Normalized jensen functional, superquadracity and related inequalities

Shoshana Abramovich; Silvestru S. Dragomir

doi:10.1007/978-3-7643-8773-0_20

Normalized jensen functional, superquadracity and related inequalities

Shoshana Abramovich, Silvestru S. Dragomir

Department of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

Abstract

In this paper we generalize the inequality (Formula Presented) where (Formula Presented) obtained by S.S. Dragomir for convex functions. We show that for the class of functions that we call superquadratic, strictly positive lower bounds of J_n (f, x, p)—mJ_n (f, x, q) and strictly negative upper bounds of J_n (f, x, p)-MJ_n (f, x, q) exist when the functions are also nonnegative. We also provide cases where we can improve the bounds m and M for convex functions and superquadratic functions. Finally, an inequality related to the Čebyšev functional and superquadracity is also given.

Original language	English
Title of host publication	International Series of Numerical Mathematics
Publisher	Springer Science and Business Media Deutschland GmbH
Pages	217-228
Number of pages	12
DOIs	https://doi.org/10.1007/978-3-7643-8773-0_20
State	Published - 2009

Publication series

Name	International Series of Numerical Mathematics
Volume	157
ISSN (Print)	0373-3149
ISSN (Electronic)	2296-6072

Bibliographical note

Publisher Copyright:
© Birkhäuser Verlag Basel/Switzerland 2008.

Keywords

Convex functions
Jensen Steffensen inequality
Jensen inequality
Superquadratic functions
Čebyšev inequality

ASJC Scopus subject areas

Numerical Analysis
Control and Optimization
Applied Mathematics

Access to Document

10.1007/978-3-7643-8773-0_20

Cite this

@inbook{35b09b57ae074b87a87c0c1f7d2aa055,

title = "Normalized jensen functional, superquadracity and related inequalities",

abstract = "In this paper we generalize the inequality (Formula Presented) where (Formula Presented) obtained by S.S. Dragomir for convex functions. We show that for the class of functions that we call superquadratic, strictly positive lower bounds of Jn (f, x, p)—mJn (f, x, q) and strictly negative upper bounds of Jn (f, x, p)-MJn (f, x, q) exist when the functions are also nonnegative. We also provide cases where we can improve the bounds m and M for convex functions and superquadratic functions. Finally, an inequality related to the {\v C}eby{\v s}ev functional and superquadracity is also given.",

keywords = "Convex functions, Jensen Steffensen inequality, Jensen inequality, Superquadratic functions, {\v C}eby{\v s}ev inequality",

author = "Shoshana Abramovich and Dragomir, {Silvestru S.}",

note = "Publisher Copyright: {\textcopyright} Birkh{\"a}user Verlag Basel/Switzerland 2008.",

year = "2009",

doi = "10.1007/978-3-7643-8773-0_20",

language = "English",

series = "International Series of Numerical Mathematics",

publisher = "Springer Science and Business Media Deutschland GmbH",

pages = "217--228",

booktitle = "International Series of Numerical Mathematics",

address = "Germany",

}

Normalized jensen functional, superquadracity and related inequalities. / Abramovich, Shoshana; Dragomir, Silvestru S.
International Series of Numerical Mathematics. Springer Science and Business Media Deutschland GmbH, 2009. p. 217-228 (International Series of Numerical Mathematics; Vol. 157).

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

TY - CHAP

T1 - Normalized jensen functional, superquadracity and related inequalities

AU - Abramovich, Shoshana

AU - Dragomir, Silvestru S.

N1 - Publisher Copyright: © Birkhäuser Verlag Basel/Switzerland 2008.

PY - 2009

Y1 - 2009

N2 - In this paper we generalize the inequality (Formula Presented) where (Formula Presented) obtained by S.S. Dragomir for convex functions. We show that for the class of functions that we call superquadratic, strictly positive lower bounds of Jn (f, x, p)—mJn (f, x, q) and strictly negative upper bounds of Jn (f, x, p)-MJn (f, x, q) exist when the functions are also nonnegative. We also provide cases where we can improve the bounds m and M for convex functions and superquadratic functions. Finally, an inequality related to the Čebyšev functional and superquadracity is also given.

AB - In this paper we generalize the inequality (Formula Presented) where (Formula Presented) obtained by S.S. Dragomir for convex functions. We show that for the class of functions that we call superquadratic, strictly positive lower bounds of Jn (f, x, p)—mJn (f, x, q) and strictly negative upper bounds of Jn (f, x, p)-MJn (f, x, q) exist when the functions are also nonnegative. We also provide cases where we can improve the bounds m and M for convex functions and superquadratic functions. Finally, an inequality related to the Čebyšev functional and superquadracity is also given.

KW - Convex functions

KW - Jensen Steffensen inequality

KW - Jensen inequality

KW - Superquadratic functions

KW - Čebyšev inequality

UR - http://www.scopus.com/inward/record.url?scp=84979078373&partnerID=8YFLogxK

U2 - 10.1007/978-3-7643-8773-0_20

DO - 10.1007/978-3-7643-8773-0_20

M3 - Chapter

AN - SCOPUS:84979078373

T3 - International Series of Numerical Mathematics

SP - 217

EP - 228

BT - International Series of Numerical Mathematics

PB - Springer Science and Business Media Deutschland GmbH

ER -

Normalized jensen functional, superquadracity and related inequalities

Abstract

Publication series

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this