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J. Nonlinear Sci. Appl. 8 (2015), 201–217
Research Article
Some inequalities of Hermite-Hadamard type for
n–times differentiable (ρ, m)–geometrically convex
functions
Fiza Zafara,∗, Humaira Kalsooma , Nawab Hussainb
a
Centre for Advanced Studies in Pure and Applied Mathematics (CASPAM), Bahauddin Zakariya University, Multan 60800, Pakistan.
b
Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia.
Communicated by Manuel De la Sen
Abstract
In this paper, some generalized Hermite–Hadamard type inequalities for n–times differentiable (ρ, m)–
geometrically convex function are established. The new inequalities recapture and give new estimates of
the previous inequalities for first differentiable functions as special cases. The estimates for trapezoid, midpoint, averaged mid-point trapezoid and Simpson’s inequalities can also be obtained for higher differentiable
c
generalized geometrically convex functions. 2015
All rights reserved.
Keywords: Hermite–Hadamard inequality, (ρ, m)–geometrically convex functions, n–times differentiable
function.
2010 MSC: 26D15, 26A15.
1. Introduction
Since the establishment of theory of convex functions in the last century by a Danish mathematician,
Jensen (1859-1925), the research on convex functions has gained much attention. However, the geometrically
convex functions only appeared in [10, 11, 12] but has now become an active domain of definition. Convex
and geometrically convex functions are used in parallel as tools to prove inequalities. The notion of geometric
convexity was introduced by Montel [6], analogous to the notion of a convex function in n variables.
Now, we restate some basic convexity domains and related results.
∗
Corresponding author
Email addresses: fizazafar@gmail.com (Fiza Zafar), humaira.k86@gmail.com (Humaira Kalsoom), nhusain@kau.edu.sa
(Nawab Hussain)
Received 2014-11-08
F. Zafar, H. Kalsoom, N. Hussain, J. Nonlinear Sci. Appl. 8 (2015), 201–217
202
Definition 1.1. A function f : I ⊆ R → R is convex, if the inequality
f (tx + (1 − t)y) ≤ tf (x) + (1 − t)f (y),
(1.1)
holds for all x, y ∈ I, and t ∈ [0, 1].
The following is well known Hermite–Hadamard inequality which holds for a convex function f : I ⊆
R → R where a, b ∈ I with a < b,
a+b
1
f(
)≤
2
b−a
Zb
f (x)dx ≤
f (a) + f (b)
.
2
(1.2)
a
As soon as an inequality appears, an attempt is made to generalize it [5]. The most earlier attempts of
refining Hermite–Hadamard inequality can be found in [2, 3, 4].
In 2004, Zhang [9] presented the following concept of geometrically convex functions.
Definition 1.2. Let f (x) be a positive function on [0, b]. If
f (xt y (1−t) ) ≤ [f (x)]t [f (y)](1−t) ,
(1.3)
holds for all x, y ∈ [0, b] and t ∈ [0, 1], then we say that the function f (x) is geometrically convex on [0, b].
In 2013, Ozdamir and Yildiz [7], presented some Ostrowski type inequalities for geometrically convex
functions involving Logarithmic mean.
Xi et al. [8] in 2012, introduced the concept of m–geometrically convex functions and presented Hermite–
Hadamard type inequalities for the generalized m–geometrically convex functions.
Definition 1.3. Let f (x) be a positive function on [0, b] and m ∈ (0, 1]. If
f (xt y m(1−t) ) ≤ [f (x)]t [f (y)]m(1−t) ,
(1.4)
holds for all x, y ∈ [0, b] and t ∈ [0, 1], then we say that the function f (x) is m–geometrically convex on
[0, b].
0
Theorem 1.4. ([8]) Let I ⊃ [0, ∞) be an open interval and f : I → (0, ∞) is differentiable. If f ∈ L([a, b])
0
and |f (x) | is decreasing and m-geometrically convex on [min{1, a}, b] for a, b ∈ [0, ∞), with a < b and
b ≥ 1, and m ∈ (0, 1], then
1
Zb
f (a) + f (b)
b − a 1 1− q 0 m
1
1
−
(1.5)
f (x)dx ≤
f (b) [G1 (α, m, q)] q ,
2
b−a
2
2
a
is valid for q ≥ 1, where
Z1
G1 (1, m, q) =
 0 qt
f (a)
|1 − 2t|  0 m  dt.
|f (b) |
0
0
Theorem 1.5. ([8]) Let I ⊃ [0, ∞) be an open interval and f : I → (0, ∞) is differentiable. If f ∈ L([a, b])
0
, |f (x)| is decreasing and m–geometrically convex on [min{1, a}, b] for a ∈ [0, ∞), b ≥ 1, and m ∈ (0, 1],
then
3
Zb
b − a 1 1− q (n) m
1
f a + b − 1
f (x)dx ≤
f (b) [G2 (1, m, q)] q ,
2
b−a
4
2
a
F. Zafar, H. Kalsoom, N. Hussain, J. Nonlinear Sci. Appl. 8 (2015), 201–217
203
is valid for q ≥ 1,where
 1 
 1q 
 0 qt  1q
0 qt
1
Z
Z2
f (a)
f (a)




G2 (1, m, q) =  t  0 m  dt +  (1 − t)  0 m  dt .
|f (b) |
|f (b) |
0
1
2
Recently, some Ostrowski type inequalities for m-geometrically convex functions are also established by
Ozdamir and Yildiz [7].
Xi et al. [8] in 2012, introduced the concept of (α, m)–geometrically convex functions and established
the generalized inequalities of this domain.
Definition 1.6. Let f (x) be a positive function on [0, b] and α, m ∈ (0, 1] × (0, 1]. If
α
α
f (xt y m(1−t) ) ≤ [f (x)]t [f (y)]m(1−t ) ,
(1.6)
holds for all x, y ∈ [0, b], and t ∈ [0, 1], then we say that the function f (x) is (α, m)–geometrically convex
on [0, b].
Remark 1.7. If α = m = 1 in (1.6), then (α, m)–geometrically convex functions become geometrically convex
functions.
Lemma 1.8. If f (x) is geometrically convex, and
g(x) = ln f (ex ),
(1.7)
then g is a convex function.
0
Theorem 1.9. ([8]) Let I ⊃ [0, ∞) be an open interval and f : I → (0, ∞) is differentiable. If f ∈ L([a, b])
0
and |f (x)| is decreasing and (α, m)-geometrically convex on [min{1, a}, b] for a ∈ [0, ∞), b ≥ 1, and
(α, m) ∈ (0, 1] × (0, 1], then
1
Zb
f (a) + f (b)
b − a 1 1− q (n) m
1
1
−
f (x)dx ≤
(1.8)
f (b) [G1 (α, m, q)] q ,
2
b−a
2
2
a
is valid for q ≥ 1, where
Z1
G1 (α, m, q) =
 0 qtα
f (a)
|1 − 2t|  0 m  dt.
|f (b) |
0
0
Theorem 1.10. ([8]) Let I ⊃ [0, ∞) be an open interval and f : I → (0, ∞) is differentiable. If f ∈
0
L([a, b]) and |f (x)| is decreasing and (α, m)–geometrically convex on [min{1, a}, b] for a ∈ [0, ∞), b ≥ 1
and (α, m) ∈ (0, 1] × (0, 1], then
3
Zb
b − a 1 1− q (n) m
1
f a + b − 1
q ,
f
(x)dx
≤
f
(b)
[G
(α,
m,
q)]
(1.9)
2
2
b−a
4
2
a
is valid for q ≥ 1, where
 1 
 1q
α
0 qt
Z2
f (a)


G2 (α, m, q) =  t  0 m  dt
|f (b) |
0

 0 qtα  1q
Z1
f (a)


+  (1 − t)  0 m  dt .
|f (b) |
1
2
F. Zafar, H. Kalsoom, N. Hussain, J. Nonlinear Sci. Appl. 8 (2015), 201–217
204
Ostrowski type inequality for (α, m)–geometrically convex functions obtained by Ozdamir and Yildiz in
[7] is stated in the form of the following theorem.
Theorem 1.11. Let I ⊂ R0 be an open interval and f : I → (0, ∞) is differentiable. If f ∈ L([a, b]) and
0
|f (x)|q is decreasing and (α, m)-geometrically convex on [a, , b] with a < b and (α, m) ∈ (0, 1] × (0, 1], then
we have
Zb
f (x) − 1
f (u)du
b−a
a
)
(
(b − x)2 0 m
(x − a)2 0 m
1
≤
f (a) M (t, q, α) +
f (b) N (t, q, α) ,
1
b−a
b−a
(p + 1) p
(1.10)
where

 0 q tα  1q
Z1
f (x)


M (t, q, α) =  t  00 q  dt
f (a)
0
and

 0 q tα  1q
Z1
f (x)


N (t, q, α) =  t  00 q  dt .
f (b)
0
Lemma 1.12. ([8]) For x, y ∈ [0, ∞) and m, t ∈ (0, 1], if x < y and y ≥ 1, then
xt y m(1−t) ≤ tx + (1 − t)y.
In this paper, we give some generalized inequalities for (ρ, m)–geometrically convex n-times differentiable
function. The special case for first differentiable (ρ, m)–geometrically convex functions is a three point
inequality of Hermite–Hadamard type which is capable of recapturing the previous results of this domain
as well as some new inequalities can be obtained as a natural consequence.
2. Main Results
Lemma 2.1. Let f be a real valued n-times differentiable mapping defined on [a, b] such that f (n) (x) be
absolutely continuous on [a, b] with α : [a, b] → [a, b] and β : [a, b] → [a, b] , α (x) ≤ x ≤ β (x) , then for all
x ∈ [a, b] , the following identity holds
n+1
Z1
(b − a)
kn (x, t) f
0
(n)
Zb
(ta + (1 − t)b)dt =
n
i
X
1 h
f (u) du −
Rk (x) f (k−1) (x) + Sk (x)
k!
(2.1)
k=1
a
where the kernel kn : [a, b] × [0, 1] → R is given by

b−β
 (t− b−a )n , t ∈ [0, b−x ]
n!
b−a
kn (x, t) =
b−α
 (t− b−a )n , t ∈ ( b−x , 1]
n!
(2.2)
b−a
Rk (x) = (β (x) − x)k + (−1)k−1 (x − α (x))k ,
Sk (x) = (α (x) − a)k f (k−1) (a) + (−1)k−1 (b − β (x))k f (k−1) (b) .
(2.3)
F. Zafar, H. Kalsoom, N. Hussain, J. Nonlinear Sci. Appl. 8 (2015), 201–217
205
Proof. Under the given conditions, the following identity was proved by Cerone and Dragomir [1]
n
Zb
Kn (x, u) f
(−1)
(n)
Zb
(u) du =
a
n
i
X
1 h
f (u) du −
Rk (x) f (k−1) (x) + Sk (x) ,
k!
(2.4)
k=1
a
where the kernel Kn : [a, b]2 → R is given by
(u−α(x))n
,
n!
(u−β(x))n
,
n!
(
Kn (x, u) =
u ∈ [a, x]
u ∈ (x, b]
Moreover, Rk (x) and Sk (x) are defined by (2.3).
Now, by considering the L. H. S. of the identity (2.4),
n
Zb
Kn (x, u) f
(−1)
(n)
Zx
(u) du =
a
(α(x) − u)n (n)
f (u) du +
n!
a
Zb
(β(x) − u)n (n)
f (u) du
n!
x
= Ia + Ib
(2.5)
Let u = ta + (1 − t)b in (2.5), then
Ia = (b − a)
n+1
Z1
(t −
b−α(x) n
b−a )
n!
f (n) (ta + (1 − t)b) dt,
b−x
b−a
and
b−x
Ib = (b − a)
n+1
Zb−a
(t −
b−β(x) n
b−a )
n!
f (n) (ta + (1 − t)b) dt.
0
Thus,
(b − a)
n+1
Z1
kn (x, t) f
(n)
Zb
(ta + (1 − t)b) dt =
n
i
X
1 h
Rk (x) f (k−1) (x) + Sk (x) ,
f (u) du −
k!
a
0
where
kn (x, t) =



(t− b−β
)n
b−a
,t
n!
n
(t− b−α
)
b−a
,t
n!
k=1
∈ [0, b−x
b−a ]
∈ ( b−x
b−a , 1]
Hence, proved.
The generalized Hermite–Hadamard type inequality for (ρ, m)–geometrically convex n–differentiable
function is stated as follows.
Theorem 2.2. Let I ⊃ [0, ∞) be an open interval and f : I → (0, ∞) is n-differentiable. Let |f (n) (x)| ∈
L ([a, b]) is decreasing and (ρ, m)-geometrically convex on [min{1, a}, b] for a ∈ [0, ∞), b ≥ 1, x ∈ [a, b] and
ρ, m ∈ (0, 1] × (0, 1], then
b
Z
n
h
i
X
1
f (u)du −
Rk (x) f (k−1) (x) + Sk (x) k!
k=1
a
n+1
q
≤
1
(b − a)
(n + 1) q (n) m
f (b) En (ρ, m, q),
(n + 1)!
(2.6)
F. Zafar, H. Kalsoom, N. Hussain, J. Nonlinear Sci. Appl. 8 (2015), 201–217
206
for q > 1,
En (ρ, m, q) =












































1
1
((n+1)(b−a)n+1 ) q
(b − β)n+1 + (β − x)n+1 + (x − α)n+1 + (α − a)n+1 ,
for µ = 1
1− 1
q
(b − β)n+1 + (β − x)n+1 + (x − α)n+1 + (α − a)n+1
ρq(b−β)
ρq(b−β)
µ b−a
n!(−1)n+1 + γ(n + 1, ln µ b−a )
(ρq ln u)n+1
ρq(b−x) n+1
P (−1)k−1 ( β−x
)n−k+1 +(−1)2k+1 ( x−α
)n−k+1
b−a
b−a
b−a
+µ
(n−k+1)!(ρq ln µ)k
k=1
! 1
ρq(b−α)
b−a
µ
+ (ρq
ln µ)n+1
n! + (−1)n+1 γ(n + 1, ln µ−
ρq(α−a)
b−a
q
,
for 0 < µ < 1











































(2.7)
1− 1
q
(b − β)n+1 + (β − x)n+1 + (x − α)n+1 + (α − a)n+1
q(b−β) q(b−β)
ρn+1 µ ρ(b−a)
n+1 n! + γ(n + 1, ln µ ρ(b−a) )
(−1)
n+1
(q ln µ)
q(b−x) n+1
P (−1)k−1 ρk ( β−x
)n−k+1 +(−1)2k+1 ρk ( x−α
)n−k+1
b−a
b−a
+µ ρ(b−a)
k
k=1
+
q(b−α)
ρn+1 µ ρ(b−a)
(q ln µ)n+1
(n−k+1)!(q ln µ)
! 1q
q(α−a)
−
n! + (−1)n+1 γ(n + 1, ln µ ρ(b−a)
,
for µ > 1
for µ, ρ > 0, where
µ=
f (n) (a)
m ,
f (n) (b)
and γ (a, x) is the lower incomplete gamma function defined as
Zx
γ (a, x) =
ta−1 e−t dt,
0
Rk (x) = (β(x) − x)k + (−1)k−1 (x − α(x))k ,
Sk (x) = (α(x) − a)k f (k−1) (a) + (−1)k−1 (b − β(x)k f (k−1) (b).
(2.8)
Proof. Applying the definition of kernel, properties of modulus and Hölder’s inequality on (2.1), we get
b
Z
n
h
i
X
1
(k−1)
f (u) du −
Rk (x) f
(x) + Sk (x) k!
k=1
a

≤ (b − a)n+1 
Z1
1− 1q  1
1
Z
q q
 |kn (x, t)| f (n) (ta + (1 − t)b) 
|kn (x, t)| dt
0

≤
(b − a)
n!
n+1
0
b−β
b−x
Zb−a
Zb−a

b−β n
 ( b − β − t)n dt +
(t −
) dt

b−a
b−a
0
b−β
b−a
F. Zafar, H. Kalsoom, N. Hussain, J. Nonlinear Sci. Appl. 8 (2015), 201–217
207
1− 1
b−α
q
Zb−a
Z1
b−α
b−α n 
(
(t −
+
− t)n dt +
) dt

b−a
b−a
b−x
b−a


×

b−α
b−a
b−β
Zb−a
b−x
( b−β
b−a
−
n!
t)n
Zb−a
q
(t −
(n)
f (ta + (1 − t)b) dt+
0
+
n!
q
(n)
f (ta + (1 − t)b) dt
b−β
b−a
b−α
Zb−a
b−β n
b−a )
( b−α
b−a
−
n!
t)n
q
(n)
f (ta + (1 − t)b) dt +
b−x
b−a
Z1
1
q
(t −
b−α n
b−a )
n!
q 
(n)
f (ta + (1 − t)b) dt
 .
b−α
b−a
(2.9)
Applying the definition of (α, m)–geometric convexity on (2.9), we obtain
b
Z
n
h
i
X
1
(k−1)
f (u) du −
Rk (x) f
(x) + Sk (x) k!
k=1
a
(b − a)
n!
≤

b−β



Zb−a
Zb−a
Zb−a
b−β
b−β n
b−α
n
(
(t −
(
− t) dt +
) dt +
− t)n dt
b−a
b−a
b−a
n+1
b−x
0
q
(t −
+
b−β
b−a
1− 1 
Z1
b−α
b−x
b−a
b−β
Zb−a
q

 ( b − β − t)n f (n) at bm(1−t) dt

b−a
b−α n 
) dt

b−a
0
b−α
b−a
b−α
b−x
Zb−a
Zb−a
q
b − β n (n) t m(1−t) q
b−α
n (n)
+
(t −
) f
ab
(
− t) f
at bm(1−t) dt
dt +
b−a
b−a
b−β
b−a
Z1
+
b−x
b−a
1
q
b − α n (n) t m(1−t) q 
(t −
) f
ab
dt
 .
b−a
b−α
b−a
Therefore, upon simplification
b
Z
n
h
i
X
1
(k−1)
f (u) du −
R
(x)
f
(x)
+
S
(x)
k
k
k!
k=1
a

≤
(b − a)
n!
n+1
b−β
b−x
b−α
Zb−a
Zb−a
Zb−a
b
−
β
b
−
β
b−α
(n) m 
n
n
) dt +
(
− t)n dt
(t −
f (b) 
 ( b − a − t) dt +
b−a
b−a
0
b−β
b−a
b−x
b−a
F. Zafar, H. Kalsoom, N. Hussain, J. Nonlinear Sci. Appl. 8 (2015), 201–217
1− 1 
q
Z1
(t −
+
208
b−α n 
) dt

b−a
b−β
ρ
Zb−a
f (n) (a) qt

b
−
β
 (
− t)n (n) m dt

f (b) b−a
0
b−α
b−a
b−x
b−α
qtρ
ρ
Zb−a
Zb−a
f (n) (a) qt
(n)
b − β n f (a) b−α
+
(t −
) (
− t)n (n) m dt
dt +
f (b) b − a f (n) (b)m b−a
b−β
b−a
Z1
+
b−α
b−a
b−x
b−a
1
q
qtρ
(n)

b − α n f (a) (t −
) dt
 .
b − a f (n) (b)m (2.10)
f (n) (a)
Let µ =
m in (2.10), we have three cases:
(f (n) (b))
Case 1: For µ = 1, (2.10) takes the form
b
Z
n
h
i
X
1
(k−1)
f (u) du −
Rk (x) f
(x) + Sk (x) k!
k=1
a

n+1
(b − a)
n!
≤
b−β
b−x
Zb−a
Zb−a
b
−
β
b−β n
(n) m 
n
(t −
) dt+
f (b) 
 ( b − a − t) dt +
b−a
0
b−β
b−a

b−α
Z1
Zb−a
b−α
b−α n 
− t)n dt +
) dt
(
(t −
.
b−a
b−a
b−x
b−a
b−α
b−a
Thus, we have
b
Z
n
h
i
X
1
(k−1)
f (u) du −
R
(x)
f
(x)
+
S
(x)
k
k
k!
k=1
a
≤
" b−β
#
β−x
x−α
α−a
(b − a)n+1 (n) m ( b−a )n+1 ( b−a )n+1 ( b−a )n+1 ( b−a )n+1
+
+
+
,
f (b)
n!
(n + 1)
(n + 1)
(n + 1)
(n + 1)
≤
(n) m
f (b) (b − β)n+1 + (β − x)n+1 + (x − α)n+1 + (α − a)n+1 .
(n + 1)!
(2.11)
qtρ
Case 2: For µ < 1, 0 < t, ρ ≤ 1, we have µ ≤ µρqt . Thus, (2.10) becomes
b
Z
n
h
i
X
1
(k−1)
f (u) du −
Rk (x) f
(x) + Sk (x) k!
k=1
a
F. Zafar, H. Kalsoom, N. Hussain, J. Nonlinear Sci. Appl. 8 (2015), 201–217
209
n+1
1
(b − a) q (n + 1) q (n) m
≤
f (b) (b − β)n+1 + (β − x)n+1 + (x − α)n+1
(n + 1)!
 b−β
b−x
Zb−a
Zb−a
1

1−
b − β n ρqt
b−β
n ρqt
q 
+(α − a)n+1
(t −
) µ dt
 ( b − a − t) µ dt +
b−a
0
b−β
b−a
b−α
b−a
Z
+
b−α
(
− t)n µρqt dt +
b−a
b−x
b−a
1
q
Z1
b − α n ρqt 
(t −
) µ dt
 .
b−a
b−α
b−a
(2.12)
Let,
b−β
Zb−a
b−β
I1 =
(
− t)n µρqt dt,
b−a
0
b−x
Zb−a
b − β n ρqt
(t −
) µ dt,
I2 =
b−a
b−β
b−a
b−α
Zb−a
b−α
I3 =
− t)n µρqt dt,
(
b−a
b−x
b−a
and
Z1
(t −
I4 =
b − α n ρqt
) µ dt.
b−a
b−α
b−a
Then, upon simplification, we have
b−β
b−β
µρq( b−a )
I1 =
γ(n + 1, ln uρq( b−a ) ),
n+1
(ρq ln u)
I2 = µ
ρq( b−x
)
b−a
n+1
X
k=1
I3 = µ
and
ρq( b−x
)
b−a
b−β
n−k+1 +
(−1)k−1 ( β−x
b−a )
(−1)n+1 n!µρq( b−a )
+
,
(ρq ln µ)n+1
(n − k + 1)! (ρq ln µ)k
n+1
X
n−k+1
(−1)2k+1 ( x−α
b−a )
b−α
n!µρq( b−a )
+
,
(ρq ln µ)n+1
(n − k + 1)! (ρq ln µ)k
k=1
b−α
α−a
(−1)n+1 µρq( b−a )
I4 =
γ(n + 1, ln µ−ρq( b−a ) ).
n+1
(ρq ln µ)
Re–substituting the values of the integrals in (2.12), we have
b
Z
n
h
i
X
1
(k−1)
f (u) du −
Rk (x) f
(x) + Sk (x) k!
k=1
a
F. Zafar, H. Kalsoom, N. Hussain, J. Nonlinear Sci. Appl. 8 (2015), 201–217
210
n+1
1
(b − a) q (n + 1) q (n) m
f (b) (b − β)n+1 + (β − x)n+1 + (x − α)n+1
(n + 1)!
≤
+(α − a)
+µ
b−β
µρq( b−a ) ρq( b−β
)
n+1
b−a )
(−1)
n!
+
γ(n
+
1,
ln
µ
(ρq ln u)n+1
1
n+1 1− q
)
ρq( b−x
b−a
n+1
X
n−k+1 + (−1)2k+1 ( x−α )n−k+1
(−1)k−1 ( β−x
b−a )
b−a
k=1
(n − k + 1)! (ρq ln µ)k
b−α
µρq( b−a ) −ρq( α−a
)
n+1
b−a
+
n!
+
(−1)
γ(n
+
1,
ln
µ
(ρq ln µ)n+1
!
! 1q
, for 0 < µ < 1.
(2.13)
qtρ
qt
ρ
Case 3: For µ > 1, 0 < t, ρ ≤ 1, we have µ ≤ µ . Thus, (2.10) becomes
b
Z
n
h
i
X
1
(k−1)
f (u) du −
R
(x)
f
(x)
+
S
(x)
k
k
k!
k=1
a
n+1
1
(b − a) q (n + 1) q (n) m
f (b) (b − β)n+1 + (β − x)n+1 + (x − α)n+1
(n + 1)!
≤

+(α − a)n+1
1− 1q
b−β
b−x
Zb−a
Zb−a

qt
b − β n qtρ
 ( b − β − t)n µ ρ dt +
(t −
) µ dt

b−a
b−a
0
b−α
b−a
Z
+
b−x
b−a
qt
b−α
(
− t)n µ ρ dt +
b−a
b−β
b−a
Z1
1
q
b − α n qtρ 
(t −
) µ dt
 .
b−a
b−α
b−a
(2.14)
Let,
b−β
Zb−a
qt
b−β
0
I1 =
(
− t)n µ ρ dt,
b−a
0
b−x
Zb−a
b − β n qtρ
0
I2 =
(t −
) µ dt,
b−a
b−β
b−a
b−α
Zb−a
qt
b−α
0
(
− t)n µ ρ dt,
I3 =
b−a
b−x
b−a
F. Zafar, H. Kalsoom, N. Hussain, J. Nonlinear Sci. Appl. 8 (2015), 201–217
and
0
Z1
(t −
I4 =
211
b − α n qtρ
) µ dt.
b−a
b−α
b−a
Then, upon simplification, we have
q(b−β)
q(b−β)
ρn+1 µ ρ(b−a)
I1 =
γ(n + 1, ln µ ρ(b−a) ),
n+1
(q ln u)
0
0
I2 = µ
q(b−x)
ρ(b−a)
0
I3 = µ
n+1
X
n−k+1
ρk (−1)k−1 ( β−x
b−a )
k=1
(n − k + 1)! (q ln µ)k
q(b−x)
ρ(b−a)
n+1
X
n−k+1
ρk (−1)2k+1 ( x−α
b−a )
k=1
(n − k + 1)! (q ln µ)k
and
q(b−β)
ρn+1 (−1)n+1 n!µ ρ(b−a)
+
,
(ρq ln µ)n+1
q(b−α)
ρn+1 n!µ ρ(b−a)
+
,
(ρq ln µ)n+1
q(b−α)
ρn+1 (−1)n+1 µ ρ(b−a)
− q ( α−a )
I4 =
γ(n + 1, ln µ ρ b−a ).
n+1
(q ln µ)
0
Re–substituting the values of the integrals in (2.14), we have
b
Z
n
h
i
X
1
(k−1)
f (u) du −
R
(x)
f
(x)
+
S
(x)
k
k
k!
k=1
a
n+1
1 (b − a) q (n + 1) q (n) b m
n+1
+ (β − x)n+1 + (x − α)n+1
f ( m ) (b − β)
(n + 1)!
≤

+ (α − a)
+µ
q(b−x)
ρ(b−a)
1
n+1 1−
n+1
X
q
q(b−β)
n+1 µ ρ(b−a)
ρ
(q ln µ)n+1
q(b−β)
n+1
ρ(b−a)
(−1)
n! + γ(n + 1, ln µ
)
n−k+1 + (−1)2k+1 ( x−α )n−k+1
(−1)k−1 ( β−x
b−a )
b−a
k
ρ
(n − k + 1)! (q ln µ)k
k=1
q(b−α)
+
ρn+1 µ ρ(b−a)
(q ln µ)n+1
!
n! + (−1)n+1 γ(n + 1, ln µ
q(α−a)
− ρ(b−a)
1
q
)  , f or µ > 1.
(2.15)
Therefore, (2.11), (2.13) and (2.15) are required inequalities.
Remark 2.3. If we take
α (x) = a,
β (x) = b,
x = a+b
2 and n = 1 in (2.10), then the inequalities for the m and (α, m)–geometrically convex functions by
Xi et al.[8] are recaptured.
The generalized inequality for m–geometrically convex n–differentiable function is stated as follows.
F. Zafar, H. Kalsoom, N. Hussain, J. Nonlinear Sci. Appl. 8 (2015), 201–217
212
Corollary 2.4. Let f, α, β, µ be defined as in Theorem 2.2. If f (n) is m–geometrically convex on [min{1,
a}, b], for m ∈ (0, 1), then for all x ∈ [a, b],
b
Z
n
h
i
X
1
(k−1)
f (u)du −
Rk (x)f
(x) + Sk (x) k!
k=1
a
≤
n+1
1
(b − a) q (n + 1) q (n) m
f (b) En (ρ, m, q),
(n + 1)!
(2.16)
for q > 1,
En (1, m, q) =































































































1
1
((n+1)(b−a)n+1 ) q
(b − β)n+1 + (β − x)n+1 + (x − α)n+1 + (α − a)n+1 ,
for µ = 1
(b − β)n+1 + (β − x)n+1 + (x − α)n+1 + (α − a)n+1
q(b−β)
×
+µ
q
µ b−a
(q ln u)n+1
q(b−β)
n+1
b−a
(−1)
n! + γ(n + 1, ln µ
)
n+1
P
(−1)k−1 ( β−x
)n−k+1 +(−1)2k+1 ( x−α
)n−k+1
b−a
b−a
q(b−x)
b−a
q(b−α)
µ b−a
(q ln µ)n+1
(n−k+1)!(q ln µ)k
k=1
+
1− 1
! 1q
q(α−a)
,
n! + (−1)n+1 γ(n + 1, ln µ− b−a
(2.17)
for 0 < µ < 1
(b − β)n+1 + (β − x)n+1 + (x − α)n+1 + (α − a)n+1
q(b−β)
×
+µ
1− 1
µ b−a
(q ln µ)n+1
q(b−β)
n+1
b−a
n!(−1)
+ γ(n + 1, ln µ
)
n+1
P
(−1)k−1 ( β−x
)n−k+1 +(−1)2k+1 ( x−α
)n−k+1
b−a
b−a
q(b−x)
b−a
q(b−α)
(n−k+1)!(q ln µ)k
k=1
b−a
+ (qµln µ)n+1
q
n! + (−1)n+1 γ(n + 1, ln µ
−
q(α−a)
b−a
! 1q
)
,
for µ > 1.
where
γ (a, x)
is the lower incomplete gamma function. Moreover,
Rk (x)
and
Sk (x)
are defined by (2.8).
Proof. Substituting ρ = 1, in (2.6)–(2.8), we have the required inequality.
The generalized three point Hermite–Hadamard type inequality for (ρ, m)–geometrically convex ndifferentiable function is stated as,
F. Zafar, H. Kalsoom, N. Hussain, J. Nonlinear Sci. Appl. 8 (2015), 201–217
213
Theorem 2.5. Let f , ρ, m, µ be defined as in Theorem 2.2, then the following inequality holds for any
r ∈ [0, 1] and for all x ∈ [a, b],
b
Z
n
X
1 h
k
k
k−1
k
f (u) du −
(1
−
r)
(b
−
x)
+
(−1)
(x
−
a)
f (k−1) (x)
k!
k=1
a
i
+rk (x − a)k f (k−1) (a) + (−1)k−1 (b − x)k f (k−1) (b) ≤
n+1
1
(b − a) q (n + 1) q (n) m
f (b) En (ρ, m, q),
(n + 1)!
(2.18)
for q > 1, and
rn+1 + 1 − rn+1
En (ρ, m, q) =
En (ρ, m, q) =
En (ρ, m, q) =
























+µ
















+





n+1
(b − x)n+1 + (x − a)
, for µ = 1
1
(2.19)
((n + 1)(b − a)n+1 ) q
rn+1 + 1 − rn+1
rρq(b−x)
µ b−a
(ρq ln µ)n+1
×
qρ(b−x)
b−a

+µ
















+




























n+1
P
(b − x)n+1 + (x − a)
n+1
1− 1
q
rqρ(b−x)
(−1)n+1 n! + γ(n + 1, ln µ b−a )
(−1)k−1 (
(1−r)(x−a) n−k+1
(1−r)(b−x) n−k+1
)
+(−1)2k+1 (
)
b−a
b−a
k
(2.20)
(n−k+1)!(ρq ln µ)
k=1
1
r(x−a)
qρ 1−
(b−a)
µ
(ρq ln µ)n+1
! q
rqρ(x−a)
−
b−a
,
n! + (−1)n+1 γ(n + 1, ln µ
)
for 0 < µ < 1
rn+1
+ 1−
rn+1
rq(b−x)
×
q(b−x)
ρ(b−a)
ρn+1 µ ρ(b−a)
(q ln µ)n+1
n+1
P
ρk
(b −
x)n+1
n+1
+ (x − a)
1− 1
q
rq(b−x)
n+1
ρ(b−a)
)
(−1)
n! + γ(n + 1, ln µ
(−1)k−1 (
k=1
(1−r)(b−x) n−k+1
(1−r)(x−a) n−k+1
)
+(−1)2k+1 (
)
b−a
b−a
k
(n−k+1)!(q ln µ)
1
q
ρ
r(x−a)
1−
(b−a)
ρn+1 µ
(q ln µ)n+1
! q
rq(x−a)
−
n! + (−1)n+1 γ(n + 1, ln µ ρ(b−a) )
,
for µ > 1
for ρ > 0.
Proof. Let α(x) and β(x) be defined as follows in Theorem 2.2,
α(x) = rx + (1 − r)a,
(2.21)
F. Zafar, H. Kalsoom, N. Hussain, J. Nonlinear Sci. Appl. 8 (2015), 201–217
214
and
β(x) = rx + (1 − r)b.
Then, we have the required result.
The generalized three point inequality for (ρ, m)–geometrically convex first differentiable function is
stated as follows.
Corollary 2.6. Let f , ρ, m, µ be defined as in Theorem 2.5, then the following inequality holds for any
r ∈ [0, 1] and x ∈ [a, b],
Zb
1
r ((x − a)f (a) + (b − x)f (b)) b − a f (u) du − (1 − r)f (x) +
b−a
a
2
≤
(b − a) q
2
−1
1− 1q
0 m
f (b) E1 (ρ, m, q),
(2.22)
for q > 1, where
E1 (ρ, m, q) =
E1 (ρ, m, q) =






























+





E1 (ρ, m, q) =


























+





r2 + (1 − r)2
(b − x)2 + (x − a)2
, for µ = 1.
1
(2(b − a)2 ) q
1− 1q
r2 + (1 − r)2 (b − x)2 + (x − a)2
rρq(b−x)
(2.23)
rρq(b−x)
rρq(b−x)
− b−a
b−a
ln µ
−1+µ
1+µ
ρq(b−x) (1−r)(b−x)
(1−r)(x−a)
2
+µ b−a
−
−
2
(b−a)ρq ln µ
(b−a)ρq ln µ
(ρq ln µ)
µ b−a
(ρq ln µ)2
×
−
rρq(b−x)
b−a
(2.24)
1
r(x−a)
ρq 1−
(b−a)
µ
(ρq ln µ)2
! q
rρq(b−x)
rρq(x−a)
rρq(x−a)
−
b−a
ln µ b−a − 1 + µ
,
1 + µ b−a
for 0 < µ < 1
1− 1q
r2 + (1 − r)2 (b − x)2 + (x − a)2
rq(b−x) rq(b−x)
rq(b−x)
rq(b−x)
− ρ(b−a)
− ρ(b−a)
ρ2 µ ρ(b−a)
1+µ
ln µ
− 1 + µ ρ(b−a)
(q ln µ)2
q(b−x) ρ(1−r)(x−a)
2ρ2
−
−
+µ ρ(b−a) ρ(1−r)(b−x)
(b−a)q ln µ
(b−a)q ln µ
(q ln µ)2
×
q
ρ2 µ ρ
r(x−a)
1−
(b−a)
(q ln µ)2
(2.25)
1
q
!
rq(b−x)
rq(x−a)
rq(x−a)
−
1 + µ ρ(b−a) ln µ ρ(b−a) − 1 + µ ρ(b−a)
,
for µ > 1.
Proof. Substituting n = 1, in (2.18)-(2.21), we have the required inequality.
Remark 2.7. For r = 0 in (2.22)-(2.25), we can get a three-point Ostrowski type inequality for (ρ, m)–
geometrically convex function.
Remark 2.8. The following are the Hermite–Hadamard type inequalities for (ρ, m)–geometrically convex
first differentiable function for different choices of r in (2.22)–(2.25).
F. Zafar, H. Kalsoom, N. Hussain, J. Nonlinear Sci. Appl. 8 (2015), 201–217
215
1. When r = 0 and x = a+b
2 in (2.22)–(2.25), then
Zb
1
a + b b − a f (u) du − f
2
a
≤
(b − a)
2
1− 1
q 1
1
f 0 (b)m [G1 (ρ, m, q)] q ,
2
for q > 1 and µ is defined as
µ=
where
G1 (ρ, m, q) =



















(2.26)
f 0 (a)
,
(f 0 (b))m
1
2,
for µ = 1
ρq
ρq
2+2µ −4µ 2
ρ2 q 2 ln2 µ
, for 0 < µ < 1
q
q
ρ2 2+2µ ρ −4µ 2ρ
q 2 ln2 µ
(2.27)
, for µ > 1
for ρ > 0.
2. When r = 1 and x = a+b
2 in (2.22)-(2.25), then
Zb
1
f (a) + f (b) b − a f (u) du +
2
a
≤
(b − a)
2
1− 1
q 1
1
f 0 (b)m [G2 (ρ, m, q)] q ,
2
for q > 1 where µ is defined as
µ=
where
f 0 (a)
,
(f 0 (b))m

1

2 , for µ = 1





ρq
2

ρq


µ −1) ln uρq −2 µ 2 −1
(


, for 0 < µ < 1
ρ2 q 2 ln2 µ
G2 (ρ, m, q) =




!
!2 

q
q

 2  ρq
ρ
2ρ

ρ
µ −1 ln µ −2 µ −1 




, for µ > 1
q 2 ln2 µ
for ρ > 0.
3. When r =
1
2
and x =
(2.28)
(2.29)
a+b
2
in (2.22)-(2.25), then
Zb
1
1
a+b
+ f (b) b − a f (u) du − 4 f (a) + 2f
2
a
1
1
(b − a) 1 1− q 0 m
≤
f (b) [G3 (ρ, m, q)] q ,
4
2
µ is defined as
µ=
f 0 (a)
,
f 0 (b)m
(2.30)
F. Zafar, H. Kalsoom, N. Hussain, J. Nonlinear Sci. Appl. 8 (2015), 201–217
and
G3 (ρ, m, q) =





















for ρ > 0.
4. When r =
1
3
and x =
1
2,
−4µρq +8µ
3ρq
4
−8µ
ρq
2
for µ = 1
ρq
+8µ 4 +4(µρq −1) ln µ
ln2 µ
ρq
4
−4
ρ2 q 2
ρ2
216
, for 0 < µ < 1
(2.31)
!
q
3q
q
q
q
q
−4µ ρ +8µ 4ρ −8µ 2ρ +8µ 4ρ +4 µ ρ −1 ln µ 4ρ −4
, for µ > 1
q 2 ln2 µ
a+b
2
in (2.22)-(2.25), then
Zb
1
a
+
b
1
+ f (b) b − a f (u) du − 6 f (a) + 4f
2
a
1
1
5(b − a) 1 1− q 0 m
f (b) [G4 (ρ, m, q)] q ,
≤
18
2
for q > 1 where µ is defined as
µ=
where
G4 (ρ, m, q) =





















f 0 (a)
,
f 0 (b)m
1
2,
9(−2µρq +4µ
5ρq
6
ρq
(2.32)
for µ = 1
ρq
−4µ 2 +4µ 6 +2(µρq −1) ln µ
5(ρ2 q 2 ln2 µ)
ρq
6
−2)
, for 0 < µ < 1
(2.33)
!
q
5q
q
q
q
q
9ρ2 −2µ ρ +4µ 6ρ −4µ 2ρ +4µ 6ρ +2 µ ρ −1 ln µ 6ρ −2
5(q 2 ln2 µ)
, for µ > 1
for ρ > 0.
3. Conclusions
Some generalized inequalities for (ρ, m)–geometrically convex and n-differentiable mappings are given
which are capable of giving bounds of the one point, two point and three point Hadamard type inequalities for
first and higher differentiable functions. The special cases recapture inequalities given by Xi et al. [8]. Some
new estimates of the average mid-point trapezoid and Simpson’s inequality are given for first differentiable
generalized geometrically convex function as special cases. The estimates for higher differentiable function
can also be obtained from (2.18)-(2.21) for all these cases.
Acknowledgements
The authors are thankful to the referee for giving valuable comments and suggestions which helped to
improve the final version of this paper.
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