)For in (15), Hermite–Hadamard inequality for strongly reciprocally convex functions is obtained .(2)If we allow in inequalities (15), we obtain the Hermite–Hadamard-type inequalities for harmonically convex functions .For further details on Hermite–Hadamard inequities, see [27–30].
Theorem 2. (Fejér-type inequality). Assume is a strongly reciprocally -convex function with modulus on ; then,holds for with and where is a nonnegative integrable function that satisfies
Proof. Since is a strongly reciprocally -convex function, then by definition for in (10), we havefor all ; suppose and in the above inequality; then, we obtainSince is nonnegative and symmetric, we haveThe above inequality is integrated with respect to over [0, 1], and then putting , we obtainAfter simplification, the above inequality becomesFor the right-hand side of (23), set and in (10); we haveIntegrating with respect to over [0, 1] and then putting , we obtainAfter simplification, we haveFrom (32) and (27), we get (23).
Remark 3. If we set in (23), the Fejér-type inequality for strongly reciprocally convex functions is obtained.
Jensen-type inequality for the aforementioned inequality is described in the next theorem.
Theorem 3. (Jensen-type inequality). If is a reciprocally strongly p-convex function with modulus , thenholds for all , with and .
Proof. Fix and such that .
Put , and suppose a function of the formsupporting at , satisfying and . Then, for every , we haveMultiplying both sides by and summing up to n, we haveSince , we havewhich completes the proof.
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare no conflicts of interest.
Hao Li analyzed all results and proofread and revised the paper, Muhammad Shoaib Saleem proposed the problem and supervised the work, Ijaz Hussain proved the results, and Muhammad Imran wrote the whole paper.
This research was supported by the Higher Education Commission of Pakistan.
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