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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

The importance of convex functions and convex sets cannot be ignored, especially in nonlinear programing [

In the theory of inequalities, the famous inequality, Hermite–Hadamard inequality was established by Jaques Hadamard [

In [

If

For more details on the Fejér inequality, see [

Mathematically, Jensen-type inequality is stated as if

This inequality has applications in probability and statistics.

The article is organized as follows: Section

This section concerns preliminaries and basic results for the strongly reciprocally

An interval

A function

Let

Let

Let

A function

(strongly reciprocally convex function; see [

Now, we are ready to introduce a new class of convexity named as strongly reciprocally

A function

If we insert

If we insert

If we insert

The following proposition expresses the algebraic property of strongly reciprocally

Let

For any

Choose

where

Let

The next lemma establishes the connection between the strong and reciprocal

Let

Let

This shows that

Conversely, if

This implies that

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

(Hermite–Hadamard-type inequality). Let

We start by the definition; set

Let

For the right side of inequality (

Integrating w.r.t

Since

From (

For

If we allow

For further details on Hermite–Hadamard inequities, see [

(Fejér-type inequality). Assume

Since

Since

The above inequality is integrated with respect to

After simplification, the above inequality becomes

For the right-hand side of (

Integrating with respect to

After simplification, we have

From (

If we set

Jensen-type inequality for the aforementioned inequality is described in the next theorem.

(Jensen-type inequality). If

Fix

Put

Multiplying both sides by

Since

In inequality (

The data used to support the findings of this study are available from the corresponding author upon request.

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.