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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=qc,qr=q¯+q¯/2,q¯−q¯/2. 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 h1,h2-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 H.H 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.

The heat transfer ratio plays an important role in the industrial and engineering sectors; in this model, the authors used the hybrid nanofluid because the heat transfer ratio of the hybrid nanofluid is more than that of the base fluid. The key objective of this research work is to boost up the heat transfer ratio, for example, not only the accomplishment of energy is enough but is also expected to regulate the feeding of energy, and this is possible only to approve the development of heat transmission liquids to the mechanism of the expenditures of energy and improvement. The current research study investigates the influence of Marangoni convection, solar radiation, and viscous dissipation on the bioconvection couple stress flow of the hybrid nanofluid over a shrinking surface. This type of flow has some important application in the industrial and engineering sectors for the purpose of cooling and heating effect. To transform the non-dimensionless form of the differential equation to the dimensionless form, the authors used the defined similarity transformation. The transformed dimensionless form of the differential equation is solved by the homotopic analysis method. The obtained important result is determined with the help of graphs which is obtained from velocity and temperature equations. The impression of different parameters such as couple stress parameter, Marangoni convection parameter, nanoparticle volume fraction, solar radiation parameter, magnetic field parameter, thermophoresis parameter, Eckert number, and Prandtl number is taken over graphs. The skin friction coefficient and Nusselt number are described in the form of tables.

Supplemental material, sj-docx-1-nvw-10.1177_09574565221093232 for Investigation of chatter prediction in end milling using morphological studies on aluminum alloy (Al6061) by Jakeer H Shaik, Rajasekhara R Mutra, Ahmed M Galal and Sinivasa R Tanneeru in Noise & Vibration Worldwide

Supplemental material, sj-docx-1-nvw-10.1177_09574565221093232 for Investigation of chatter prediction in end milling using morphological studies on aluminum alloy (Al6061) by Jakeer H Shaik, Rajasekhara R Mutra, Ahmed M Galal and Sinivasa R Tanneeru in Noise & Vibration Worldwide