On new inequalities of Hermite–Hadamard–Fejer type for harmonically convex functions via fractional integrals
Kunt et al. SpringerPlus
On new inequalities of Hermite- Hadamard-Fejer type for harmonically convex functions via fractional integrals
Mehmet Kunt 0 1 2 4
İmdat İşcan 0 1 2 3
Nazlı Yazıcı 0 1 2 4
Uğur Gözütok 0 1 2 4
0 Technical University , Trabzon
1 of Sciences , Karadeniz
2 of Mathematics , Faculty
3 Department of Mathematics, Faculty of Sciences and Arts, Giresun University , Giresun , Turkey
4 Department of Mathematics, Faculty of Sciences, Karadeniz Technical University , Trabzon , Turkey
In this paper, firstly, new Hermite-Hadamard type inequalities for harmonically convex functions in fractional integral forms are given. Secondly, Hermite-Hadamard-Fejer inequalities for harmonically convex functions in fractional integral forms are built. Finally, an integral identity and some Hermite-Hadamard-Fejer type integral inequalities for harmonically convex functions in fractional integral forms are obtained. Some results presented here provide extensions of others given in earlier works. Riemann-Liouville fractional integrals, Harmonically convex functions Mathematics Subject Classification: 26A51, 26A33, 26D10
Hermite-Hadamard inequality; Hermite-Hadamard-Fejer inequality
-
Turkey
article
*Correspondence:
Background
a, b ∈ I. The inequality
Let f : I ⊂ R → R be a convex function defined on the interval I of real numbers and
f
f
a + b
2
≤
1
b − a a
is well known in the literature as Hermite–Hadamard’s inequality
(Hadamard 1893;
Hermite 1883)
.
The most well-known inequalities related to the integral mean of a convex function f
are the Hermite Hadamard inequalities or their weighted versions, the so-called
Hermite–Hadamard-Fejér inequalities.
Fejér (1906)
established the following Fejér inequality which is the weighted
generalization of Hermite–Hadamard inequality (1):
Theorem 1 Let f : [a, b]→ R be a convex function. Then the inequality
a + b
2
a
a
b
g (x)dx ≤
f (x)g (x)dx ≤
f (a) + f (b)
2
a
holds, where g : [a, b]→ R is nonnegative, integrable and symmetric to (a + b)/2.
(1)
(2)
For some results which generalize, improve and extend the inequalities (1) and (2) see
Bombardelli and Varošanec (1869)
, İşcan
(2013a, 2014c)
,
Minculete and Mitroi (2012)
,
Sarıkaya (2012)
,
Tseng et al. (2011
).
We recall the following inequality and special functions which are known as Beta and
hypergeometric function respectively:
β x, y =
2F1(a, b; c; z) =
Γ (x)Γ y = 1 tx−1(1 − t)y−1dt,
Γ x + y 0
1 1
β(b, c − b) 0
c > b > 0, |z| < 1
(see Kilbas et al. 2006)
.
tb−1(1 − t)c−b−1(1 − zt)−adt,
x, y > 0,
Lemma 1
(Prudnikov et al. 1981; Wang et al. 2013)
For 0 < α ≤ 1 and 0 ≤ a < b we
have
aα − bα ≤ (b − a)α.
The following definitions and mathematical preliminaries of fractional calculus theory
are used further in this paper.
Definition 1
(Kilbas et al. 2006)
Let f ∈ L[a, b]. The Riemann–Liouville integrals Jaα+f
and Jbα−f of oder α > 0 with a ≥ 0 are defined by
and
Jaα+f (x) =
Jbα−f (x) =
1
Γ (α) a
1
Γ (α) x
x
b
(x − t)α−1f (t)dt,
x > a
(t − x)α−1f (t)dt,
x < b
respectively, where Γ (α) is the Gamma function defined by Γ (α) = e−t tα−1dt and
Ja0+f (x) = Jb0−f (x) = f (x). 0
Because of the wide application of Hermite–Hadamard type inequalities and fractional
integrals, many researchers extend their studies to Hermite–Hadamard type inequalities
involving fractional integrals not limited to integer integrals. Recently, more and more
Hermite–Hadamard inequalities involving fractional integrals have been obtained for
different classes of functions; see
Dahmani (2010)
, İşcan (2013b, 2014a), İşcan and Wu
(2014),
Mihai and Ion (2014)
,
Sarıkaya et al. (2013
),
Wang et al. (2012
), Wang et al. (2013).
İşcan (2014b) can defined the so-called harmonically convex functions and
established following Hermite–Hadamard type inequality for them as follows:
Definition 2 Let I ⊂ R\{0} be a real interval. A function f : I → R is said to be
harmonically convex, if
∞
f tx + (x1y− t)y
≤ tf y + (1 − t)f (x)
(3)
for all x, y ∈ I and t ∈ [0, 1]. If the inequality in (3) is reversed, then f is said to be
harmonically concave.
Theorem 2
(İşcan 2014b)
Let f : I ⊂ R\{0} → R be a harmonically convex function
and a, b ∈ I. If f ∈ L[a, b] then the following inequalities holds:
f (a) +2 f (b) .
Latif et al. (2015
) gave the following definition:
Definition 3 A function g : [a, b] ⊆ R\{0} → R is said to be harmonically symmetric
with respect to 2ab/a + b if
f
f
(4)
(5)
g (x) = g
with α > 0 and h(x) = 1/x, x ∈ b1 , a1 .
g 1t dt
with α > 0 and h(x) = 1/x, x ∈ b1 , a1 .
Proof Since f is a harmonically convex function on [a, b], we have for all t ∈ [0, 1]
Multiplying both sides of (7) by 2tα−1, then integrating the resulting inequality with
respect to t over 0, 12 , we obtain
≤
=
0
Setting x = tb+(1−t)a and dx =
ab
dx
Then multiplying both sides of (8) by tα−1 and integrating the resulting inequality with
respect to t over 0, 21 , we obtain
i.e.
α
ab
b − a
21−α f (a) + f (b)
≤ α 2 .
The proof (...truncated)