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Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 18 (2017), No. 2, pp. 837–849 DOI: 10.18514/MMN.2017.1831
SOME NEW HERMITE-HADAMARD TYPE INTEGRALINEQUALITIES FOR FUNCTIONS WHOSE nth DERIVATIVES
ARE LOGARITHMICALLY RELATIVE h�PREINVEX
SABIR HUSSAIN AND SOBIA RAFEEQ
Received 28 October, 2015
Abstract. The authors have introduced the concept of logarithmically relative h�preinvex func-tion which is a generalized form of previously known concepts [9, 11], and try to establish somenew Hermite-Hadamard type integral inequalities for functions whose absolute values of nthderivatives are logarithmically relative h�preinvex.
2010 Mathematics Subject Classification: 34B10; 34B15
Keywords: Hermite-Hadamard inequality, Invex set, Preinvex function, logarithmically h�preinvexfunction, Beta function, Incomplete beta function
1. INTRODUCTION AND PRELIMINARIES
In recent years, researchers have paid a lot of attention to study and investigatethe theory of convex functions due to its extensive applications in different fields ofpure and applied sciences. Weir et al. [15] had given a significant generalization ofconvex functions by introducing preinvex functions. Varosanec[12], introduced theh�convex functions. Noor et al. [9] have introduced the concept of logarithmic-ally h�preinvex functions, generalizing the concept of logarithmically s�preinvexfunctions, logarithmically P�preinvex functions and logarithmically Q�preinvexfunctions.
There are many inequalities for convex functions but the most famous is the Hermite-Hadamard inequality, due to its rich geometrical significance and applications. Hermit-Hadamard inequlaities for the convex functions and their variant forms are availablein the literature [3–11, 13, 14].
Motivated by the works of Noor et al. [9], Fulga et al. [1], and using the concept oflogarithmically relative h�preinvex functions, we have derived some new Hermite-Hadamard type inequalites.
The paper is arranged in such a way that after this Introduction,in Section 1, theauthors have defined the logarithmically relative h�preinvex functions and its relat-ing consequent concepts, in section 2, the authors have discussed their main results
c 2017 Miskolc University Press
838 SABIR HUSSAIN AND SOBIA RAFEEQ
regarding Hermite-Hadamard type integral inequalities, using the concept of Young’sinquality, Holder’s inequality, and power mean’s inequality.
Let M be a nonempty closed set in Rn: Let f W M ! .0;1/ be a continuousfunction and let � W Rn�Rn! Rn be a continuous bifunction.
Definition 1 ([1]). A set M � Rn is said to be a relative invex (or g�invex) withrespect to the map �; if there exists a function g W Rn! Rn such that,
g.a/C t� .g.b/;g.a// 2M I 8 a;b 2 Rn W g.a/;g.b/ 2M; t 2 Œ0;1�:
Definition 2. LetM �Rn be a relative invex set with respect to �: Then ��relativepath, Pg.a/Ig.c/; joining the points g.a/ and g.c/D g.a/C�.g.b/;g.a// is definedas:
Pg.a/Ig.c/ D fg.d/jg.d/D g.a/C t� .g.b/;g.a// I a;b 2M; t 2 Œ0;1�g:
Remark 1. For gD I; I an identity function, definition 2 coincides with definitionof ��path [14].
Definition 3. Let h W J ! R; where .0;1/ � J be an interval in RI let h¤ 0 forall values of J I let M � Rn be a relative invex set with respect to �: A functionf W M ! .0;1/ is said to be logarithmically relative h�preinvex (or log�.g;h/preinvex) with respect to �; if
f .g.a/C t� .g.b/;g.a///� Œf .g.a//�h.1�t/Œf .g.b//�h.t/; (1.1)
for a;b 2 Rn such that g.a/;g.b/ 2M; t 2 .0;1/:f is logarithmically relative h�preconcave (or log�.g;h/ preconcave) with re-
spect to � whenever the inequality sign in .1:1/ is reversed.
Remark 2. For g.t/D I; I an identity function, definition 3 coincides with defin-ition of logarithmically h�preinvex function[9].
Example 1. Let M D Œ�2;�1�SŒ1;2�; then obviously M is relative invex for
g W R! RCSf0g and � W R�R! R respectively defined by:
g.x/D
(x2 for jxj �
p2
2 for jxj >p2:
�.x;y/D
(x�y for x:y > 02�y for x:y < 0:
Consider, f WM ! .0;1/ defined by
f .x/D arctan.xC2Cp3/ and h.t/D ts;
where s � 0: Then f is logarithmically relative h�preinvex function with respect to�:
Remark 3. For x Dp2; y D 1; t D 1
2; s D 1; the above function f is not logar-
ithmically relative h�preinvex function.
HERMITE-HADAMARD TYPE INEQUALITIES 839
Definition 4. A function f WM ! .0;1/ is said to be logarithmically relatives�preinvex (or log�.g;s/ preinvex) with respect to �; if
f .g.a/C t� .g.b/;g.a///� Œf .g.a//�.1�t/s
Œf .g.b//�ts
;
for a;b 2 Rn such that g.a/;g.b/ 2M; t 2 Œ0;1�; s 2 .0;1�:
Remark 4. (1) For g.t/ D I; I an identity function, definition 4 coincideswith definition of logarithmically s�preinvex function [9].
(2) For h.t/D ts; definition 3 coincides with definition 4.
Definition 5. A function f WM ! .0;1/ is said to be logarithmically relativeP�preinvex (or log�.g;P / preinvex) with respect to �; if
f .g.a/C t� .g.b/;g.a///� Œf .g.a//�Œf .g.b//�;
for a;b 2 Rn such that g.a/;g.b/ 2M; t 2 Œ0;1�:
Remark 5. (1) For g.t/D I; I an identity function, definition 5 coincideswith definition of logarithmically P�preinvex function [9].
(2) For h.t/D 1; definition 3 coincides with definition 5.
Definition 6. A function f WM ! .0;1/ is said to be logarithmically relativeQ�preinvex (or log�.g;Q/ preinvex) with respect to �; if
f .g.a/C t� .g.b/;g.a///� Œf .g.a//�1
1�t Œf .g.b//�1t ;
for a;b 2 Rn such that g.a/;g.b/ 2M; t 2 .0;1/:
Remark 6. (1) For g.t/D I; I an identity function, definition 6 coincideswith definition of logarithmically Q�preinvex function[9].
(2) For h.t/D 1t; definition 3 coincides with definition 6.
Example 2. Let M D Œ�2;�1�SŒ1;2�; then obviously M is relative invex for
g W R! RCSf0g and � W R�R! R respectively defined by
g.x/D
(x2 for jxj �
p2
1 for jxj >p2:
�.x;y/D x�y:
Consider, f WM ! .0;1/ defined by f .x/D arctan.xC2C�/ for � > 0 and h.t/D1t: Then f is logarithmically relative Q�preinvex function with respect to �:
Definition 7. The beta function is defined as:
ˇ.x;y/D
Z 1
0
tx�1.1� t /y�1dt; x;y > 0:
Definition 8. The incomplete beta function is defined as:
ˇ´.x;y/D
Z ´
0
tx�1.1� t /y�1dt; 0� ´� 1I x;y > 0:
840 SABIR HUSSAIN AND SOBIA RAFEEQ
2. RESULTS
For establishing some new Hermite-Hadamard type integral inequalities for n�timesdifferentiable functions such that jf .n/j are logarithmically relative h�preinvex func-tions, we need the following result, which is a generalization of a result proved byWang et al. [14, Lemma1].
Lemma 1 ([2]). Let A�R be an open relative invex subset with respect to � WR�R!R; with a;b 2A and �.g.a/;g.b//¤ 0: If f WR!R is n�times differentiablefunction and f .n/ is integrable on the ��relative path Pg.b/Ig.c/; then
Œ� .g.a/;g.b//�n
2nC2�nŠ
Z 1
0
.n�2t/tn�1�f .n/
�g.b/C
1� t
2� .g.a/;g.b//
�Cf .n/
�g.b/C
2� t
2� .g.a/;g.b//
��dt
D1
2
�f .g.b//Cf .g.b/C� .g.a/;g.b///
2Cf
�2g.b/C� .g.a/;g.b//
2
���
1
� .g.a/;g.b//
Z g.b/C�.g.a/;g.b//
g.b/
f .g.x//dg.x/
�
n�1XkD2
Œ� .g.a/;g.b//�k.k�1/
2kC2� .kC1/Š
�f .k/ .g.b//Cf .k/
�2g.b/C� .g.a/;g.b//
2
��(2.1)
where the above summation is zero for nD 1;2
Before going to our main results, we make the following assumptions:
(1) fAn WD j�.g.a/;g.b//jnnŠ�2nC2
(2) eBn;k.t/ WD ˇf .n/ .g.b//ˇ� ˇf .n/.g.a//
f .n/.g.b//
ˇh.k�t2/
(3) h.1Ct2/Ch.1�t
2/D 1D h. t
2/Ch.2�t
2/.
Theorem 1. Let A � R be a relative invex subset with respect to � W R�R!R; with a;b 2 A and �.g.a/;g.b// ¤ 0: Suppose that f W A! .0;1/ is n�timesdifferentiable function and f .n/ is integrable on the ��relative path Pg.b/Ig.c/; suchthat jf .n/j is logarithmically relative h�preinvex on AI let qk > 1; k D 1;2; thenˇ
1
2
�f .g.b//Cf .g.b/C� .g.a/;g.b///
2Cf
�2g.b/C� .g.a/;g.b//
2
���
1
� .g.a/;g.b//
Z g.b/C�.g.a/;g.b//
g.b/
f .g.u//dg.u/
HERMITE-HADAMARD TYPE INEQUALITIES 841
�
n�1XkD2
Œ� .g.a/;g.b//�k.k�1/
.kC1/Š�2kC2
�f .k/ .g.b//Cf .k/
�2g.b/C� .g.a/;g.b//
2
��ˇˇ
�
8<:fA1P2
kD11qk
h.qk�1/
2
2qk�1CR 10
�eB1;k.t/�qkdti
for nD 1
fAnP2kD1
1qk
"n
nqkqk�1
C1�.qk�1/
2
qk .n�1/
qk�1C1
ˇ 2n
�nqk�1qk�1
; 2qk�1qk�1
�CR 10
�eBn;k.t/�qkdt
#for n� 2:
(2.2)
Proof. Since g.b/C t�.g.a/;g.b// 2 A for every t 2 .0;1/; by Lemma 1 andYoung’s inequality, we haveˇ
1
2
�f .g.b//Cf .g.b/C� .g.a/;g.b///
2Cf
�2g.b/C� .g.a/;g.b//
2
���
1
� .g.a/;g.b//
Z g.b/C�.g.a/;g.b//
g.b/
f .g.u//dg.u/
�
n�1XkD2
Œ� .g.a/;g.b//�k.k�1/
.kC1/Š�2kC2
�f .k/ .g.b//Cf .k/
�2g.b/C� .g.a/;g.b//
2
��ˇˇ
�j� .g.a/;g.b//jn
nŠ�2nC2
Z 1
0
ˇ.n�2t/tn�1
ˇ �ˇf .n/
�g.b/C
1� t
2� .g.a/;g.b//
�ˇC
ˇf .n/
�g.b/C
2� t
2� .g.a/;g.b//
�ˇ�dt
�j� .g.a/;g.b// jn
nŠ�2nC2�
Z 1
0
�q1�1
q1
ˇ.n�2t/tn�1
ˇ q1q1�1
C1
q1
ˇf .n/
�g.b/C
1� t
2� .g.a/;g.b//
�ˇq1
Cq2�1
q2
ˇ.n�2t/tn�1
ˇ q2q2�1
C1
q2
ˇf .n/
�g.b/C
2� t
2� .g.a/;g.b//
�ˇq2�dt (2.3)
Using logarithmically relative h�preinvexity of jf .n/j; we haveZ 1
0
ˇf .n/
�g.b/C
1� t
2� .g.a/;g.b//
�ˇq1
dt
�
Z 1
0
ˇf .n/ .g.b//
ˇh. 1Ct2/ ˇf .n/ .g.a//
ˇh. 1�t2/!q1
dt
D
Z 1
0
0@ˇf .n/ .g.b//ˇ ˇˇf .n/ .g.a//f .n/ .g.b//
ˇˇh. 1�t
2/1Aq1
dt (2.4)
842 SABIR HUSSAIN AND SOBIA RAFEEQ
SimilarlyZ 1
0
ˇf .n/
�g.b/C
2� t
2� .g.a/;g.b//
�ˇq2
dt
�
Z 1
0
0@ˇf .n/ .g.b//ˇ ˇˇf .n/ .g.a//f .n/ .g.b//
ˇˇh. 2�t
2/1Aq2
dt (2.5)
Also,
Z 1
0
ˇ.n�2t/tn�1
ˇ qkqk�1 dt D
8<:qk�12qk�1
for nD 1
n
nqkqk�1
C1
2
qk.n�1/
qk�1C1
ˇ 2n
�nqk�1qk�1
; 2qk�1qk�1
�for n� 2:
(2.6)
A combination of .2:3/-.2:6/ yields the desired result. �
Corollary 1. Under the conditions of theorem 1 for q1 D q2; we haveˇ1
2
�f .g.b//Cf .g.b/C� .g.a/;g.b///
2Cf
�2g.b/C� .g.a/;g.b//
2
���
1
� .g.a/;g.b//
Z g.b/C�.g.a/;g.b//
g.b/
f .g.u//dg.u/
�
n�1XkD2
Œ� .g.a/;g.b//�k.k�1/
.kC1/Š�2kC2
�f .k/ .g.b//Cf .k/
�2g.b/C� .g.a/;g.b//
2
��ˇ
�
8<:eA1
q2
h2.q2�1/
2
2q2�1CR 10
P2kD1
�eB1;k.t/�q2dti
for nD 1
eAn
q2
"n
nq2q2�1
C1�.q2�1/
2
q2.n�1/
q2�1
ˇ 2n
�nq2�1q2�1
; 2q2�1q2�1
�CR 10
P2kD1
�eBn;k.t/�q2dt
#for n� 2:
Theorem 2. Let A � R be a relative invex subset with respect to � W R�R!R; with a;b 2 A and �.g.a/;g.b// ¤ 0: Suppose that f W A! .0;1/ is n�timesdifferentiable function and f .n/ is integrable on the ��relative path Pg.b/Ig.c/; suchthat jf .n/j is logarithmically relative h�preinvex on AI let q3 > 1; thenˇ
1
2
�f .g.b//Cf .g.b/C� .g.a/;g.b///
2Cf
�2g.b/C� .g.a/;g.b//
2
���
1
� .g.a/;g.b//
Z g.b/C�.g.a/;g.b//
g.b/
f .g.u//dg.u/
�
n�1XkD2
Œ� .g.a/;g.b//�k.k�1/
.kC1/Š�2kC2
�f .k/ .g.b//Cf .k/
�2g.b/C� .g.a/;g.b//
2
��ˇˇ
HERMITE-HADAMARD TYPE INEQUALITIES 843
�
8<:eA1
q3
h.q3�1/
2
2q3�1CR 10
�P2kD1
eB1;k.t/�q3
dti
for nD 1eAn
q3
"n
nq3q3�1
C1�.q3�1/
2
q3.n�1/
q3�1C1
ˇ 2n
�nq3�1q3�1
; 2q3�1q3�1
�CR 10
�P2kD1
eBn;k.t/�q3
dt
#for n� 2:
(2.7)
Proof. Since g.b/C t�.g.a/;g.b// 2A for every t 2 .0;1/; by Lemma 1, Young’sinequality and the logarithmically relative h�preinvexity of jf .n/j; we haveˇ
1
2
�f .g.b//Cf .g.b/C� .g.a/;g.b///
2Cf
�2g.b/C� .g.a/;g.b//
2
���
1
� .g.a/;g.b//
Z g.b/C�.g.a/;g.b//
g.b/
f .g.u//dg.u/
�
n�1XkD2
Œ� .g.a/;g.b//�k.k�1/
.kC1/Š�2kC2
�f .k/ .g.b//Cf .k/
�2g.b/C� .g.a/;g.b//
2
��ˇˇ
�j� .g.a/;g.b//jn
nŠ�2nC2
Z 1
0
ˇ.n�2t/tn�1
ˇ �ˇf .n/
�g.b/C
1� t
2� .g.a/;g.b//
�ˇC
ˇf .n/
�g.b/C
2� t
2� .g.a/;g.b//
�ˇ�dt
�j� .g.a/;g.b// jn
nŠ�2nC2�
Z 1
0
�q3�1
q3
ˇ.n�2t/tn�1
ˇ q3q3�1
C1
q3
�ˇf .n/
�g.b/C
1� t
2� .g.a/;g.b//
�ˇC
ˇf .n/
�g.b/C
2� t
2� .g.a/;g.b//
�ˇ�q3�dt
�j� .g.a/;g.b// jn
nŠ�2nC2�q3�
Z 1
0
�.q3�1/
ˇ.n�2t/tn�1
ˇ q3q3�1
C
0@ 2XkD1
ˇf .n/ .g.b//
ˇ ˇˇf .n/ .g.a//f .n/ .g.b//
ˇˇh.k�t
2/1Aq3
375dt�
Corollary 2. Under the conditions of theorem 2 for nD 2; we haveˇ1
2
�f .g.b//Cf .g.b/C� .g.a/;g.b///
2Cf
�2g.b/C� .g.a/;g.b//
2
���
1
� .g.a/;g.b//
Z g.b/C�.g.a/;g.b//
g.b/
f .g.u//dg.u/
ˇˇ
844 SABIR HUSSAIN AND SOBIA RAFEEQ
�fA2q3
242 q3q3�1 � .q3�1/�ˇ
�2q3�1
q3�1;2q3�1
q3�1
�C
Z 1
0
2XkD1
eB2;k.t/!q3
dt
35Theorem 3. Let A � R be a relative invex subset with respect to � W R�R!
R; with a;b 2 A and �.g.a/;g.b// ¤ 0: Suppose that f W A! .0;1/ is n�timesdifferentiable function and f .n/ is integrable on the ��relative path Pg.b/Ig.c/; suchthat jf .n/j is logarithmically relative h�preinvex on AI let q3 > 1; and q3 � r > 0;thenˇ
1
2
�f .g.b//Cf .g.b/C� .g.a/;g.b///
2Cf
�2g.b/C� .g.a/;g.b//
2
���
1
� .g.a/;g.b//
Z g.b/C�.g.a/;g.b//
g.b/
f .g.u//dg.u/
�
n�1XkD2
Œ� .g.a/;g.b//�k.k�1/
.kC1/Š�2kC2
�f .k/ .g.b//Cf .k/
�2g.b/C� .g.a/;g.b//
2
��ˇˇ
�
8<ˆ:
eA1
q3
h.q3�1/
2
2q3�r�1CR 10 j.n�2t/t
n�1jr�P2
kD1eB1;k.t/�q3
dti
for nD 1
eAn
q3
"n
n.q3�r/
q3�1C1�.q3�1/
2
.q3�r/.n�1/
q3�1C1
ˇ 2n
�nq3�nrCr�1
q3�1; 2q3�r�1
q3�1
�CR 10 j.n�2t/t
n�1jr�P2
kD1eBn;k.t/�q3
dti
for n� 2:
(2.8)
Proof. Since g.b/C t�.g.a/;g.b// 2A for every t 2 .0;1/; by Lemma 1, Young’sinequality and the logarithmically relative h�preinvexity of jf .n/j, we haveˇ
1
2
�f .g.b//Cf .g.b/C� .g.a/;g.b///
2Cf
�2g.b/C� .g.a/;g.b//
2
���
1
� .g.a/;g.b//
Z g.b/C�.g.a/;g.b//
g.b/
f .g.u//dg.u/
�
n�1XkD2
Œ� .g.a/;g.b//�k.k�1/
.kC1/Š�2kC2
�f .k/ .g.b//Cf .k/
�2g.b/C� .g.a/;g.b//
2
��ˇˇ
�j� .g.a/;g.b//jn
nŠ�2nC2
Z 1
0
ˇ.n�2t/tn�1
ˇ �ˇf .n/
�g.b/C
1� t
2� .g.a/;g.b//
�ˇC
ˇf .n/
�g.b/C
2� t
2� .g.a/;g.b//
�ˇ�dt
�j� .g.a/;g.b// jn
nŠ�2nC2
Z 1
0
�q3�1
q3
ˇ.n�2t/tn�1
ˇ q3�r
q3�1 C1
q3
ˇ.n�2t/tn�1
ˇr
HERMITE-HADAMARD TYPE INEQUALITIES 845
�
�ˇf .n/
�g.b/C
1� t
2� .g.a/;g.b//
�ˇC
ˇf .n/
�g.b/C
2� t
2� .g.a/;g.b//
�ˇ�q3�dt
�j� .g.a/;g.b// jn
nŠ�2nC2�q3�
Z 1
0
�.q3�1/
ˇ.n�2t/tn�1
ˇ q3�r
q3�1
Cˇ.n�2t/tn�1
ˇr�
0@ 2XkD1
ˇf .n/ .g.b//
ˇ ˇˇf .n/ .g.a//f .n/ .g.b//
ˇˇh.k�t
2/1Aq3
375dt�
Corollary 3. Under the conditions of theorem 3 for r D q3; we have
ˇ1
2
�f .g.b//Cf .g.b/C� .g.a/;g.b///
2Cf
�2g.b/C� .g.a/;g.b//
2
���
1
� .g.a/;g.b//
Z g.b/C�.g.a/;g.b//
g.b/
f .g.u//dg.u/
�
n�1XkD2
Œ� .g.a/;g.b//�k.k�1/
.kC1/Š�2kC2
�f .k/ .g.b//Cf .k/
�2g.b/C� .g.a/;g.b//
2
��ˇˇ
�
8<:eA1
q3
h.q3�1/C
R 10
�j.n�2t/tn�1j
P2kD1
eB1;k.t/�q3
dti
for nD 1eAn
q3
hn.q3�1/
2ˇ 2
n.1;1/C
R 10
�j.n�2t/tn�1j
P2kD1
eBn;k.t/�q3
dti
for n� 2:
Theorem 4. Let A � R be a relative invex subset with respect to � W R�R!R; with a;b 2 A and �.g.a/;g.b// ¤ 0: Suppose that f W A! .0;1/ is n�timesdifferentiable function and f .n/ is integrable on the ��relative path Pg.b/Ig.c/; suchthat jf .n/j is logarithmically relative h�preinvex on AI let q3 > 1; and then
ˇ1
2
�f .g.b//Cf .g.b/C� .g.a/;g.b///
2Cf
�2g.b/C� .g.a/;g.b//
2
���
1
� .g.a/;g.b//
Z g.b/C�.g.a/;g.b//
g.b/
f .g.u//dg.u/
�
n�1XkD2
Œ� .g.a/;g.b//�k.k�1/
.kC1/Š�2kC2
�f .k/ .g.b//Cf .k/
�2g.b/C� .g.a/;g.b//
2
��ˇˇ
846 SABIR HUSSAIN AND SOBIA RAFEEQ
�
8<:eA1P2
kD1
�R 10
�eB1;k.t/�q3dt� 1
q3
�q3�12q3�1
�1� 1q3 for nD 1
eAnP2kD1
�R 10
�eBn;k.t/�q3dt� 1
q3
"n
nq3q3�1
C1
2
q3.n�1/
q3�1C1ˇ 2
n
�nq3�1q3�1
; 2q3�1q3�1
�#1� 1q3
for n� 2:
(2.9)
Proof. Since g.b/C t�.g.a/;g.b//2A for every t 2 .0;1/; by Lemma 1, Holder’sinequality and the logarithmically relative h�preinvexity of jf .n/j; we haveˇ
1
2
�f .g.b//Cf .g.b/C� .g.a/;g.b///
2Cf
�2g.b/C� .g.a/;g.b//
2
���
1
� .g.a/;g.b//
Z g.b/C�.g.a/;g.b//
g.b/
f .g.u//dg.u/
�
n�1XkD2
Œ�.g.a/;g.b//�k.k�1/
.kC1/Š�2kC2
�f .k/ .g.b//Cf .k/
�2g.b/C� .g.a/;g.b//
2
��ˇˇ
�j� .g.a/;g.b// jn
nŠ�2nC2
Z 1
0
ˇ.n�2t/tn�1
ˇ �ˇf .n/
�g.b/C
1� t
2� .g.a/;g.b//
�ˇC
ˇf .n/
�g.b/C
2� t
2� .g.a/;g.b//
�ˇ�dt
�j� .g.a/;g.b// jn
nŠ�2nC2
�Z 1
0
ˇ.n�2t/tn�1
ˇ q3q3�1 dt
�1� 1q3
�
8<:�Z 1
0
ˇf .n/
�g.b/C
1� t
2� .g.a/;g.b//
�ˇq3
dt
� 1q3
C
�Z 1
0
ˇf .n/
�g.b/C
2� t
2� .g.a/;g.b//
�ˇq3
dt
� 1q3
9=;�j� .g.a/;g.b// jn
nŠ�2nC2
�Z 1
0
ˇ.n�2t/tn�1
ˇ q3q3�1 dt
�1� 1q3
�
2XkD1
0B@Z 1
0
0@ˇf .n/ .g.b//ˇ ˇˇf .n/ .g.a//f .n/ .g.b//
ˇˇh.k�t
2/1Aq3
dt
1CA1
q3
�
Corollary 4. Under the conditions of theorem 4 for nD 2; we have
HERMITE-HADAMARD TYPE INEQUALITIES 847ˇ1
2
�f .g.b//Cf .g.b/C� .g.a/;g.b///
2Cf
�2g.b/C� .g.a/;g.b//
2
���
1
� .g.a/;g.b//
Z g.b/C�.g.a/;g.b//
g.b/
f .g.u//dg.u/
ˇˇ
� 2eA2 2XkD1
�Z 1
0
�eB2;k.t/�q3dt
� 1q3�ˇ
�2q3�1
q3�1;2q3�1
q3�1
��1� 1q3
Theorem 5. Let A � R be a relative invex subset with respect to � W R�R!R; with a;b 2 A and �.g.a/;g.b// ¤ 0: Suppose that f W A! .0;1/ is n�timesdifferentiable function and f .n/ is integrable on the ��relative path Pg.b/Ig.c/; suchthat jf .n/j is logarithmically relative h�preinvex on AI let q3 � 1; thenˇ
1
2
�f .g.b//Cf .g.b/C� .g.a/;g.b///
2Cf
�2g.b/C� .g.a/;g.b//
2
���
1
� .g.a/;g.b//
Z g.b/C�.g.a/;g.b//
g.b/
f .g.u//dg.u/
�
n�1XkD2
Œ� .g.a/;g.b//�k.k�1/
.kC1/Š�2kC2
�f .k/ .g.b//Cf .k/
�2g.b/C� .g.a/;g.b//
2
��ˇˇ
�
8<ˆ:eA1
P2kD1
�R 10 j.n�2t/t
n�1j
�eB1;k.t/�q3
dt
� 1q3
21� 1
q3
for nD 1
eAnP2kD1
�R 10 j.n�2t/t
n�1j�eBn;k.t/�q3
dt� 1
q3�n�1nC1
�1� 1q3 for n� 2:
Proof. By using the weighted power mean inequality and the similar techniquesused in theorem 4, we can prove this theorem. �
Corollary 5. Under the conditions of theorem 5 for q3 D 1; we haveˇ1
2
�f .g.b//Cf .g.b/C� .g.a/;g.b///
2Cf
�2g.b/C� .g.a/;g.b//
2
���
1
� .g.a/;g.b//
Z g.b/C�.g.a/;g.b//
g.b/
f .g.u//dg.u/
�
n�1XkD2
Œ� .g.a/;g.b//�k.k�1/
.kC1/Š�2kC2
�f .k/ .g.b//Cf .k/
�2g.b/C� .g.a/;g.b//
2
��ˇˇ
�
(eA1 P2kD1
R 10 j.n�2t/t
n�1j eB1;k.t/dt for nD 1eAn P2kD1
R 10 j.n�2t/t
n�1j eBn;k.t/dt for n� 2:
848 SABIR HUSSAIN AND SOBIA RAFEEQ
Remark 7. If g.t/D I; where I is an identity function, then our results coincidethe results for logarithmically h�preinvex functions.
ACKNOWLEDGEMENT
The authors are grateful to the anonymous reviewers and editor for their so-calledinsights.
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HERMITE-HADAMARD TYPE INEQUALITIES 849
Authors’ addresses
Sabir HussainUniversity of Engineering and Technology, Department of Mathematics, Lahore, PakistanE-mail address: [email protected]
Sobia RafeeqUniversity of Engineering and Technology, Department of Mathematics, Lahore, PakistanE-mail address: [email protected]
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