On co-ordinated quasi-convex functions

M. Emin Özdemir; Ahmet Ocak Akdemir; Çetin Yıldız

Czechoslovak Mathematical Journal (2012)

  • Volume: 62, Issue: 4, page 889-900
  • ISSN: 0011-4642

Abstract

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A function f : I → ℝ , where I ⊆ ℝ is an interval, is said to be a convex function on I if f ( t x + ( 1 - t ) y ) ≤ t f ( x ) + ( 1 - t ) f ( y ) holds for all x , y ∈ I and t ∈ [ 0 , 1 ] . There are several papers in the literature which discuss properties of convexity and contain integral inequalities. Furthermore, new classes of convex functions have been introduced in order to generalize the results and to obtain new estimations. We define some new classes of convex functions that we name quasi-convex, Jensen-convex, Wright-convex, Jensen-quasi-convex and Wright-quasi-convex functions on the co-ordinates. We also prove some inequalities of Hadamard-type as Dragomir’s results in Theorem 5, but now for Jensen-quasi-convex and Wright-quasi-convex functions. Finally, we give some inclusions which clarify the relationship between these new classes of functions.

How to cite

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Özdemir, M. Emin, Akdemir, Ahmet Ocak, and Yıldız, Çetin. "On co-ordinated quasi-convex functions." Czechoslovak Mathematical Journal 62.4 (2012): 889-900. <http://eudml.org/doc/246668>.

@article{Özdemir2012,
abstract = {A function $f\colon I\rightarrow \mathbb \{R\}$, where $I\subseteq \mathbb \{R\}$ is an interval, is said to be a convex function on $I$ if \[ f( tx+( 1-t) y) \le tf( x) +(1-t) f( y) \] holds for all $x,y\in I$ and $t\in [ 0,1] $. There are several papers in the literature which discuss properties of convexity and contain integral inequalities. Furthermore, new classes of convex functions have been introduced in order to generalize the results and to obtain new estimations. We define some new classes of convex functions that we name quasi-convex, Jensen-convex, Wright-convex, Jensen-quasi-convex and Wright-quasi-convex functions on the co-ordinates. We also prove some inequalities of Hadamard-type as Dragomir’s results in Theorem 5, but now for Jensen-quasi-convex and Wright-quasi-convex functions. Finally, we give some inclusions which clarify the relationship between these new classes of functions.},
author = {Özdemir, M. Emin, Akdemir, Ahmet Ocak, Yıldız, Çetin},
journal = {Czechoslovak Mathematical Journal},
keywords = {co-ordinate; quasi-convex; Wright-quasi-convex; Jensen-quasi-convex; quasi-convex; Wright-quasi-convex; Jensen-quasi-convex},
language = {eng},
number = {4},
pages = {889-900},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On co-ordinated quasi-convex functions},
url = {http://eudml.org/doc/246668},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Özdemir, M. Emin
AU - Akdemir, Ahmet Ocak
AU - Yıldız, Çetin
TI - On co-ordinated quasi-convex functions
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 4
SP - 889
EP - 900
AB - A function $f\colon I\rightarrow \mathbb {R}$, where $I\subseteq \mathbb {R}$ is an interval, is said to be a convex function on $I$ if \[ f( tx+( 1-t) y) \le tf( x) +(1-t) f( y) \] holds for all $x,y\in I$ and $t\in [ 0,1] $. There are several papers in the literature which discuss properties of convexity and contain integral inequalities. Furthermore, new classes of convex functions have been introduced in order to generalize the results and to obtain new estimations. We define some new classes of convex functions that we name quasi-convex, Jensen-convex, Wright-convex, Jensen-quasi-convex and Wright-quasi-convex functions on the co-ordinates. We also prove some inequalities of Hadamard-type as Dragomir’s results in Theorem 5, but now for Jensen-quasi-convex and Wright-quasi-convex functions. Finally, we give some inclusions which clarify the relationship between these new classes of functions.
LA - eng
KW - co-ordinate; quasi-convex; Wright-quasi-convex; Jensen-quasi-convex; quasi-convex; Wright-quasi-convex; Jensen-quasi-convex
UR - http://eudml.org/doc/246668
ER -

References

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