Inequalities for averages of quasiconvex and superquadratic functions

S. Abramovich; L. E. Persson

doi:10.7153/mia-19-40

Inequalities for averages of quasiconvex and superquadratic functions

S. Abramovich
, L. E. Persson

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

For n ε ℤ+ we consider the difference B_n-1 (f)-B_n(f):= 1/a_n ^n-1/η_i=0 f(ai/a_n-1)-1/a_n+1ⁿη_i=0f(a_i/a_n) where the sequences{a_i} and {a_i-a_i-1} are increasing. Some lower bounds are derived when f is 1-quasiconvex and when f is a closely related superquadratic function. In particular, by using some fairly new results concerning the so called "Jensen gap", these bounds can be compared. Some applications and related results about A_n-1 (f)-A_n(f):= 1/a_n ^n-1/η_i=0 f(ai/a_n-1)-1/a_n+1ⁿη_i=0f(a_i/a_n) are also included.

Original language	English
Pages (from-to)	535-550
Number of pages	16
Journal	Mathematical Inequalities and Applications
Volume	19
Issue number	2
DOIs	https://doi.org/10.7153/mia-19-40
State	Published - Apr 2016

Keywords

Convexity
Differences of averages
Inequalities
Lower bounds.
Refinements
Superquadracity
γ-quasiconvexity

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.7153/mia-19-40

Cite this

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title = "Inequalities for averages of quasiconvex and superquadratic functions",

abstract = "For n ε ℤ+ we consider the difference Bn-1 (f)-Bn(f):= 1/an n-1/ηi=0 f(ai/an-1)-1/an+1nηi=0f(ai/an) where the sequences\{ai\} and \{ai-ai-1\} are increasing. Some lower bounds are derived when f is 1-quasiconvex and when f is a closely related superquadratic function. In particular, by using some fairly new results concerning the so called {"}Jensen gap{"}, these bounds can be compared. Some applications and related results about An-1 (f)-An(f):= 1/an n-1/ηi=0 f(ai/an-1)-1/an+1nηi=0f(ai/an) are also included.",

keywords = "Convexity, Differences of averages, Inequalities, Lower bounds., Refinements, Superquadracity, γ-quasiconvexity",

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T1 - Inequalities for averages of quasiconvex and superquadratic functions

AU - Abramovich, S.

AU - Persson, L. E.

PY - 2016/4

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N2 - For n ε ℤ+ we consider the difference Bn-1 (f)-Bn(f):= 1/an n-1/ηi=0 f(ai/an-1)-1/an+1nηi=0f(ai/an) where the sequences{ai} and {ai-ai-1} are increasing. Some lower bounds are derived when f is 1-quasiconvex and when f is a closely related superquadratic function. In particular, by using some fairly new results concerning the so called "Jensen gap", these bounds can be compared. Some applications and related results about An-1 (f)-An(f):= 1/an n-1/ηi=0 f(ai/an-1)-1/an+1nηi=0f(ai/an) are also included.

AB - For n ε ℤ+ we consider the difference Bn-1 (f)-Bn(f):= 1/an n-1/ηi=0 f(ai/an-1)-1/an+1nηi=0f(ai/an) where the sequences{ai} and {ai-ai-1} are increasing. Some lower bounds are derived when f is 1-quasiconvex and when f is a closely related superquadratic function. In particular, by using some fairly new results concerning the so called "Jensen gap", these bounds can be compared. Some applications and related results about An-1 (f)-An(f):= 1/an n-1/ηi=0 f(ai/an-1)-1/an+1nηi=0f(ai/an) are also included.

KW - Convexity

KW - Differences of averages

KW - Inequalities

KW - Lower bounds.

KW - Refinements

KW - Superquadracity

KW - γ-quasiconvexity

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DO - 10.7153/mia-19-40

M3 - Article

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VL - 19

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EP - 550

JO - Mathematical Inequalities and Applications

JF - Mathematical Inequalities and Applications

IS - 2

ER -

Inequalities for averages of quasiconvex and superquadratic functions

Abstract

Keywords

ASJC Scopus subject areas

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