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In this paper, we introduce an affine geometric quantity and call it Orlicz mixed chord integral by defining a new Orlicz chord addition, which generalizes the mixed chord integrals to Orlicz space. The Minkoswki and Brunn-Minkowski inequalities for the Orlicz mixed chord integrals are established. The new inequalities in special cases yield $L_{p}$-Minkowski and Brunn-Minkowski inequalities for the chord integrals. The related concepts and inequalities of $L_{p}$-mixed chord integrals are derived. As an application, a new isoperimetric inequality for the chord integrals is given. As extensions, Orlicz multiple mixed chord integrals and Orlicz-Aleksandrov-Fenchel inequality for the Orlicz multiple mixed chord integrals are also derived here for the first time.

In the paper, we introduce a new partial derivative call it β-partial derivatives as the most natural extensions of the limit definitions of the partial derivative and the β-derivative, which obeys classical properties including: continuity, linearity, product rule, quotient rule, power rule, chain rule and vanishing derivatives for constant functions. As applications, we establish some new Opial-Traple type inequalities for the β-partial derivatives.

In the present paper we establish some new Opial-Wirtinger’s type inequalities involving Katugampola partial derivatives. These new results in special cases yield Agarwal and Pang’s, Traple’s and Pachpatte’s inequalities and provide new estimates on inequality of this type.

Abstract In the paper, we give a new generalization of Gardner–Hartenstine inequality and establish its integral form. As applications, we combine an important inequality and give some broader improvements.

Abstract In the paper, we introduce two concepts of Katugampola conformable partial derivatives and α-conformable integrals. As applications, we establish Opial type inequalities for Katugampola conformable partial derivatives and α-conformable integrals. The new inequalities in special cases yield some of the recent results on inequality of this type.

Abstract In the paper, we first improve the radial Blaschke and harmonic Blaschke additions and introduce the p-radial Blaschke and p-harmonic Blaschke additions. Following this, Dresher type inequalities for the radial Blaschke-Minkowski homomorphisms with respect to p-radial Blaschke and p-harmonic Blaschke additions are established.

In this paper, the classical dual mixed volume of star bodies V˜(K1,⋯,Kn) and dual Aleksandrov–Fenchel inequality are extended to the Orlicz space. Under the framework of dual Orlicz-Brunn-Minkowski theory, we put forward a new affine geometric quantity by calculating first order Orlicz variation of the dual mixed volume, and call it Orlicz multiple dual mixed volume. We generalize the fundamental notions and conclusions of the dual mixed volume and dual Aleksandrov-Fenchel inequality to an Orlicz setting. The classical dual Aleksandrov-Fenchel inequality and dual Orlicz-Minkowski inequality are all special cases of the new dual Orlicz-Aleksandrov-Fenchel inequality. The related concepts of Lp-dual multiple mixed volumes and Lp-dual Aleksandrov-Fenchel inequality are first derived here. As an application, the dual Orlicz–Brunn–Minkowski inequality for the Orlicz harmonic addition is also established.

Our main aim is to generalize the mean dual affine quermassintegrals to the Orlicz space. Under the framework of dual Orlicz-Brunn-Minkowski theory, we introduce a new affine geometric quantity by calculating the first Orlicz variation of the mean dual affine quermassintegrals and call it the Orlicz mean dual affine quermassintegral. The fundamental notions and conclusions of the mean dual affine quermassintegrals and the Minkowski and Brunn-Minkowski inequalities for them are extended to an Orlicz setting. The related concepts and inequalities of dual Orlicz mixed volumes are also included in our conclusions. The new Orlicz isoperimetric inequalities in special case yield the Lp-dual Minkowski inequality and Brunn-Minkowski inequality for the mean dual affine quermassintegrals, which also imply the dual Orlicz-Minkowski inequality and dual Orlicz-Brunn-Minkowski inequality.

Our main aim is to generalize the classical mixed volume V(K1,…,Kn) and Aleksandrov-Fenchel inequality to the Orlicz space. In the framework of Orlicz-Brunn-Minkowski theory, we introduce a new affine geometric quantity by calculating the Orlicz first-order variation of the mixed volume and call it Orlicz multiple mixed volume of convex bodies K1,…,Kn, and Ln, denoted by Vφ(K1,…,Kn,Ln), which involves (n+1) convex bodies in Rn. The fundamental notions and conclusions of the mixed volume and Aleksandrov-Fenchel inequality are extended to an Orlicz setting. The related concepts and inequalities of Lp-multiple mixed volume Vp(K1,…,Kn,Ln) are also derived. The Orlicz-Aleksandrov-Fenchel inequality in special cases yields Lp-Aleksandrov-Fenchel inequality, Orlicz-Minkowski inequality, and Orlicz isoperimetric type inequalities. As application, a new Orlicz-Brunn-Minkowski inequality for Orlicz harmonic addition is established, which implies Orlicz-Brunn-Minkowski inequalities for the volumes and quermassintegrals.

In the article we establish some new Opial’s type inequalities involving higher order partial derivatives. These new inequalities, in special cases, yield Agarwal-Pang’s, Pachpatte’s and Das’s type inequalities.

In the paper, we give some new improvements of the Kantorovich type inequalities by using Popoviciu’s, Hölder’s, Bellman’s and Minkowski’s inequalities. These results in special case yield Hao’s, reverse Cauchy’s and Minkowski’s inequalities.

We establish new inequalities similar to Hardy-Pachpatte-Copson’s type inequalities. These results in special cases yield some of the recent results.

In the present paper, we establish some new inequalities similar to Hilbert’s type inequalities. Our results provide some new estimates to these types of inequalities.

We establish a new Laypunov-type inequality for two nonlinear systems of partial differential equations and the discrete analogue is also established. As application, boundness of the two-dimensional Emden-Fowler-type equation is proved.

Some new inequalities similar to Hilbert's type inequality involving series of nonnegative terms are established.

The main purpose of the present paper is to establish some new Grüss integral inequalities in n independent variables. Our results in special cases yield some of the recent results on Pachpatte's, Mitrinović's, and Ostrowski's inequalities, and provide new estimates on such types of inequalities.

We establish some new Opial-type inequalities involving functions of two and many independent variables. Our results in special cases yield some of the recent results on Opial's inequality and also provide new estimates on inequalities of this type.

In the paper, our main aim is to generalize the qth dual volume to Lp space, and introduce pqth -dual mixed volume by calculating the first order variation of qth dual volumes. We establish the Lpq-Minkowski inequality for pqth-dual mixed volumes and Lpq-Brunn-Minkowski inequality for the qth-dual volumes, respectively. The new inequalities in special case yield some new dual inequalities for the qth-dual volumes. [ABSTRACT FROM AUTHOR]

In this paper, we establish an affine Orlicz log-Minkowki inequality for the affine quermassintegrals by introducing new concepts of affine measures and Orlicz mixed affine measures, and using the newly established Orlicz affine Minkowski inequality for the affine quermassintegrals. The affine Orlicz log-Minkowski inequality in special case yields Lp-affine log-Minkowski inequality. The affine log-Minkowski inequality is also derived. [ABSTRACT FROM AUTHOR]