DOI: 10.1155/2023/8865992 ISSN: 2314-4785

# Some Novel Estimates of Integral Inequalities for a Generalized Class of Harmonical Convex Mappings by Means of Center-Radius Order Relation

Waqar Afzal, Khurram Shabbir, Mubashar Arshad, Joshua Kiddy K. Asamoah, Ahmed M. Galal In interval analysis, integral inequalities are determined based on different types of order relations, including pseudo, fuzzy, inclusion, and various other partial order relations. By developing a link between center-radius (CR) order relations, it seeks to develop a theory of inequalities with novel estimates. A (CR)-order relation relationship differs from traditional interval-order relationships in that it is calculated as follows:
$q=\u2329{q}_{c},{q}_{r}\u232a=\u2329\overline{q}+\underset{\xaf}{q}/2,\overline{q}-\underset{\xaf}{q}/2\u232a$
. There are several advantages to using this ordered relationship, including the fact that the inequality terms deduced from it yield much more precise results than any other partial-order relation defined in the literature. This study introduces the concept of harmonical
$\left({h}_{1},{h}_{2}\right)$
-convex functions associated with the center-radius order relations, which is very novel in literature. Applied to uncertainty, the center-radius order relation is an effective tool for studying inequalities. Our first step was to establish the Hermite−Hadamard
$\left(\mathcal{H}.\mathcal{H}\right)$
inequality and then to establish Jensen inequality using these notions. We discuss a few exceptional cases that could have practical applications. Moreover, examples are provided to verify the applicability of the theory developed in the present study.