On Hermite-Hadamard type integral inequalities for preinvex and log-preinvex functions

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arXiv:1203.4759v1 [math.FA] 21 Mar 2012

FOR PREINVEX AND LOG-PREINVEX FUNCTIONS

MEHMET ZEKI SARIKAYA, HAKAN BOZKURT, AND NECMETTIN ALP

Abstract. In this paper, we extend some estimates of the left hand side of a Hermite- Hadamard type inequality for nonconvex functions whose derivatives absolute values are preinvex and log-preinvex.

1. Introduction

The following inequality is well-known in the literature as Hermite-Hadamard inequality: Let f : I ⊂ R → R be a convex function on an interval I of real numbers and a, b ∈ I with a < b. Then the following holds

(1.1) f a + b 2 ≤ 1 b − a b Z a f (x) dx ≤ f (a) + f (b) 2 .

Both inequalities hold in the reversed direction if the function f is concave. The inequalities (1.1) have become an important cornerstone in mathematical analysis and optimization and many uses of these inequalities have been discovered in a variety of settings. Recently, Hermite-Hadamard type inequality has been the subject of intensive research. For recent results, refinements, counterparts, generalizations and new Hadamard’s-type inequalities, see ([1], [2], [8]-[11], [16]-[21]).

In [8] some inequalities of Hermite-Hadamard type for differentiable convex map-pings connected with the left part of (1.1) were proved using the following lemma: Lemma 1. Let f : I◦_{⊂ R → R, be a differentiable mapping on I}◦_{, a, b ∈ I}◦ _(I◦ _is

the interior of I) with a < b. If f′_{∈ L ([a, b]), then we have}

(1.2) 1 b − a Z b a f (x)dx − f a + b 2 = (b − a) " Z 12 0 tf′_{(ta + (1 − t)b)dt +} Z 1 1 2 (t − 1) f′_{(ta + (1 − t)b)dt} # .

One more general result related to (1.2) was established in [9]. The main result in [8] is as follows:

2000 Mathematics Subject Classification. 26D07, 26D10, 26D99.

Key words and phrases. Hermite-Hadamard’s inequalities, non-convex functions, invex sets, H¨older’s inequality.

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Theorem 1. Let f : I ⊂ R → R, be a differentiable mapping on I◦_{, a, b ∈ I with}

a < b. If the mapping |f′_{| is convex on [a, b], then}

(1.3) 1 b − a Z b a f (x)dx − f a + b 2 ≤ b − a 4 |f′_{(a)| + |f}′_(b)| 2 .

It is well known that convexity has been playing a key role in mathematical pro-gramming, engineering, and optimization theory. The generalization of convexity is one of the most important aspects in mathematical programming and optimization theory. There have been many attempts to weaken the convexity assumptions in the literature, (see, [1], [2], [8]-[11], [16]-[21]). A significant generalization of convex functions is that of invex functions introduced by Hanson in [12]. Ben-Israel and Mond [14] introduced the concept of preinvex functions, which is a special case of invexity. Pini [15] introduced the concept of prequasiinvex functions as a gener-alization of invex functions. Noor [5]-[7] has established some Hermite-Hadamard type inequalities for preinvex and log-preinvex functions. In recent papers Barani, Ghazanfari, and Dragomir in [3] presented some estimates of the right hand side of a Hermite- Hadamard type inequality in which some preinvex functions are in-volved. His class of nonconvex functions include the classical convex functions and its various classes as special cases. For some recent results related to this nonconvex functions, see the papers ([4]-[7], [12]-[15]).

2. Preliminaries

Let f : K → R, and η(., .) : K × K → R , where K is a nonempty closed set in Rn, be continuous functions. First of all, we recall the following well known results and concepts, see [4]-[7] [13] and the references theirin

Definition 1. Let u ∈ K. Then the set K is said to be invex at u with respect to η(., .), if

u + tη(v, u) ∈ K, ∀u, v ∈ K, t ∈ [0, 1] .

K is said to be an invex set with respect to η, if K is invex at each u ∈ K. The invex set K is also called η-connected set.

Remark 1. We would like to mention that the Definition 1 of an invex set has a clear geometric interpretation. This definition essentially says that there is a path starting from a point u which is contained in K. We do not require that the point v should be one of the end points of the path. This observation plays an important role in our analysis. Note that, if we demand that v should be an end point of the path for every pair of points, u, v ∈ K, then η(v, u) = v − u and consequently invexity reduces to convexity. Thus, it is true that every convex set is also an ˙Invex set with respect to η(v, u) = v − u, but the converse is not necessarily true, see [4]-[7] and the references therein.

Definition 2. The function f on the invex set K is said to be preinvex with respect to η, if

f (u + tη(v, u)) ≤ (1 − t) f (u) + tf (v) , ∀u, v ∈ K, t ∈ [0, 1] .

The function f is said to be preconcave if and only if −f is preinvex. Note that every convex function is an preinvex function, but the converse is not true.

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Definition 3. The function f on the invex set K is said to be logarithmic preinvex with respect to η, such that

f (u + tη(v, u)) ≤ (f (u))1−t(f (v))t, u, v ∈ K, t ∈ [0, 1] where f (.) > 0.

Now we define a new definition for prequasiinvex functions as follows:

Definition 4. The function f on the invex set K is said to be prequasiinvex with respect to η, if

f (u + tη(v, u)) ≤ max {f (u) , f (v)} , u, v ∈ K, t ∈ [0, 1] . From the above definitions, we have

f (u + tη(v, u)) ≤ (f (u))1−t(f (v))t ≤ (1 − t) f (u) + tf (v) ≤ max {f (u) , f (v)} .

We also need the following assumption regarding the function η which is due to Mohan and Neogy [13]:

Condition C_{Let K ⊆ R be an open invex subset with respect to η : K×K → R.} For any x, y ∈ K and any t ∈ [0, 1] ,

η(y, y + tη(x, y)) = −tη(x, y) η(x, y + tη(x, y)) = (1 − t) η(x, y).

Note that for every x, y ∈ K and every t1, t2∈ [0, 1] from Condition C, we have

(2.1) η(y + t2η(x, y), y + t1η(x, y)) = (t2− t1) η(x, y).

In [5], Noor proved the Hermite-Hadamard inequality for the preinvex functions as follows:

Theorem 2. Let f : K = [a, a + η(b, a)] → (0, ∞) be an preinvex function on the interval of real numbers K0 _{(the interior of K) and a, b ∈ K}0 _{with a < a + η(b, a).}

Then the following inequality holds: (2.2) f 2a + η(b, a) 2 ≤ 1 η(b, a) a+η(b,a) Z a f (x) dx ≤ f (a) + f (a + η(b, a)) 2 ≤ f (a) + f (b) 2 .

In [3], Barani, Gahazanfari and Dragomir proved the following theorems. Theorem 3. Let A ⊆ R be an open invex subset with respect to η : A × A → R. Suppose that f : A → R is a diferentiable function.Assume p ∈ R with p > 1. If |f′_|p−1p _{is prequasiinvex on A then, for every a, b ∈ A the following inequality holds}

f (a) + f (a + η(b, a)) 2 − 1 η(a, b) Z b+η(b,a) b f (x)dx ≤ η(b, a) 2(p + 1)1p h supn|f′_(a)|p−1p _{, |f}′_(b)|p−1p oi p p−1

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Theorem 4. Let A ⊆ R be an open invex subset with respect to η : A × A → R. Suppose that f : A → R is a diferentiable function. If |f′_{| is prequasiinvex on A}

then, for every a, b ∈ A the following inequality holds

f (a) + f (a + η(b, a)) 2 − 1 η(a, b) Z b+η(b,a) b f (x)dx ≤ η(b, a) 4 max {|f ′_{(a)| , |f}′_(b)|}

In this article, using functions whose derivatives absolute values are preinvex and log-preinvex, we obtained new inequalities releted to the left side of Hermite-Hadamard inequality for nonconvex functions.

3. Hermite-Hadamard type inequalities for preinvex functions We shall start with the following refinements of the Hermite-Hadamard inequal-ity for preinvex functions. Firstly, we give the following results connected with the left part of (2.2):

Theorem 5. Let K ⊆ R be an open invex subset with respect to η : K × K → R. Suppose that f : K → R is a diferentiable function. If |f′_{| is preinvex on K, then,}

for every a, b ∈ K the following inequality holds:

(3.1) 1 η(b, a) Z a+η(b,a) a f (x)dx − f 2a + η(b, a) 2 ≤ η(b, a) 8 [|f ′_{(a)| + |f}′_(b)|]

Proof. Suppose that a, a+ η(b, a) ∈ K. Since K is invex with respect to η, for every t ∈ [0, 1], we have a + η(b, a) ∈ K. Integrating by parts implies that

Z 12 0 tf′_{(a + tη(b, a))dt +} Z 1 1 2 (t − 1)f′_{(a + tη(b, a))dt} (3.2) = tf (a + tη(b, a)) η(b, a) 12 0 + (t − 1)f (a + tη(b, a)) η(b, a) 1 1 2 − 1 η(b, a) Z 1 0 f (a + tη(b, a))dt = 1 η(b, a)f 2a + η(b, a) 2 − 1 [η(b, a)]2 Z a+η(b,a) a f (x)dx.

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By preinvex function of |f′_{| and (3.2), we have} 1 η(b, a) Z a+η(b,a) a f (x)dx − f 2a + η(b, a) 2 ≤ η(b, a) " Z 12 0 t |f′_{(a + tη(b, a))| dt +} Z 1 1 2 (1 − t) |f′_{(a + tη(b, a))| dt} # ≤ η(b, a) " Z 12 0 t [(1 − t) |f′_{(a)| + t |f}′_{(b)|] dt +} Z 1 1 2 (1 − t) [(1 − t) |f′_{(a)| + t |f}′_{(b)|] dt} # ≤ η(b, a) |f′(a)| + |f′(b)| 8 .

The proof is completed.

Theorem 6. Let K ⊆ R be an open invex subset with respect to η : K × K → R. Suppose that f : K → R is a diferentiable function. Assume p ∈ R with p > 1. If |f′_|p−1p _{is preinvex on K then, for every a, b ∈ K the following inequality holds}

1 η(b, a) Z a+η(b,a) a f (x)dx − f 2a + η(b, a) 2 (3.3) ≤ η(b, a) 16 4 p + 1 1_p 3 |f′_(a)|p−1p _{+ |f}′_(b)|p−1p p−1 p +|f′_(a)|p−1p _{+ 3 |f}′_(b)|p−1p p−1 p .

Proof. Suppose that a, a + η(b, a) ∈ K. By assumption, H¨older’s inequality and (3.2) in the proof of Theorem 5, we have

1 η(b, a) Z a+η(b,a) a f (x)dx − f 2a + η(b, a) 2 ≤ η(b, a) " Z 12 0 t |f′_{(a + tη(b, a))| dt +} Z 1 1 2 (1 − t) |f′_{(a + tη(b, a))| dt} # ≤ η(b, a)   Z 12 0 tp_dt !p1 _Z 1 2 0 |f′_{(a + tη(b, a))|}p−1p _dt !p−1p + Z 1 1 2 (1 − t)pdt !p1 _Z 1 1 2 |f′_{(a + tη(b, a))|}p−1p _dt !p−1p   ≤ η(b, a) 21+1p(p + 1) 1 p   Z 12 0 h (1 − t) |f′_(a)|p−1p _{+ t |f}′_(b)|p−1p i_dt !p−1p + Z 1 1 2 h (1 − t) |f′_(a)|p−1p _{+ t |f}′_(b)|p−1p i dt !p−1_p   = η(b, a) 16 ₄ p + 1 1p 3 |f′_(a)|p−1p _{+ |f}′_(b)|p−1p p−1_p +|f′_(a)|p−1p _{+ 3 |f}′_(b)|p−1p p−1_p

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which completes the proof.

Theorem 7. Under the assumptaions of Theorem 6. Then, for every a, b ∈ K the following inequality holds

1 η(b, a) Z a+η(b,a) a f (x)dx − f 2a + η(b, a) 2 (3.4) ≤ η(b, a) 16 ₄ p + 1 1p (3p−1p + 1) [|f′(a)| + |f′(b)|] .

Proof. We consider the inequality (3.3) i.e.

1 η(b, a) Z a+η(b,a) a f (x)dx − f 2a + η(b, a) 2 ≤ η(b, a) 16 ₄ p + 1 1p 3 |f′_(a)|p−1p _{+ |f}′_(b)|p−1p p−1_p +|f′_(a)|p−1p _{+ 3 |f}′_(b)|p−1p p−1_p . Let a1 = 3 |f′(a)| p p−1_{, b}₁ _{= |f}′_(b)|p−1p _{, a}₂ _{= |f}′_(a)|p−1p _{, b}₂ _{= 3 |f}′_(b)|p−1p _{. Here}

0 < (p − 1) /p < 1, for p > 1. Using the fact that,

n X k=1 (ak+ bk) s ≤ n X k=1 as k+ n X k=1 bs k For (0 ≤ s < 1), a1, a2, ..., an≥ 0, b1, b2, ..., bn≥ 0, we obtain η(b, a) 16 ₄ p + 1 1p 3 |f′_(a)|p−1p _{+ |f}′_(b)|p−1p p−1 p +|f′_(a)|p−1p _{+ 3 |f}′_(b)|p−1p p−1 p ≤ η(b, a) 16 ₄ p + 1 1p (3p−1p + 1) [|f′(a)| + |f′(b)|]

which completed proof.

Theorem 8. _{Let K ⊆ R be an open invex subset with respect to η : K × K → R.} Suppose that f : K → R is a diferentiable function. Assume q ∈ R with q ≥ 1. If |f′_|q _{is preinvex on K then, for every a, b ∈ K the following inequality holds}

1 η(b, a) Z a+η(b,a) a f (x)dx − f 2a + η(b, a) 2 (3.5) ≤ η(b, a) 8 " 2 |f′_(a)|q_{+ |f}′_(b)|q 3 1q + |f′(a)| q + 2 |f′_(b)|q 3 1q# .

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Proof. Suppose that a, a+ η(b, a) ∈ K. By assumption, using the well known power mean inequality and (3.2) in the proof of Theorem 5, we have

1 η(b, a) Z a+η(b,a) a f (x)dx − f 2a + η(b, a) 2 ≤ η(b, a) " Z 12 0 t |f′_{(a + tη(b, a))| dt +} Z 1 1 2 (1 − t) |f′_{(a + tη(b, a))| dt} # ≤ η(b, a)   Z 12 0 tdt !_p1 Z 12 0 t |f′_{(a + tη(b, a))|}q_dt !1_q + Z 1 1 2 (1 − t) dt !1p _Z ₁ 1 2 (1 − t) |f′_{(a + tη(b, a))|}q_dt !1q  ≤ η(b, a) 8p1   Z 1₂ 0 t(1 − t) |f′_(a)|q_{+ t |f}′_(b)|q_dt !1q + Z 1 1 2 (1 − t)(1 − t) |f′_(a)|q_{+ t |f}′_(b)|q_dt !q1  = η(b, a) 8 " 2 |f′_(a)|q_{+ |f}′_(b)|q 3 1q + |f′(a)| q + 2 |f′_(b)|q 3 1q# ,

where 1_p+1_q = 1. The proof is completed.

Theorem 9. Under the assumptions of Theorem 8. Then the following inequality holds: (3.6) 1 η(b, a) Z a+η(b,a) a f (x)dx − f 2a + η(b, a) 2 ≤η(b, a) 8 ( 21q + 1 3q1 ) [|f′_{(a)| + |f}′_(b)|]

Proof. We consider the inequality (3.5), i.e.

1 η(b, a) Z a+η(b,a) a f (x)dx − f 2a + η(b, a) 2 ≤ η(b, a) 8 " 2 |f′_(a)|q_{+ |f}′_(b)|q 3 1q + |f′(a)| q + 2 |f′_(b)|q 3 1q# .

Let a1 = 2 |f′(a)|q/3, b1 = |f′(b)|q/3, a2 = |f′(a)|q/3, b2 = 2 |f′(b)|q/3. Here

0 < 1/q < 1, for q ≥ 1. Using the fact that

n X k=1 (ak+ bk)s≤ n X k=1 as k+ n X k=1 bs k.

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For (0 ≤ s < 1), a1, a2, ..., an≥ 0, b1, b2, ..., bn≥ 0, we obtain η(b, a) 8 " 2 |f′_(a)|q_{+ |f}′_(b)|q 3 1q + |f′(a)| q + 2 |f′_(b)|q 3 1q# ≤ η(b, a) 8 ( 21q + 1 31q ) [|f′_{(a)| + |f}′_{(b)|] .} 4. Hermite-Hadamard type inequalities for log-preinvex function

In this section, we shall continue with the following refinements of the Hermite-Hadamard inequality for log-preinvex functions and we give some results connected with the left part of (2.2):

Theorem 10. _{Let K ⊆ R be an open invex subset with respect to η : K × K → R.} Suppose that f : K → R is a diferentiable function. If |f′_{| is log-preinvex on K}

then, for every a, b ∈ K the following inequality holds 1 η(b, a) Z a+η(b,a) a f (x)dx − f 2a + η(b, a) 2 ≤ η(b, a) |f′(b)| 1 2 _{− |f}′_(a)|12

log |f′_{(b)| − log |f}′_(a)|

!2

Proof. Suppose that a, a + η(b, a) ∈ K. By assumption and (3.2) in the proof of Theorem 5, integrating by parts implies that

1 η(b, a) Z a+η(b,a) a f (x)dx − f 2a + η(b, a) 2 ≤ η(b, a) " Z 12 0 t |f′_{(a + tη(b, a))| dt +} Z 1 1 2 (1 − t) |f′_{(a + tη(b, a))| dt} # ≤ η(b, a) " Z 12 0 t |f′_(a)|1−t_|f′_(b)|t_{dt +} Z 1 1 2 (1 − t) |f′_(a)|1−t_|f′_(b)|t_dt # = η(b, a) " Z 12 0 |f′_{(a)| t} |f′(b)| |f′_(a)| t dt + Z 1 1 2 (1 − t) |f′_(b)| |f′(b)| |f′_(a)| 1−t dt # = η(b, a)   |f′_(a)|

log |f′_{(b)| − log |f}′_(a)|

"

− 1

log |f′_{(b)| − log |f}′_(a)|

|f′_(b)| |f′_(a)| t# 1 2 0 + " 1

log |f′_{(b)| − log |f}′_(a)|

|f′_(b)| |f′_(a)| t#1 1 2   = η(b, a) " −2 |f′_(a)|12_|f′_(b)|12

(log |f′_{(b)| − log |f}′_(a)|)2 +

|f′_(a)|

(log |f′_{(b)| − log |f}′_(a)|)2

+ |f′(a)|

(log |f′_{(b)| − log |f}′_(a)|)2

#

= η(b, a) "

|f′_(b)|12 _{− |f}′_(a)|12

log |f′_{(b)| − log |f}′_(a)|

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which completes the proof.

Theorem 11. Let K ⊆ R be an open invex subset with respect to η : K × K → R. Suppose that f : K → R is a diferentiable function. Assume q ∈ R with q ≥ 1. If |f′_|q _{is log-preinvex on K then, for every a, b ∈ K the following inequality holds}

1 η(b, a) Z a+η(b,a) a f (x)dx − f 2a + η(b, a) 2 ≤ η(b, a)   |f′_(a)|12 21p(p + 1) 1 p_q1q |f′_(b)|q2 − |f′(a)| q 2

log |f′_{(b)| − log |f}′_(a)|

!1_q .

Proof. By H¨older inequality and (3.2) in the proof of Theorem 5, we have

1 η(b, a) Z η(b,a) a f (x)dx − f 2a + η(b, a) 2 ≤ η(b, a) " Z 12 0 t |f′_{(a + tη(b, a))| dt +} Z 1 1 2 (1 − t) |f′_{(a + tη(b, a))| dt} # ≤ η(b, a)   Z 12 0 tp_dt !p1 _Z 1 2 0 |f′_{(a + tη(b, a))|}q !1q dt + Z 1 1 2 (1 − t)p !p1 _Z ₁ 1 2 |f′_{(a + tη(b, a))|}q_dt !1q  ≤ η(b, a)   Z 12 0 tp_dt !_p1 Z 12 0 |f′_(a)|1−t_|f′_(b)|tq_dt !1_q + Z 1 1 2 (1 − t)p !1p _Z ₁ 1 2 |f′_(a)|1−t_|f′_(b)|tq_dt !1q  = η(b, a)   |f′_(a)|12 21p(p + 1) 1 p_q1q |f′_(b)|q2 − |f′(a)| q 2

log |f′_{(b)| − log |f}′_(a)|

!1_q 

where 1_p+1_q = 1.

Now, we give the followig results connected with the left part of (1.1) for classical log-convex functions.

Corollary 1. Under the assumptions of Theorem 10 with η(b, a) = b − a, then the following inequality holds:

1 b − a Z b a f (x)dx − f a + b 2 ≤ (b − a) |f′(b)| 1 2 − |f′(a)| 1 2

log |f′_{(b)| − log |f}′_(a)|

!2 .

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Corollary 2. Under the assumptions of Theorem 11 with η(b, a) = b − a, then the following inequality holds:

1 b − a Z b a f (x)dx − f a + b 2 ≤ (b − a)   |f′_(a)|12 2p1(p + 1) 1 p_q1q |f′_(b)|q2 − |f′(a)| q 2

log |f′_{(b)| − log |f}′_(a)|

!_q1 .

5. An extension to several variables functions

In this section, we shall extend the Corollary 1 and Corollary 2 to functions of several variables defined on invex subsets of Rn

Let K ⊆ Rn_{be an invex set with respect to η : K × K → R}n_{. For every x, y ∈ K}

the η-path Pxv joining the points x and v := x + η(y, x) is defined as follows

Pxv= {z : z = x + tη(y, x) : t ∈ [0, 1]} .

Proposition 1. Let K ⊆ Rn _{be an invex set with respect to η : K × K → R}n _and

f : K → R is a functio. Suppose that η satisfies Condition C on K. Then for every x, y ∈ K the function f is log-preinvex with respect to η on η-path Pxv if and only

if the function ϕ : [0, 1] → R defined by

ϕ (t) := f (x + tη(y, x)) , is log-convex on [0, 1] .

Proof. Suppose that ϕ is log-convex on [0, 1] and z1 := x + t1η(y, x) ∈ Pxv, z2:=

x + t2η(y, x) ∈ Pxv. Fix λ ∈ [0, 1]. By (2.1), we have

f (z1+ λη(z2, z1)) = f (x + ((1 − λ) t1+ λt2) η(y, x))

= ϕ ((1 − λ) t1+ λt2)

≤ [ϕ (t1)](1−λ)[ϕ (t2)]λ

= [f (z1)](1−λ)[f (z2)]λ

Hence, f is log-preinvex with respect to η on η-path Pxv.

Conversely, let x, y ∈ K and the function f be log-preinvex with respect to η on η-path Pxv. Suppose that t1, t2∈ [0, 1]. Then, for every λ ∈ [0, 1] we have

ϕ ((1 − λ) t1+ λt2) = f (x + ((1 − λ) t1+ λt2) η(y, x))

= f (x + t1η(y, x) + λη(x + t2η(y, x), x + t1η(y, x)))

≤ [f (x + t1η(y, x))](1−λ)[f (x + t2η(y, x))]λ

= [ϕ (t1)](1−λ)[ϕ (t2)]λ

Therefore, ϕ is log-convex on [0, 1].

The following Teorem is a generalization of Corollary 1. Theorem 12. _{Let K ⊆ R}n

be an invex set with respect to η : K × K → Rn _and

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every x, y ∈ K the function f is log-preinvex with respect to η on η-path Pxv. Then,

for every a, b ∈ (0, 1) with a < b the following inequality holds 1 b − a Z b a Z s 0 f (x + tη(y, x)) dt ds − Z a+b2 0 f (x + sη(y, x)) ds (5.1) ≤ (b − a) " [f (x + bη(y, x))]12 − [f (x + aη(y, x))] 1 2

log f (x + bη(y, x)) − log f (x + aη(y, x)) #2

.

Proof. Let x, y ∈ K and a, b ∈ (0, 1) with a < b. Since f is log-preinvex with respect to η on η-path Pxvby Proposition 1 the function ϕ : [0, 1] → R+defined by

ϕ (t) := f (x + tη(y, x)) ,

is log-convex on [0, 1]. Now, we define the function φ : [0, 1] → R+_{as follows}

φ (t) := Z t 0 ϕ (s) ds = Z t 0 f (x + sη(y, x)) ds.

Obviously for every t ∈ (0, 1) we have

φ′_{(t) = ϕ (t) = f (x + tη(y, x)) ≥ 0}

hence,φ′(t)

= φ′(t). Applying Corollary 1 to the function φ implies that 1 b − a Z b a φ (t) dt − φ a + b 2 ≤ (b − a)   φ′_(b) 1 2 ₋_φ′_(a) 1 2 log φ′_(b) − log φ′_(a)   2

and we deduce that (5.1) holds.

Remark 2. Let ϕ (t) : [0, 1] → R+_{be a function and q a positive real number, then}

ϕ is log-convex if and only if the function ϕq_{(t) : [0, 1] → R}+ _{is log-convex. ˙Indeed}

for every x, y ∈ [0, 1] it is easy to see that h

[ϕ (x)]1−t[ϕ (y)]ti

q

= [ϕq(x)]1−t[ϕq(y)]t Therefore if t ∈ [0, 1] , we have

ϕ (tx + (1 − t)y) ≤ [ϕ (x)]1−t[ϕ (y)]tif and only if ϕq_{(tx + (1 − t)y) ≤ [ϕ}q_(x)]1−t_[ϕq_(y)]t

. The following Theorem is a generalization Corollory 2 to functions several

vari-ables.

Theorem 13. Let K ⊆ Rn _{be an invex set with respect to η : K × K → R}n _and

f : K → R+ _{is a function. Suppose that η satisfies condition C on K. Then for}

every x, y ∈ K the function f is log-preinvex with respect to η on η-path Pxv. Then,

for every p > 1 and a, b ∈ (0, 1) with a < b the following inequality holds 1 b − a Z b a Z s 0 f (x + tη(y, x)) dt ds − Z a+b2 0 f (x + sη(y, x)) ds (5.2) ≤ (b − a)   [f (x + aη(y, x))]12 2p1(p + 1) 1 p_q 1 q [f (x + bη(y, x))]q2 _{− [f (x + aη(y, x))]} q 2

log f (x + bη(y, x)) − log f (x + aη(y, x)) !1q  where 1 p+ 1 q = 1.

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Proof. Let x, y ∈ K and a, b ∈ (0, 1) with a < b. Suppose that φ and ϕ are the functions whixh are defined in the Theorem 12. Since φ′

: [0, 1] → R+ is log-convex on [0, 1], by Remark 2 the function

φ′

q

is also is log-convex on [0, 1]. Now, by applying Corollary 2 to function φ we get

1 b − a Z b a φ(x)dx − φ a + b 2 ≤ (b − a)   φ′_(a) 1 2 21p(p + 1) 1 p_q 1 q φ′_(b) q 2 ₋_φ′_(a) q 2 log φ′_(b) − log φ′_(a) ! 1 q 

and we deduce that (5.2) holds. The proof is complete.

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Department of Mathematics, Faculty of Science and Arts, D¨uzce University, D¨ uzce-TURKEY

E-mail address: sarikayamz@gmail.com E-mail address: insedi@yahoo.com E-mail address: placenn@gmail.com