Jensen Functional, Quasi-Arithmetic Mean and Sharp Converses of Hölder’s Inequalities

Slavko Simić; Vesna Todorčević

doi:10.3390/math9233104

Jensen Functional, Quasi-Arithmetic Mean and Sharp Converses of Hölder’s Inequalities

Mathematics ◽

10.3390/math9233104 ◽

2021 ◽

Vol 9 (23) ◽

pp. 3104

Author(s):

Slavko Simić ◽

Vesna Todorčević

Keyword(s):

Generating Function ◽

Hölder’S Inequality ◽

Arithmetic Mean ◽

Hölder's Inequality ◽

Power Means ◽

Jensen Functional ◽

Exact Bounds ◽

Stolarsky Means

In this article, we give sharp two-sided bounds for the generalized Jensen functional Jn(f,g,h;p,x). Assuming convexity/concavity of the generating function h, we give exact bounds for the generalized quasi-arithmetic mean An(h;p,x). In particular, exact bounds are determined for the generalized power means in terms from the class of Stolarsky means. As a consequence, some sharp converses of the famous Hölder’s inequality are obtained.

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Global Bounds for the Generalized Jensen Functional with Applications

Symmetry ◽

10.3390/sym13112105 ◽

2021 ◽

Vol 13 (11) ◽

pp. 2105

Author(s):

Slavko Simić ◽

Bandar Bin-Mohsin

Keyword(s):

Power Mean ◽

Jensen Functional ◽

Exact Bounds ◽

Harmonic Means ◽

Stolarsky Means ◽

Generalized Arithmetic

In this article we give sharp global bounds for the generalized Jensen functional Jn(g,h;p,x). In particular, exact bounds are determined for the generalized power mean in terms from the class of Stolarsky means. As a consequence, we obtain the best possible global converses of quotients and differences of the generalized arithmetic, geometric and harmonic means.

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Jensen's Functionals on Time Scales

Journal of Function Spaces and Applications ◽

10.1155/2012/384045 ◽

2012 ◽

Vol 2012 ◽

pp. 1-17 ◽

Cited By ~ 1

Author(s):

Matloob Anwar ◽

Rabia Bibi ◽

Martin Bohner ◽

Josip Pečarić

Keyword(s):

Time Scales ◽

Hölder’S Inequality ◽

Hölder's Inequality ◽

Power Means

We consider Jensen’s functionals on time scales and discuss its properties and applications. Further, we define weighted generalized and power means on time scales. By applying the properties of Jensen’s functionals on these means, we obtain several refinements and converses of Hölder’s inequality on time scales.

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Jessen's functional, its properties and applications

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.2478/v10309-012-0015-6 ◽

2012 ◽

Vol 20 (1) ◽

pp. 225-248

Author(s):

Neda Lovričević ◽

Josip Pečarić ◽

Mario Krnić

Keyword(s):

Linear Functional ◽

Geometric Mean ◽

Hölder’S Inequality ◽

Young’S Inequality ◽

Hölder's Inequality ◽

Power Means ◽

Young's Inequality

AbstractIn this paper we consider Jessen's functional, defined by means of a positive isotonic linear functional, and investigate its properties. Derived results are then applied to weighted generalized power means, which yields extensions of some recent results, known from the literature. In particular, we obtain the whole series of refinements and converses of numerous classical inequalities such as the arithmetic-geometric mean inequality, Young's inequality and Hölder's inequality

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Hilbert-type inequalities for time scale nabla calculus

Advances in Difference Equations ◽

10.1186/s13662-020-03079-w ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

H. M. Rezk ◽

Ghada AlNemer ◽

H. A. Abd El-Hamid ◽

Abdel-Haleem Abdel-Aty ◽

Kottakkaran Sooppy Nisar ◽

...

Keyword(s):

Time Scale ◽

Hölder’S Inequality ◽

Hölder's Inequality ◽

Arbitrary Time ◽

Hilbert Type

Abstract This paper deals with the derivation of some new dynamic Hilbert-type inequalities in time scale nabla calculus. In proving the results, the basic idea is to use some algebraic inequalities, Hölder’s inequality, and Jensen’s time scale inequality. This generalization allows us not only to unify all the related results that exist in the literature on an arbitrary time scale, but also to obtain new outcomes that are analytical to the results of the delta time scale calculation.

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