Abstract
The theory of symmetry has a significant influence in many research areas of mathematics. The class of symmetric functions has wide connections with other classes of functions. Among these, one is the class of convex functions, which has deep relations with the concept of symmetry. In recent years, the Schur convexity, convex geometry, probability theory on convex sets, and Schur geometric and harmonic convexities of various symmetric functions have been extensively studied topics of research in inequalities. The present attempt provides novel portmanteauHermite–Hadamard–Jensen–Mercer-type inequalities for convex functions that unify continuous and discrete versions into single forms. They come as a result of using Riemann–Liouville fractional operators with the joint implementations of the notions of majorization theory and convex functions. The obtained inequalities are in compact forms, containing both weighted and unweighted results, where by fixing the parameters, new and old versions of the discrete and continuous inequalities are obtained. Moreover, some new identities are discovered, upon employing which, the bounds for the absolute difference of the two left-most and right-most sides of the main results are established.
Highlights
Publisher’s Note: MDPI stays neutralMathematical inequalities have successfully extended their influence to various fields of science and engineering, and they are accepted and taught as some of the most applicable disciplines of mathematics
The Hermite–Hadamard inequality [3], Jensen’s inequality [4], the Jensen–Mercer inequality [5], the Ostrowski inequality [6], and the Fejér inequality [7] are some names that are immensely popular with researchers
New portmanteauinequalities containing both continuous and discrete versions were successfully introduced for the first time in the present field of research
Summary
Mathematical inequalities have successfully extended their influence to various fields of science and engineering, and they are accepted and taught as some of the most applicable disciplines of mathematics. In the field of inequalities, up to now, there are two main concepts (which are continuous and discrete) where mathematicians are conducting research independently In both cases, researchers have been developing generalized or unified inequalities using (sometimes) generalized integral operators and sometimes a generalized type of convexity, or sometimes, they use both [24]. They provide a unique platform to researchers working with different integrals or convex functions In this stage, there is a necessary notion whose applications can lead us to the inequalities that are a mixture or combination of both discrete and continuous versions. The main results of the present paper are organized as follows: In Theorem 2, the generalized fractional portmanteauform of the Hermite–Hadamard–Jensen–Mercer-type inequalities is obtained using Riemann–Liouville fractional integrals. The conclusion of the whole research work is presented
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