Vol. 17, No.1, April 2009, pp 53-69 ISSN 1222-5657, ISBN 978-973-88255-5-0, www.hetfalu.ro/octogon
53
Hermite-Hadamard and Fej´ er Inequalities for Wright-Convex Functions
Vlad Ciobotariu-Boer 4
ABSTRACT.In this paper, we establish several inequalities of Hermite-Hadamard and Fej´er type for Wright-convex functions.
1. INTRODUCTION
Throughout this paper we will consider a real-valued convex functionf, defined on a nonempty intervalI ⊂R, and a, b∈I, witha < b.
In the conditions above, we have:
f
a+b 2
≤ 1 b−a ·
Zb
a
f(x)dx≤ f(a) +f(b)
2 . (1.1)
The inequalities (1.1) are known as the Hermite-Hadamard inequalities (see [10], [17], [20]). In [9], Fej´er established the following weighted generalization of the inequalities (1.1):
f
a+b 2
· Zb
a
p(x)dx≤ Zb
a
f(x)p(x)dx≤ f(a) +f(b)
2 ·
Z b
a
p(x)dx, (1.2) wherep: [a, b]→Ris a nonnegative, integrable, and symmetric about x= a+b2 . The last inequalities are known as the Fej´er inequalities.
In recent years, many extensions, generalizations, applications and similar results of the inequalities (1.1) and (1.2) were deduced (see [1]-[4], [6]-[8], [12]-[16], [18], [21], [22]).
In [5], Dragomir established the following theorem, which is a refinement of the first inequality of (1.1):
4Received: 12.03.2009
2000Mathematics Subject Classification. 26D15.
Key words and phrases. Hermite- Hadamard inequalities, Fej´er inequalities, Wright- convex functions.
Theorem 1.1. If H is defined on [0,1] by
H(t) = 1 b−a·
Zb
a
f
tx+ (1−t)·a+b 2
dx,
where the function f is convex on [a, b], then H is convex, nondecreasing on [0,1], and for allt∈[0,1], we have
f
a+b 2
=H(0)≤H(t)≤H(1) = 1 b−a·
Zb
a
f(x)dx. (1.3) In [19], Yang and Hong established the following theorem which is a
refinement of the second inequality of (1.1):
Theorem 1.2. If F is defined on [0,1] by
F(t) = 1 2(b−a) ·
Zb
a
f
1 +t
2 ·a+1−t 2 ·x
+f
1 +t
2 ·b+1−t 2 ·x
dx,
where the function f is convex on [a, b], then F is convex, nondecreasing on [0,1], and for allt∈[0,1], we have
1 b−a·
Z b
a
f(x)dx=F(0)≤F(t)≤F(1) = f(a) +f(b)
2 . (1.4)
In [20], Yang and Tseng established the following theorem, which refines the inequality (1.2):
Theorem 1.3. If P, Qare defined on [0,1] by
P(t) = Zb
a
f
tx+ (1−t)·a+b 2
·p(x)dx
and
Q(t) = 1 2·
Zb
a
f
1 +t
2 ·a+ 1−t 2 ·x
·p
x+a 2
+
+f
1 +t
2 ·b+1−t 2 ·x
·p
x+b 2
dx,
where the function f is convex on [a, b], then P, Q are convex and increasing on [0,1], and for allt∈[0,1]:
f
a+b 2
· Zb
a
p(x)dx=P(0)≤P(t)≤P(1) = Zb
a
f(x)p(x)dx (1.5) and
Zb
a
f(x)·p(x)dx=Q(0)≤Q(t)≤Q(1) = f(a) +f(b)
2 ·
Zb
a
p(x)dx, (1.6)
wherep: [a, b]→Ris nonnegative, integrable and symmetric aboutx= a+b2 . In the following, we recall the definition of a Wright-convex function:
Definition 1.1. (see [15]) We say thatf : [a, b]→R is aWright-convex function if for all x, y∈[a, b] withx < y and δ ≥0, so that x+δ∈[a, b], we have:
f(x+δ) +f(y)≤f(y+δ) +f(x).
Denoting the set of all convex functions on [a, b] by K([a, b]) and the set of all Wright-convex functions on [a, b] by W([a, b]), then K([a, b])⊂W([a, b]), the inclusion being strict (see [14], [15]).
Next, we give a theorem that characterizes Wright-convex functions (see [18]):
Theorem 1.4. Iff : [a, b]→R, then the following statements are equivalent:
(i) f ∈W([a, b]);
(ii) for alls, t, u, v∈[a, b] withs≤t≤u≤v and t+u=s+v, we have f(t) +f(u)≤f(s) +f(v).
In [18], Tseng, Yang and Dragomir established the following theorems for Wright-convex functions, related to the inequalities (1.1):
Theorem 1.5. Letf ∈W([a, b])∩L1[a, b]. Then, the inequalities (1.1) hold.
Theorem 1.6. Let f ∈W([a, b])∩L1[a, b] and letH be defined as in Theorem 1.1. Then H∈W([0,1]) is nondecreasing on [0,1], and the inequality (1.3) holds for all t∈[0,1].
Theorem 1.7. Let f ∈W([a, b])∩L1[a, b] and letF be defined as in Theorem 1.2. Then F ∈W([0,1]) is nondecreasing on [0,1], and the inequality (1.4) holds for all t∈[0,1].
In [12], Ming-In-Ho established the following theorems for Wright-convex functions related to the inequalities (1.2):
Theorem 1.8. Let f :W([a, b])∩L1[a, b] and letp defined as in Theorem 1.3. Then the inequalities (1.2) hold.
Theorem 1.9. Let p, P, Qbe defined as in Theorem 1.3. Then
P, Q∈W([0,1]) are nondecreasing on [0,1], and the inequalities (1.5) and (1.6) hold for allt∈[0,1].
In [3], we established the following theorems for convex functions related to inequalitites (1.2) and (1.2):
Theorem 1.10. IfR, S are defined on [0,1] by
R(t) = 1 b−a·
Zb
a
f
1 +t
2 ·a+b
2 +1−t 2 ·x
dx and
S(t) = 1 2(b−a) ·
Zb
a
f
a+b
2 −t·b−x 2
+f
a+b
2 +t·x−a 2
dx,
where the function f is convex on [a, b], then R is convex, nonincreasing on [0,1] andS is convex, nondecreasing on [0,1], and for all t∈[0,1], we have:
f
a+b 2
=R(1)≤R(t)≤R(0) = 1 b−a·
Zb
a
f
a+b 4 +x
2
dx≤
≤ 1 2 ·f
a+b 2
+ 1
2(b−a) · Zb
a
f(x)dx≤ 1 b−a·
Zb
a
f(x)dx (1.7) and
f
a+b 2
=S(0)≤S(t)≤S(1) = 1 b−a·
Zb
a
f(x)dx. (1.8) Theorem 1.11. IfT, U are defined on [0,1] by
T(t) = Zb
a
f
1 +t
2 ·a+b
2 +1−t 2 ·x
·p(x)dx
and
U(t) = 1 2·
Zb
a
f
a+b
2 −t·b−x 2
·p
x+a 2
+
+f
a+b
2 +t·x−a 2
·p
x+b 2
dx,
where the function f is convex on [a, b] and p is defined on [a, b] as in Theorem 1.3, thenT is convex, nonincreasing on [0,1] andU is convex, nondecreasing on [0,1], and for all t∈[0,1] we have:
f
a+b 2
· Zb
a
p(x)dx=T(1)≤T(t)≤T(0) = Zb
a
f
a+b 4 + x
2
·p(x)dx≤
≤ 1 2 ·f
a+b 2
· Zb
a
p(x)dx+1 2 ·
Zb
a
f(x)p(x)dx≤ Zb
a
f(x)p(x)dx (1.9) and
f
a+b 2
· Zb
a
p(x)dx=U(0)≤U(t)≤U(1) = Zb
a
f(x)p(x)dx. (1.10)
In this paper, we establish some results related to Theorem 1.10 and Theorem 1.11 for Wright-convex functions.
MAIN RESULTS
Theorem 2.1. Let f ∈W([a, b])∩L1[a, b] and letR be defined as in Theorem 1.10. Then,R∈W([0,1]) is nonincreasing on [0,1], and the inequalities (1.7) hold for allt∈[0,1]
Proof. If s, t, u, v∈[0,1] withs≤t≤u≤v and t+u=s+v, then, for all x∈
a,a+b2
, we have a≤ 1 +s
2 ·a+b
2 +1−s
2 ·x≤ 1 +t
2 ·a+b
2 +1−t 2 ·x≤
≤ 1 +u
2 ·a+b
2 +1−u
2 ·x≤ 1 +v
2 ·a+b
2 +1−v
2 ·x≤ a+b 2 and, for allx∈a+b
2 , b
, we have a+b
2 ≤ 1 +v
2 ·a+b
2 +1−v
2 ·x≤ 1 +u
2 ·a+b
2 +1−u 2 ·x≤
≤ 1 +t
2 ·a+b
2 + 1−t
2 ·x≤ 1 +s
2 ·a+b
2 +1−s
2 ·x≤b.
Denoting
s1:= 1 +s
2 ·a+b
2 +1−s 2 ·x, t1:= 1 +t
2 ·a+b
2 + 1−t 2 ·x, u1:= 1 +u
2 ·a+b
2 +1−u 2 ·x, v1 := 1 +v
2 ·a+b
2 + 1−v 2 ·x, we note that forx∈
a,a+b2
,s1, t1, u1, v1∈ a,a+b2
with s1 ≤t1 ≤u1 ≤v1 and t1+u1 =s1+v1. Sincef ∈W([a, b]), taking into account the Theorem 1.4, we deduce:
f(t1) +f(u1)≤f(s1) +f(v1) for allx∈
a,a+b 2
. (2.1)
Denoting
s2:= 1 +v
2 ·a+b
2 +1−v 2 ·x, t2 := 1 +u
2 ·a+b
2 + 1−u 2 ·x, u2 := 1 +t
2 ·a+b
2 +1−t 2 ·x, v2:= 1 +s
2 ·a+b
2 + 1−s 2 ·x, forx∈a+b
2 , b
, we note that s2, t2, u2, v2 ∈a+b
2 , b
withs2 ≤t2≤u2 ≤v2
and t2+u2 =s2+v2. Sincef ∈W([a, b]), taking into account the Theorem 1.4, we obtain:
f(t2) +f(u2)≤f(s2) +f(v2) for allx∈ a+b
2 , b
. (2.2)
Integrating the inequality (2.1) over xon a,a+b2
, the inequality (2.2) overx on a+b
2 , b
and adding the obtained inequalities and multiplying the result by b−1a, we find:
R(t) +R(u)≤R(s) +R(v), namely R∈W([0,1]).
In order to prove the monotonicity ofR∈W([0,1]), we consider 0≤t1 < t2≤1. Then, we have:
a≤ 1 +t1
2 ·a+b
2 +1−t1
2 ·x≤ 1 +t2
2 ·a+b
2 + 1−t2 2 ·x≤
≤ 1 +t2 2 ·a+b
2 +1−t2
2 ·(a+b−x)≤ 1 +t1 2 ·a+b
2 +1−t1
2 ·(a+b−x)≤ a+b 2 for all x∈
a,a+b2 and a+b
2 ≤ 1 +t1 2 ·a+b
2 +1−t1
2 ·(a+b−x)≤ 1 +t2 2 ·a+b
2 +1−t2
2 ·(a+b−x)≤
≤ 1 +t2
2 ·a+b
2 + 1−t2
2 ·x≤ 1 +t1
2 ·a+b
2 +1−t1
2 ·x≤b
for all x∈a+b
2 , b . Considering
s3 := 1 +t1
2 ·a+b
2 +1−t1
2 ·x, t3 := 1 +t2
2 ·a+b
2 + 1−t2
2 ·x, u3 := 1 +t2
2 ·a+b
2 +1−t2
2 ·(a+b−x), v3 := 1 +t1
2 ·a+b
2 +1−t1
2 ·(a+b−x), for all x∈
a,a+b2
, we note that s3, t3, u3, v3 ∈ a,a+b2
with
s3≤t3 ≤u3≤v3 and t3+u3 =s3+v3. Applying Theorem 1.4, we find:
f(t3) +f(u3)≤f(s3) +f(v3) for allx∈
a,a+b 2
. (2.3)
Denoting
s4 := 1 +t1
2 ·a+b
2 +1−t1
2 ·(a+b−x), t4 := 1 +t2
2 ·a+b
2 +1−t2
2 ·(a+b−x), u4 := 1 +t2
2 ·a+b
2 +1−t2 2 ·x, v4 := 1 +t1
2 ·a+b
2 + 1−t1 2 ·x, for all x∈a+b
2 , b
, we note that s4, t4, u4, v4∈a+b
2 , b with
s4≤t4 ≤u4≤v4 and t4+u4 =s4+v4. Since f ∈W([a, b]), taking into account Theorem 1.4, we obtain:
f(t4) +f(u4)≤f(s4) +f(v4) for allx∈ a+b
2 , b
. (2.4)
Integrating the inequality (2.3) over xon a,a+b2
, the inequality (2.4) overx on a+b
2 , b
, adding the obtained inequalities and multiplying the result by
1
b−a, we deduce
2R(t2)≤2R(t1), namely R is nonincreasing on the interval [0,1].
Now, we note that x≤ a+b
4 +x
2 ≤ a+b 4 +x
2 ≤ a+b
2 for allx∈
a,a+b 2
(2.5) and
a+b
2 ≤ a+b 4 +x
2 ≤ a+b 4 +x
2 ≤x for allx∈ a+b
2 , b
. (2.6)
Sincef ∈W([a, b]), taking into account the Theorem 1.4, we have, from (2.5) and (2.6):
2·f
a+b 4 + x
2
≤f
a+b 2
+f(x), for allx∈[a, b]. (2.7) Multiplying the inequality (2.7) by 2(b1−a) and integrating the obtained result overx on [a, b], we deduce:
1 b−a·
Zb
a
f
a+b 4 +x
2
dx≤ 1 2 ·f
a+b 2
+ 1
2(b−a) Zb
a
f(x)dx. (2.8)
The monotonicity ofR on [0,1], the inequality (2.8) and the first inequality of (1.1) for Wright-convex functions, imply the inequalities (1.7) for
Wright-convex functions.
Remark 2.1. The inequalities (1.7) refine the first inequality of (1.1) for Wright-convex functions.
Theorem 2.2. Let f ∈W([a, b])∩L1[a, b] and letS be defined as in Theorem 1.10. Then S∈W([0,1]) is nondecreasing on [0,1], and the inequalities (1.8) hold for allt∈[0,1].
Proof. If s, t, u, v∈[0,1] withs≤t≤u≤v and t+u=s+v, then, for all x∈[a, b], we have
a≤ a+b
2 −v·b−x
2 ≤ a+b
2 −u·b−x
2 ≤
≤ a+b
2 −t·b−x
2 ≤ a+b
2 −s·b−x
2 ≤ a+b
2 (2.9)
and
a+b
2 ≤ a+b
2 +s·x−a
2 ≤ a+b
2 +t·x−a
2 ≤
≤ a+b
2 +u·x−a
2 ≤ a+b
2 +v·x−a
2 ≤b. (2.10)
Considering
s5:= a+b
2 −v·b−x
2 , t5 := a+b
2 −u·b−x
2 , u5 := a+b
2 −t·b−x 2 , v5 := a+b
2 −s·b−x 2 in (2.9), we note that s5, t5, u5, v5∈
a,a+b2
, withs5 ≤t5≤u5 ≤v5 and t5+u5 =s5+v5. Since f ∈W([a, b]), taking into account the Theorem 1.4, we find
f(t5) +f(u5)≤f(s5) +f(v5) for allx∈[a, b]. (2.11) Putting
s6:= a+b
2 +s·x−a
2 , t6 := a+b
2 +t·x−a
2 , u6 := a+b
2 +u·x−a 2 , v6 := a+b
2 +v·x−a 2 in (2.10), we note that s6, t6, u6, v6∈a+b
2 , b
, with s6 ≤t6 ≤u6≤v6 and t6+u6 =s6+v6. Since f ∈W([a, b]), taking into account the Theorem 1.4, we have
f(t6) +f(u6)≤f(s6) +f(v6) for allx∈[a, b]. (2.12) Adding the inequalities (2.11) and (2.12), integrating overx on [a, b] and multiplying by 2(b1−a) the obtained result, we deduce
S(t) +S(u)≤S(s) +S(v),
namely S∈W([0,1]).
In order to prove the monotonicity ofS on the interval [0,1], we take 0≤t1 < t2≤1. Then, for alx∈[a, b], we have
a≤ a+b
2 −t2·b−x
2 ≤ a+b
2 −t1·b−x
2 ≤ a+b
2 +t1·b−x
2 ≤
≤ a+b
2 +t2·b−x
2 ≤b. (2.13)
Considering
s7 := a+b
2 −t2·b−x
2 , t7 := a+b
2 −t1·b−x
2 , u7 := a+b
2 +t1·b−x 2 , v7:= a+b
2 +t2·b−x 2
in (2.13), we note that s7, t7, u7, v7∈[a, b], with s7 ≤t7≤u7 ≤v7 and t7+u7 =s7+v7. Since f ∈W([a, b]), taking into account the Theorem 1.4, we deduce
f(t7) +f(u7)≤f(s7) +f(v7) for allx∈[a, b]. (2.14) Integrating the last inequality over x on [a, b], we obtain
Zb
a
f
a+b
2 −t1·b−x 2
dx+
Zb
a
f
a+b
2 +t1·b−x 2
dx≤
≤ Zb
a
f
a+b
2 −t2·b−x 2
dx+
Zb
a
f
a+b
2 +t2·b−x 2
dx or
Zb
a
f
a+b
2 −t1·b−x 2
dx+
Zb
a
f
a+b
2 +t1·x−a 2
dx≤
≤ Zb
a
f
a+b
2 −t2·b−x 2
dx+
Zb
a
f
a+b
2 +t2·x−a 2
dx. (2.15)
The inequality (2.15) is equivalent toS(t1)≤S(t2), namelyS is nondecreasing on the interval [0,1].
The monotonicity ofS implies the inequalities (1.8) for Wright-convex functions.
Remark 2.2. The inequalities (1.8) refine the first inequality of (1.1) for Wright-convex functions.
Theorem 2.3. Let f ∈W([a, b])∩L1[a, b] and letT be defined as in Theorem 1.11. Then T ∈W([0,1]) is nonincreasing on [0,1] and the inequalities (1.9) hold for allt∈[0,1].
Proof. If s, t, u, v∈[0,1] withs≤t≤u≤v and t+u=s+v, then the inequalities (2.1) and (2.2) hold. Multiplying those inequalities byp(x), integrating the obtained results: the first one overx on
a,a+b2
and the second one over x ona+b
2 , b
, multiplying by 12 and adding the found inequalities, we deduce
T(t) +T(u)≤T(s) +T(v), namely T ∈W([0,1]).
In order to prove the monotonicity ofT, we take 0≤t1< t2 ≤1. Then, the inequalities (2.3) and (2.4) hold. Multiplying those relations by p(x) and integrating the obtained results, we may write:
Z a+b2
a
f
1 +t2
2 ·a+b
2 +1−t2 2 ·x
p(x)dx+
+ Z a+b2
a
f
1 +t2
2 ·a+b
2 +1−t2
2 ·(a+b−x)
p(a+b−x)dx≤
≤ Z a+b2
a
f
1 +t1
2 ·a+b
2 +1−t1
2 ·x
p(x)dx+
+ Z a+b2
a
f
1 +t1
2 ·a+b
2 +1−t1
2 ·(a+b−x)
p(a+b−x)dx (2.16) and
Z b
a+b 2
f
1 +t2
2 ·a+b
2 +1−t2
2 ·x
p(x)dx+
+ Z b
a+b 2
f
1 +t2
2 ·a+b
2 +1−t2
2 ·(a+b−x)
p(a+b−x)dx≤
≤ Z b
a+b 2
f
1 +t1
2 ·a+b
2 + 1−t1
2 ·x
p(x)dx+
+ Z b
a+b 2
f
1 +t1
2 ·a+b
2 +1−t1
2 ·(a+b−x)
p(a+b−x)dx. (2.17) Adding the inequalities (2.16) and (2.17), we find 2·T(t2)≤2·T(t1),
namely T is nonincreasing on the interval [0,1].
Since f ∈W([a, b]), taking into account the inequality (2.7), we deduce f
a+b 4 +x
2
·p(x)≤ 1 2·f
a+b 2
·p(x) +1
2 ·f(x)·p(x)
for allx∈[a, b]. (2.18)
Integrating the inequality (2.18) overx on [a, b], we find Zb
a
f
a+b 4 + x
2
·p(x)dx≤ 1 2 ·f
a+b 2
Zb
a
p(x)dx+
+1 2·
Zb
a
f(x)·p(x)dx. (2.19)
The monotonicity ofT on [0,1], the inequality (2.19) and the first inequality of (1.2) for Wright-convex functions imply the inequalities (1.9) for
Wright-convex functions.
Remark 2.3. If we set p(x)≡1(x∈[a, b])in Theorem 2.3, then we find Theorem 2.1.
Theorem 2.4, Letf ∈W([a, b])∩L1[a, b] and let U be defined as in Theorem 1.11. Then U ∈W([0,1]) is nondecreasing on [0,1], and the inequalities (1.10) hold for allt∈[0,1].
Proof. If 0≤s≤t≤u≤v≤1 and t+u=s+v, then, for all x∈[a, b], the inequalities (2.11) and (2.12) hold.
Multiplying (2.11) byp x+a2
and integrating the obtained result overx on [a, b], we have
Zb
a
f
a+b
2 −u·b−x 2
·p
x+a 2
dx+
Zb
a
f
a+b
2 −t·b−x 2
·
·p
x+a 2
dx≤
Zb
a
f
a+b
2 −v·b−x 2
·p
x+a 2
dx+
+ Zb
a
f
a+b
2 −s·b−x 2
·p
x+a 2
dx. (2.20)
Multiplying (2.12) byp x+b2
and integrating the obtained result overx on [a, b], we have
Zb
a
f
a+b
2 +t·x−a 2
·p
x+b 2
dx+
Zb
a
f
a+b
2 +u·x−a 2
·p
x+b 2
dx≤
≤ Zb
a
f
a+b
2 +s·x−a 2
·p
x+b 2
dx+
+ Zb
a
f
a+b
2 +v·x−a 2
·p
x+b 2
dx. (2.21)
Adding the inequalities (2.20) and (2.21) and multiplying the result by 12, we find U(t) +U(u)≤U(s) +U(v), namely U ∈W([0,1]).
Next, we take 0≤t1 < t2≤1. Then, the inequality (2.14) holds.
Multiplying the inequality (2.14) byp x+a2
and integrating the obtained result over xon [a, b], we have
Zb
a
f
a+b
2 −t1·b−x 2
·p
x+a 2
dx+
Zb
a
f
a+b
2 +t1·b−x 2
·
·p
x+a 2
dx≤
Zb
a
f
a+b
2 −t2·b−x 2
·p
x+a 2
dx+
+ Zb
a
f
a+b
2 +t2·b−x 2
·p
x+a 2
dx or
Zb
a
f
a+b
2 −t1·b−x 2
·p
x+a 2
dx+
Zb
a
f
a+b
2 +t1·x−a 2
·
·p
2a+b−x 2
dx≤
Zb
a
f
a+b
2 −t2·b−x 2
·p
x+a 2
dx+
+ Zb
a
f
a+b
2 +t2·x−a 2
·p
2a+b−x 2
dx.
Using the symmetry ofp about x= a+b2 in the last inequality, we deduce Zb
a
f
a+b
2 −t1·b−x 2
·p
x+a 2
dx+
Zb
a
f
a+b
2 +t1·x−a 2
·
·p
x+b 2
dx≤
Zb
a
f
a+b
2 −t2·b−x 2
·p
x+a 2
dx+
+ Zb
a
f
a+b
2 +t2·x−a 2
·p
x+b 2
dx
which, multiplied by 12, givesU(t1)≤U(t2), namelyU is nondecreasing on the interval [0,1].
From the monotonicity of U, we deduce the inequalities (1.10) for Wright-convex functions.
Remark 2.4. The Theorem 2.4 is a weighted generalization of Theorem 2.2.
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“Avram Iancu” Secondary School, Cluj-Napoca, Romania
vlad ciobotariu@yahoo.com
