The Proofs of product inequalities in a generalized vector space (original) (raw)

The Proofs of Product Inequalities in Vector Spaces

European Journal of Pure and Applied Mathematics, 2018

In this paper, we introduce the proofs of product inequalities:u v ≤ u + v , for all u, v ∈ [0, 2], and u + v ≤ u v , for allu, v ∈ [2, ∞). The first product inequality u v ≤ u + v holds forany two vectors in the interval [0, 1] in Holder’s space and also valid anytwo vectors in the interval [1, 2] in the Euclidean space. On the otherhand, the second product inequality u + v ≤ u v ∀u, v ∈ [2, ∞)only in Euclidean space. By applying the first product inequality to theL p spaces, we observed that if f : Ω → [0, 1], and g : Ω → R, thenf p g p ≤ f p + g p . Also, if f, g : Ω → R, then f p + g p ≤f p g p .

On Pečarić-Rajić-Dragomir-Type Inequalities in Normed Linear Spaces

Journal of Inequalities and Applications, 2009

We establish some generalizations of the recent Pečarić-Rajić-Dragomir-type inequalities by providing upper and lower bounds for the norm of a linear combination of elements in a normed linear space. Our results provide new estimates on inequalities of this type.

On Functional Inequalities Associated with

2016

In the paper, the equivalence of the functional inequality 2f (x) + f (y) + f (−y) − f (x − y) ≤ f (x + y) (x, y ∈ G) and the Drygas functional equation f (x + y) + f (x − y) = 2f (x) + f (y) + f (−y) (x, y ∈ G) is proved for functions f : G → E where (G, +) is an abelian group, (E, < •, • >) is an inner product space, and the norm is derived from the inner product in the usual way.

Some Inequalities in Classical Spaces with Mixed Norms

2002

We consider some inequalities in such classical Banach Function Spaces as Lorentz, Marcinkiewicz, and Orlicz spaces. Our aim is to explore connections between the norm of a function of two variables on the product space and the mixed norm of the same function, where mixed norm is calculated in function spaces on coordinate spaces, first in one variable, then in the other. This issue is motivated by various problems of functional analysis and theory of functions. We will currently mention just geometry of spaces of vector-valued functions and embedding theorems for Sobolev and Besov spaces generated by metrics which differ from L p. Our main results are actually counterexamples for Lorentz spaces versus the natural intuition that arises from the easier case of Orlicz spaces (Section 2). In the Appendix we give a proof for the Kolmogorov-Nagumo theorem on change of order of mixed norm calculation in its most general form. This result shows that L p is the only space where it is possible to change this order.

INEQUALITIES OF GR¨ USS TYPE INVOLVING THE p-HH-NORMS IN THE CARTESIAN PRODUCT SPACE

Inequalities in estimating a type ofČebyšev functional involving the p -HH-norms are obtained by applying the known results by Grüss, Ostrowski,Čebyšev, and Lupaş. Some of these inequalities are proven to be sharp. In 1998, Dragomir and Fedotov considered a generaliseď Cebyšev functional, in order to approximate the Riemann-Stieltjes integral. In this paper, some sharp bounds for the generalisedČebyšev functional with convex integrand and monotonically increasing integrator are established as well. An application for theČebyšev functional involving the p -HH-norms is also considered; and the bounds are proven to be sharp. and the product (·, ·) p p−HH (·, ·) q q−HH , for any p, q 1 . This difference, however, : 26D15, 46B20, 46C50.

On Generalizations of Grüss Inequality in Inner Product Spaces and Applications

Journal of Inequalities and Applications, 2010

Using a Kurepa's result for Gramians, we achieve refinements of well-known generalizations of Grüss inequality in inner product spaces. These results are further applied in L 2 a, b to derive improvements of some published trapezoid-Grüss and Ostrowki-Grüss type inequalities. Refinements of the discrete version of Grüss inequality as well as a reverse of the Schwarz inequality are also given.