Superadditivity, Monotonicity, and Exponential Convexity of the Petrović-Type Functionals

We consider functionals derived from Petrović-type inequalities and establish their superadditivity, subadditivity, and monotonicity properties on the corresponding real n-tuples. By virtue of established results we also define some related functionals and investigate their properties regarding exponential convexity. Finally, the general results are then applied to some particular settings

Weighted Popoviciu type inequalities via generalized Montgomery identities

We obtained useful identities via generalized Montgomery identities, by which the inequality of Popoviciu for convex functions is generalized for higher order convex functions. We investigate the bounds for the identities related to the generalization of the Popoviciu inequality using inequalities for the Čebyšev functional. Some results relating to the Grüss and Ostrowski type inequalities are constructed. Further, we also construct new families of exponentially convex functions and Cauchy-type means by looking at linear functionals associated with the obtained inequalities

Combinatorial extensions of Popoviciu\u27s inequality via Abel-Gontscharoff polynomial with applications in information theory

We establish new refinements and improvements of Popoviciu’s inequality for n-convex functions using Abel-Gontscharoff interpolating polynomial along with the aid of new Green functions. We construct new inequalities for n-convex functions and compute new upper bounds for Ostrowski and Grüss type inequalities. As an application of our work in information theory, we give new estimations for Shannon, Relative and Zipf-Mandelbrot entropies using generalized Popoviciu’s inequality

NON-SYMMETRIC STOLARSKY MEANS

Abstract. In this paper we construct n -exponentially convex functions and exponentially convex functions using the functional defined as the difference of the right parts of the HermiteHadamard inequality, for different classes of functions. Applying these results on some starshaped functions, we derive non-symmetric means of Stolarsky type

Superadditivity, Monotonicity, and Exponential Convexity of the Petrović-Type Functionals

We consider functionals derived from Petrović-type inequalities and establish their superadditivity, subadditivity, and monotonicity properties on the corresponding real n-tuples. By virtue of established results we also define some related functionals and investigate their properties regarding exponential convexity. Finally, the general results are then applied to some particular settings

Some Hermite-Hadamard and midpoint type inequalities in symmetric quantum calculus

The Hermite-Hadamard inequalities are common research topics explored in different dimensions. For any interval $[\mathrm{b_{0}}, \mathrm{b_{1}}]\subset\Re$, we construct the idea of the Hermite-Hadamard inequality, its different kinds, and its generalization in symmetric quantum calculus at $\mathrm{b_{0}}\in[\mathrm{b_{0}}, \mathrm{b_{1}}]\subset\Re$. We also construct parallel results for the Hermite-Hadamard inequality, its different types, and its generalization on other end point $\mathrm{b_{1}}$, and provide some examples as well. Some justification with graphical analysis is provided as well. Finally, with the assistance of these outcomes, we give a midpoint type inequality and some of its approximations for convex functions in symmetric quantum calculus

Hermite–Hadamard type inequalities for multiplicatively harmonic convex functions

Abstract In this work, the notion of a multiplicative harmonic convex function is examined, and Hermite–Hadamard inequalities for this class of functions are established. Many inequalities of Hermite–Hadamard type are also taken into account for the product and quotient of multiplicative harmonic convex functions. In addition, new multiplicative integral-based inequalities are found for the quotient and product of multiplicative harmonic convex and harmonic convex functions. In addition, we provide certain upper limits for such classes of functions. The obtained results have been verified by providing examples with included graphs. The findings of this study may encourage more research in several scientific areas

New Variants of Quantum Midpoint-Type Inequalities

Recently, there has been a strong push toward creating and expanding quadrature inequalities in quantum calculus. In order to investigate various avenues for quantum inquiry, a number of quantum extensions of midpoint estimations are studied. The goal of this research article is to discover novel quantum midpoint-type inequalities that are twice qξ2-differentiable for (α,m)-convex functions. Firstly, we obtain novel identity for qξ2-integral by employing quantum calculus tools. Then by using the auxiliary identity, we formulate new bounds by taking into account the known quantum Hölder and Power mean inequalities. An example is provided with a graphical representation to show the validity of obtaining results. The outcomes of this study clarify and expand earlier research on midpoint-type inequalities. Analytic inequalities of this type as well as particularly related strategies have applications for various fields where symmetry plays an important role

New Quantum Mercer Estimates of Simpson–Newton-like Inequalities via Convexity

Recently, developments and extensions of quadrature inequalities in quantum calculus have been extensively studied. As a result, several quantum extensions of Simpson’s and Newton’s estimates are examined in order to explore different directions in quantum studies. The main motivation of this article is the development of variants of Simpson–Newton-like inequalities by employing Mercer’s convexity in the context of quantum calculus. The results also give new quantum bounds for Simpson–Newton-like inequalities through Hölder’s inequality and the power mean inequality by employing the Mercer scheme. The validity of our main results is justified by providing examples with graphical representations thereof. The obtained results recapture the discoveries of numerous authors in quantum and classical calculus. Hence, the results of these inequalities lead us to the development of new perspectives and extensions of prior results

New generalizations of Popoviciu type inequalities via new green functions and Fink’s identity

We formulate new identities involving new Green functions. Inequality of Popoviciu, which was improved by Vasić and Stanković (1976), is generalized by using newly introduced Green functions. We utilize Fink’s identity along with new Green’s function to generalize the known Popoviciu’s inequality from convex functions to higher order convex functions. Then we construct linear functionals from the generalized identities and formulate the monotonicity of these functionals utilizing the recent theory of inequalities for n-convex functions at a point. New upper bounds of Grüss and Ostrowski type are computed. Keywords: Popoviciu inequality, Fink’s identity, Abel–Gontscharoff interpolating polynomial, New Green functions, Grüss upper bounds, Ostrowski type bound