Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 507560, 11 pages
doi:10.1155/2010/507560
Research Article

The Hermite-Hadamard Type Inequality of GA-Convex Functions and Its Application

Xiao-Ming Zhang, Yu-Ming Chu, and Xiao-Hui Zhang

Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China

Received 7 April 2009; Revised 29 January 2010; Accepted 1 February 2010

Academic Editor: Andrea Laforgia

Copyright © 2010 Xiao-Ming Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We established a new Hermit-Hadamard type inequality for GA-convex functions. As applications, we obtain two new Gautschi type inequalities for gamma function.

1. Introduction

Let 𝑓 be a convex (concave) function on [ 𝑎 , 𝑏 ] ⊆ ℝ ; the well-known Hermite-Hadamard's inequality [1] can be expressed as

𝑓 𝑎 + 𝑏 2 1 ≤ ( ≥ ) 𝑏 − 𝑎 𝑏 𝑎 𝑓 ( 𝑡 ) 𝑑 𝑡 ≤ ( ≥ ) 𝑓 ( 𝑎 ) + 𝑓 ( 𝑏 ) 2 . ( 1 . 1 )

Recently, Hermite-Hadamard's inequality has been the subject of intensive research. In particular, many improvements, generalizations, and applications for the Hermite-Hadamard's inequality can be found in the literature [2–20].

Let 𝐼 ⊆ ( 0 , ∞ ) be an interval; a real-valued function 𝑓 ∶ 𝐼 → ℝ is said to be GA-convex (concave) on 𝐼 if 𝑓 ( 𝑥 𝛼 𝑦 1 − 𝛼 ) ≤ ( ≥ ) 𝛼 𝑓 ( 𝑥 ) + ( 1 − 𝛼 ) 𝑓 ( 𝑦 ) for all 𝑥 , 𝑦 ∈ 𝐼 and 𝛼 ∈ [ 0 , 1 ] .

In [21], Anderson et al. discussed the GA and related kinds of convexity; some applications to special functions were presented.

For 𝑏 > 𝑎 > 0 , let √ 𝐺 ( 𝑎 , 𝑏 ) = 𝑎 𝑏 , 𝐿 ( 𝑎 , 𝑏 ) = ( 𝑏 − 𝑎 ) / ( l o g 𝑏 − l o g 𝑎 ) , 𝐼 ( 𝑎 , 𝑏 ) = 1 / 𝑒 ( 𝑏 𝑏 / 𝑎 𝑎 ) 1 / ( 𝑏 − 𝑎 ) , and 𝐴 ( 𝑎 , 𝑏 ) = ( 𝑎 + 𝑏 ) / 2 be the geometric, logarithmic, identric, and arithmetic means of 𝑎 and 𝑏 , respectively. Then

m i n { 𝑎 , 𝑏 } < 𝐺 ( 𝑎 , 𝑏 ) < 𝐿 ( 𝑎 , 𝑏 ) < 𝐼 ( 𝑎 , 𝑏 ) < 𝐴 ( 𝑎 , 𝑏 ) < m a x { 𝑎 , 𝑏 } . ( 1 . 2 )

The first purpose of this paper is to establish the following new Hermite-Hadamard type inequality for GA-convex (concave) functions.

Theorem 1.1. If 𝑏 > 𝑎 > 0 and 𝑓 ∶ [ 𝑎 , 𝑏 ] → ℝ is a differentiable GA-convex (concave) function, then 1 𝑓 ( 𝐼 ( 𝑎 , 𝑏 ) ) ≤ ( ≥ ) 𝑏 − 𝑎 𝑏 𝑎 𝑓 ( 𝑡 ) 𝑑 𝑡 ≤ ( ≥ ) 𝑏 − 𝐿 ( 𝑎 , 𝑏 ) 𝑏 − 𝑎 𝑓 ( 𝑏 ) + 𝐿 ( 𝑎 , 𝑏 ) − 𝑎 𝑏 − 𝑎 𝑓 ( 𝑎 ) . ( 1 . 3 )

For real and positive values of 𝑥 , the Euler gamma function Γ and its logarithmic derivative 𝜓 , the so-called digamma function, are defined by

Γ ( 𝑥 ) = 0 + ∞ 𝑡 𝑥 − 1 𝑒 − 𝑡 Γ 𝑑 𝑡 , 𝜓 ( 𝑥 ) = ( 𝑥 ) . Γ ( 𝑥 ) ( 1 . 4 )

The ratio Γ ( 𝑠 ) / Γ ( 𝑟 ) ( 𝑠 > 𝑟 > 0 ) has attracted the attention of many mathematicians and physicists. Gautschi [22] first proved that

𝑛 1 − 𝑠 < Γ ( 𝑛 + 1 ) [ ] Γ ( 𝑛 + 𝑠 ) < e x p ( 1 − 𝑠 ) 𝜓 ( 𝑛 + 1 ) ( 1 . 5 ) for 0 < 𝑠 < 1 and 𝑛 = 1 , 2 , 3 … .

A strengthened upper bound was given by Erber [23]:

Γ ( 𝑛 + 1 ) Γ < ( 𝑛 + 𝑠 ) 4 ( 𝑛 + 𝑠 ) ( 𝑛 + 1 ) 1 − 𝑠 4 𝑛 + ( 𝑠 + 1 ) 2 . ( 1 . 6 )

In [24], Kečkić and Vasić established the following double inequality for 𝑏 > 𝑎 > 0 :

𝑏 𝑏 − 1 𝑎 𝑎 − 1 𝑒 𝑎 − 𝑏 < Γ ( 𝑏 ) < 𝑏 Γ ( 𝑎 ) 𝑏 − 1 / 2 𝑎 𝑎 − 1 / 2 𝑒 𝑎 − 𝑏 . ( 1 . 7 )

In [25], Kershaw obtained

e x p ( 1 − 𝑠 ) 𝜓 𝑥 + 𝑠 1 / 2 < Γ ( 𝑥 + 1 ) Γ ( 𝑥 + 𝑠 ) < e x p ( 1 − 𝑠 ) 𝜓 𝑥 + 𝑠 + 1 2 , 𝑠 𝑥 + 2 1 − 𝑠 < Γ ( 𝑥 + 1 ) Γ < 1 ( 𝑥 + 𝑠 ) 𝑥 − 2 + 1 𝑠 + 4 1 / 2 1 − 𝑠 ( 1 . 8 ) for 𝑥 > 0 and 0 < 𝑠 < 1 .

In [26], Zhang and Chu proved

l o g Γ ( 𝑏 ) − l o g Γ ( 𝑎 ) > 𝑏 − 𝑎 𝑏 − 𝐿 ( 𝑎 , 𝑏 ) 𝑏 − 𝑎 𝜓 ( 𝑏 ) + 𝐿 ( 𝑎 , 𝑏 ) − 𝑎 𝑏 − 𝑎 𝜓 ( 𝑎 ) ( 1 . 9 ) for all 𝑏 > 𝑎 > 0 .

In [27], Zhang and Chu presented

𝜓 ( 𝐿 ( 𝑎 , 𝑏 ) ) < l o g Γ ( 𝑏 ) − l o g Γ ( 𝑎 ) 𝑏 − 𝑎 < 𝜓 ( 𝐿 ( 𝑎 , 𝑏 ) ) + l o g 𝐼 ( 𝑎 , 𝑏 ) 𝐿 ( 𝑎 , 𝑏 ) ( 1 . 1 0 ) for all 𝑏 > 𝑎 > 0 .

The second purpose of this paper is to establish the following two new Gautschi type inequalities by using Theorem 1.1.

Theorem 1.2. If 𝑏 > 𝑎 > 0 , then 𝐼 𝜓 ( 𝐼 ( 𝑎 , 𝑏 ) ) − ( 𝑎 , 𝑏 ) − 𝐿 ( 𝑎 , 𝑏 ) − 𝐼 2 𝐼 ( 𝑎 , 𝑏 ) 𝐿 ( 𝑎 , 𝑏 ) 2 ( 𝑎 , 𝑏 ) − 𝐺 2 ( 𝑎 , 𝑏 ) 1 2 𝐼 2 ( 𝑎 , 𝑏 ) 𝐺 2 ≤ ( 𝑎 , 𝑏 ) l o g Γ ( 𝑏 ) − l o g Γ ( 𝑎 ) 𝑏 − 𝑎 ≤ 𝜓 ( 𝐼 ( 𝑎 , 𝑏 ) ) − 𝐼 ( 𝑎 , 𝑏 ) − 𝐿 ( 𝑎 , 𝑏 ) . 2 𝐼 ( 𝑎 , 𝑏 ) 𝐿 ( 𝑎 , 𝑏 ) ( 1 . 1 1 )

Theorem 1.3. If 𝑏 > 𝑎 > 0 , then 𝑏 − 𝐿 ( 𝑎 , 𝑏 ) 𝐿 𝑏 − 𝑎 𝜓 ( 𝑏 ) + ( 𝑎 , 𝑏 ) − 𝑎 𝐿 𝑏 − 𝑎 𝜓 ( 𝑎 ) + 2 ( 𝑎 , 𝑏 ) − 𝐺 2 ( 𝑎 , 𝑏 ) 2 𝐿 ( 𝑎 , 𝑏 ) 𝐺 2 ≤ ( 𝑎 , 𝑏 ) l o g Γ ( 𝑏 ) − l o g Γ ( 𝑎 ) ≤ 𝑏 − 𝑎 𝑏 − 𝐿 ( 𝑎 , 𝑏 ) 𝜓 𝐿 𝑏 − 𝑎 ( 𝑏 ) + ( 𝑎 , 𝑏 ) − 𝑎 𝜓 𝐿 𝑏 − 𝑎 ( 𝑎 ) + 2 ( 𝑎 , 𝑏 ) − 𝐺 2 ( 𝑎 , 𝑏 ) 2 𝐿 ( 𝑎 , 𝑏 ) 𝐺 2 + 𝐿 ( 𝑎 , 𝑏 ) ( 𝑎 , 𝑏 ) 𝐴 ( 𝑎 , 𝑏 ) − 𝐺 2 ( 𝑎 , 𝑏 ) 6 𝐺 4 . ( 𝑎 , 𝑏 ) ( 1 . 1 2 )

2. Lemmas

In order to establish our main results we need several lemmas, which we present in this section.

Lemma 2.1. One has ∑ ∞ 𝑛 = 1 1 / 𝑛 3 < 1 . 2 0 3 .

Proof. Simple computations lead to ∞ 𝑛 = 1 1 𝑛 3 = 2 0 𝑛 = 1 1 𝑛 3 + ∞ 𝑛 = 2 1 1 𝑛 3 < 2 0 𝑛 = 1 1 𝑛 3 + ∞ 𝑛 = 2 1 1 = ( 𝑛 − 1 ) 𝑛 ( 𝑛 + 1 ) 2 0 𝑛 = 1 1 𝑛 3 + ∞ 𝑛 = 2 1 1 − 1 2 𝑛 ( 𝑛 − 1 ) = 2 𝑛 ( 𝑛 + 1 ) 2 0 𝑛 = 1 1 𝑛 3 + 1 2 × 2 0 × 2 1 = 1 . 2 0 2 ⋯ < 1 . 2 0 3 . ( 2 . 1 )

Lemma 2.2 (see [28, Lemma 2 . 1 ]). If 𝑥 > 0 , then 𝜓 1 ( 𝑥 ) = 𝑥 + 1 2 𝑥 2 + 𝑚 𝑛 = 1 ( − 1 ) 𝑛 − 1 𝐵 𝑛 𝑥 2 𝑛 + 1 + ( − 1 ) 𝑚 𝜃 1 𝐵 𝑚 + 1 𝑥 2 𝑚 + 3 , 𝜓 ( 2 . 2 ) 1 ( 𝑥 ) = − 𝑥 2 − 1 𝑥 3 + 𝑚 𝑛 = 1 ( − 1 ) 𝑛 ( 2 𝑛 + 1 ) 𝐵 𝑛 𝑥 2 𝑛 + 2 + ( − 1 ) 𝑚 + 1 𝜃 2 ( 2 𝑚 + 3 ) 𝐵 𝑚 + 1 𝑥 2 𝑚 + 4 , ( 2 . 3 ) where 0 < 𝜃 1 , 𝜃 2 < 1 , 𝑚 ≥ 1 , 𝑚 ∈ ℕ , 𝐵 1 = 1 / 6 ,   𝐵 2 = 1 / 3 0 ,   𝐵 3 = 1 / 4 2 ,   𝐵 4 = 1 / 3 0 , … .

Lemma 2.3. Suppose that 𝐼 ⊆ ( 0 , ∞ ) is an interval and 𝑓 ∶ 𝐼 → ℝ is a real-valued function. If 𝑓 is second-order differentiable on 𝐼 , then 𝑓 is GA-convex (concave) on 𝐼 if and only if 𝑓 ( 𝑥 ) + 𝑥 𝑓 ( 𝑥 ) ≥ ( ≤ ) 0 ( 2 . 4 ) for all 𝑥 ∈ 𝐼 .

Proof. Lemma 2.3 follows easily from the basic properties of convex (concave) functions and the fact that 𝑓 is GA-convex (concave) on 𝐼 if and only if 𝑔 ( 𝑥 ) = 𝑓 ( 𝑒 𝑥 ) is convex (concave) on 𝐽 = { l o g 𝑥 ∶ 𝑥 ∈ 𝐼 } .

Lemma 2.4 (see [29, Theorem 3 ]). If 𝑥 > 0 , then 0 < 𝑥 2 𝜓 ( 𝑥 + 1 ) + 𝑥 3 𝜓 1 ( 𝑥 + 1 ) < 2 . ( 2 . 5 )

Lemma 2.5. 𝜓 ( 𝑥 ) + 1 / 2 𝑥 is GA-concave on ( 0 , ∞ ) .

Proof. Differentiating the well-known identity Γ ( 𝑥 + 1 ) = 𝑥 Γ ( 𝑥 ) we get 𝜓 1 ( 𝑥 + 1 ) = − 𝑥 2 + 𝜓 𝜓 ( 𝑥 ) , 2 ( 𝑥 + 1 ) = 𝑥 3 + 𝜓 ( 𝑥 ) . ( 2 . 6 ) From inequalities (2.5) and (2.6) we have 𝑥 2 𝜓 ( 𝑥 ) + 𝑥 3 𝜓 1 ( 𝑥 ) + 2 < 0 . ( 2 . 7 ) Inequality (2.7) leads to 1 𝜓 ( 𝑥 ) + 2 𝑥 1 + 𝑥 𝜓 ( 𝑥 ) + 2 𝑥 = 1 𝑥 2 𝑥 2 𝜓 ( 𝑥 ) + 𝑥 3 𝜓 1 ( 𝑥 ) + 2 < 0 . ( 2 . 8 )

Therefore, Lemma 2.5 follows from (2.8) and Lemma 2.3.

Lemma 2.6. 𝜓 ( 𝑥 ) + 1 / 2 𝑥 + 1 / 1 2 𝑥 2 is GA-convex on ( 0 , ∞ ) .

Proof. Simple computation leads to 1 𝜓 ( 𝑥 ) + + 1 2 𝑥 1 2 𝑥 2 1 + 𝑥 𝜓 ( 𝑥 ) + + 1 2 𝑥 1 2 𝑥 2 = 𝜓 ( 𝑥 ) + 𝑥 𝜓 1 ( 𝑥 ) + 2 𝑥 2 + 1 3 𝑥 3 . ( 2 . 9 ) From (2.9) and Lemma 2.3 we know that we need only to prove that 𝜓 ( 𝑥 ) + 𝑥 𝜓 1 ( 𝑥 ) + 2 𝑥 2 + 1 3 𝑥 3 ≥ 0 . ( 2 . 1 0 )
We divide the proof into three cases.
Case 1. √ 𝑥 ∈ [ 5 / 2 , ∞ ) . Taking 𝑚 = 2 in (2.2) and 𝑚 = 3 in (2.3) we get 𝜓 1 ( 𝑥 ) > 𝑥 + 1 2 𝑥 2 + 1 6 𝑥 3 − 1 3 0 𝑥 5 , 𝜓 ( 2 . 1 1 ) 1 ( 𝑥 ) > − 𝑥 2 − 1 𝑥 3 − 1 2 𝑥 4 + 1 6 𝑥 6 − 1 6 𝑥 8 . ( 2 . 1 2 ) Inequalities (2.11) and (2.12) together with √ 𝑥 ≥ 5 / 2 lead to 𝜓 ( 𝑥 ) + 𝑥 𝜓 1 ( 𝑥 ) + 2 𝑥 2 + 1 3 𝑥 3 > 2 1 5 𝑥 7 𝑥 2 − 5 4 ≥ 0 . ( 2 . 1 3 ) Case 2. √ 𝑥 ∈ [ 1 , 5 / 2 ) . It is well-known that l o g Γ ( 𝑥 ) = − 𝛾 𝑥 + ∞ 𝑘 = 1 𝑥 𝑘 𝑥 − l o g 1 + 𝑘 − l o g 𝑥 , ( 2 . 1 4 ) where 𝛾 = 0 . 5 7 7 2 1 5 ⋯ is Euler's constant.
Differentiating (2.14) we get 𝜓 ′ ( 𝑥 ) = ∞ 𝑘 = 0 1 ( 𝑘 + 𝑥 ) 2 , 𝜓 ( 2 . 1 5 ) ( 𝑥 ) = − ∞ 𝑘 = 0 2 ( 𝑘 + 𝑥 ) 3 . ( 2 . 1 6 )
We clearly see that 𝑥 2 ( 𝑘 − 𝑥 ) / ( 𝑘 + 𝑥 ) 3 is increasing in √ [ 1 , 5 / 2 ) for 𝑘 ≥ 3 ; hence (2.15) and (2.16) lead to 𝑥 3 𝜓 ( 𝑥 ) + 𝑥 4 𝜓 ( 𝑥 𝑥 ) + 2 + 1 3 = ∞ 𝑘 = 0 𝑥 3 ( 𝑘 − 𝑥 ) ( 𝑘 + 𝑥 ) 3 + 𝑥 2 + 1 3 = 1 3 − 𝑥 2 + 𝑥 3 − 𝑥 4 ( 1 + 𝑥 ) 3 + 𝑥 3 ( 2 − 𝑥 ) ( 2 + 𝑥 ) 3 + 𝑥 ∞ 𝑘 = 3 𝑥 2 ( 𝑘 − 𝑥 ) ( 𝑘 + 𝑥 ) 3 ≥ 1 3 − 𝑥 2 + 𝑥 3 − 𝑥 4 ( 1 + 𝑥 ) 3 + 𝑥 3 ( 2 − 𝑥 ) ( 2 + 𝑥 ) 3 + 𝑥 ∞ 𝑘 = 3 𝑘 − 1 ( 𝑘 + 1 ) 3 = 1 3 − 𝑥 2 + 𝑥 3 − 𝑥 4 ( 1 + 𝑥 ) 3 + 𝑥 3 ( 2 − 𝑥 ) ( 2 + 𝑥 ) 3 + 𝑥 ∞ 𝑘 = 3 1 ( 𝑘 + 1 ) 2 − 2 ( 𝑘 + 1 ) 3 . ( 2 . 1 7 )
It follows from inequality (2.17), Lemma 2.1, ∑ ∞ 𝑛 = 1 1 / 𝑛 2 = 𝜋 2 / 6 , and √ 𝑥 ∈ [ 1 , 5 / 2 ) that 𝑥 3 𝜓 ′ ( 𝑥 ) + 𝑥 4 𝜓 𝑥 ( 𝑥 ) + 2 + 1 3 > 1 3 − 𝑥 2 + 𝑥 3 − 𝑥 4 ( 1 + 𝑥 ) 3 + 𝑥 3 ( 2 − 𝑥 ) ( 2 + 𝑥 ) 3 𝜋 + 𝑥 2 6 1 − 1 − 4 − 1 9 1 − 2 1 . 2 0 3 − 1 − 8 − 1 = 1 2 7 3 + 𝑥 3 − 𝑥 4 ( 1 + 𝑥 ) 3 + 𝑥 3 ( 2 − 𝑥 ) ( 2 + 𝑥 ) 3 > 1 − 𝑥 × 0 . 2 9 8 1 ⋯ 3 − 0 . 2 9 8 2 𝑥 + 1 − 𝑥 ( 1 + 1 / 𝑥 ) 3 + 2 − 𝑥 ( 1 + 2 / 𝑥 ) 3 > 1 3 √ − 0 . 2 9 8 2 × 5 2 + √ 1 − 5 / 2 √ 1 + 2 / 5 3 + √ 2 − 5 / 2 2 7 = 0 . 0 1 5 2 4 ⋯ > 0 . ( 2 . 1 8 ) Case 3. 𝑥 ∈ ( 0 , 1 ) . Since ( 𝑘 − 𝑥 ) / ( 𝑘 + 𝑥 ) 3 is decreasing in [ 0 , 1 ] for 𝑘 ≥ 1 , hence (2.15) and (2.16) imply that 𝑥 3 𝜓 ′ ( 𝑥 ) + 𝑥 4 𝜓 ( 𝑥 𝑥 ) + 2 + 1 3 = 𝑥 3 ∞ 𝑘 = 1 𝑘 − 𝑥 ( 𝑘 + 𝑥 ) 3 + 1 3 − 𝑥 2 ≥ 1 3 − 𝑥 2 + 𝑥 3 ∞ 𝑘 = 1 𝑘 − 1 ( 𝑘 + 1 ) 3 = 1 3 − 𝑥 2 + 𝑥 3 ∞ 𝑘 = 1 1 ( 1 + 𝑘 ) 2 − 2 ( 𝑘 + 1 ) 3 . ( 2 . 1 9 ) From (2.19), Lemma 2.1, ∑ ∞ 𝑛 = 1 1 / 𝑛 2 = 𝜋 2 / 6 , and 𝑥 ∈ ( 0 , 1 ) we get 𝑥 3 𝜓 ′ ( 𝑥 ) + 𝑥 4 𝜓 𝑥 ( 𝑥 ) + 2 + 1 3 > 1 3 − 𝑥 2 + 𝑥 3 𝜋 2 6 = 1 − 1 − 2 ( 1 . 2 0 3 − 1 ) 3 − 𝑥 2 + 𝑥 3 > 1 × 0 . 2 3 8 9 3 4 ⋯ 3 − 𝑥 2 + 0 . 2 3 8 𝑥 3 . ( 2 . 2 0 )
It is not difficult to verify that m i n [ ] 𝑥 ∈ 0 , 1 1 3 − 𝑥 2 + 0 . 2 3 8 𝑥 3 = 1 3 − 1 2 1 1 . 4 2 8 + 0 . 2 3 8 1 1 . 4 2 8 3 = 0 . 0 5 4 3 9 0 ⋯ > 0 . ( 2 . 2 1 )
Therefore, inequality (2.10) follows from (2.20) and (2.21).

3. Proof of Theorems 1.1, 1.2, and 1.3

Proof of Theorem 1.1. Suppose that 𝑓 is a GA-convex function. For any fixed 𝑐 ∈ ( 𝑎 , 𝑏 ) , if 𝑥 ∈ [ 𝑐 , 𝑏 ] , then 𝑔 ( 𝑡 ) = 𝑓 ( 𝑒 𝑡 ) is convex on [ l o g 𝑐 , l o g 𝑥 ] and 𝑔 ( l o g 𝑥 ) − 𝑔 ( l o g 𝑐 ) l o g 𝑥 − l o g 𝑐 ≥ 𝑔 ′ ( l o g 𝑐 ) . ( 3 . 1 )
Inequality (3.1) implies that 𝑓 ( 𝑥 ) − 𝑓 ( 𝑐 ) ≥ 𝑐 ( l o g 𝑥 − l o g 𝑐 ) 𝑓 ′ ( 𝑐 ) . ( 3 . 2 )
Let ∫ ℎ ( 𝑥 ) = 𝑥 𝑐 𝑓 ( 𝑡 ) 𝑑 𝑡 − ( 𝑥 − 𝑐 ) 𝑓 ( 𝑐 ) − 𝑐 [ 𝑥 ( l o g 𝑥 − l o g 𝑐 ) − ( 𝑥 − 𝑐 ) ] 𝑓 ′ ( 𝑐 ) , then inequality (3.2) leads to that ℎ ′ ( 𝑥 ) = 𝑓 ( 𝑥 ) − 𝑓 ( 𝑐 ) − 𝑐 ( l o g 𝑥 − l o g 𝑐 ) 𝑓 ′ ( 𝑐 ) ≥ 0 for 𝑥 ∈ [ 𝑐 , 𝑏 ] . Hence ℎ ( 𝑏 ) ≥ ℎ ( 𝑐 ) = 0 , namely, 𝑏 𝑐 = 𝑓 ( 𝑡 ) 𝑑 𝑡 ≥ ( 𝑏 − 𝑐 ) 𝑓 ( 𝑐 ) + 𝑐 ( 𝑏 l o g 𝑏 − 𝑏 l o g 𝑐 − 𝑏 + 𝑐 ) 𝑓 ′ ( 𝑐 ) ( 𝑏 − 𝑐 ) 𝑓 ( 𝑐 ) + 𝑐 ( l o g 𝑏 − l o g 𝑐 ) ( 𝑏 − 𝐿 ( 𝑐 , 𝑏 ) ) 𝑓 ′ ( 𝑐 ) . ( 3 . 3 )
Using a similar method we get 𝑐 𝑎 𝑓 ( 𝑡 ) 𝑑 𝑡 ≥ ( 𝑐 − 𝑎 ) 𝑓 ( 𝑐 ) − 𝑐 ( l o g 𝑐 − l o g 𝑎 ) ( 𝐿 ( 𝑎 , 𝑐 ) − 𝑎 ) 𝑓 ′ ( 𝑐 ) . ( 3 . 4 )
Let 𝑐 = 𝐼 ( 𝑎 , 𝑏 ) , then ( l o g 𝑏 − l o g 𝑐 ) ( 𝑏 − 𝐿 ( 𝑐 , 𝑏 ) ) = ( l o g 𝑐 − l o g 𝑎 ) ( 𝐿 ( 𝑎 , 𝑐 ) − 𝑎 ) = 𝐼 ( 𝑎 , 𝑏 ) − 𝑎 𝑏 𝐿 . ( 𝑎 , 𝑏 ) ( 3 . 5 )
From inequalities (3.3) and (3.4) together with (3.5) we clearly see that 𝑏 𝑎 𝑓 ( 𝑡 ) 𝑑 𝑡 ≥ ( 𝑏 − 𝑎 ) 𝑓 ( 𝐼 ( 𝑎 , 𝑏 ) ) . ( 3 . 6 )
Next for any 𝑥 ∈ [ 𝑎 , 𝑏 ] , let 𝑦 = ( l o g 𝑥 − l o g 𝑎 ) / ( l o g 𝑏 − l o g 𝑎 ) , then 0 ≤ 𝑦 ≤ 1 and 𝑥 = 𝑎 1 − 𝑦 𝑏 𝑦 . From the definition of GA-convex function and the transformation to variable of integration we get 𝑏 𝑎 𝑓 ( 𝑥 ) 𝑑 𝑥 = 1 0 𝑓 𝑎 1 − 𝑦 𝑏 𝑦 𝑑 𝑎 1 − 𝑦 𝑏 𝑦 ≤ 𝑎 1 0 [ ] 𝑑 𝑏 ( 1 − 𝑦 ) 𝑓 ( 𝑎 ) + 𝑦 𝑓 ( 𝑏 ) 𝑎 𝑦 = 𝑎 1 0 [ 𝑓 ] 𝑑 𝑏 ( 𝑎 ) + ( 𝑓 ( 𝑏 ) − 𝑓 ( 𝑎 ) ) 𝑦 𝑎 𝑦 = ( 𝑏 − 𝑎 ) 𝑓 ( 𝑎 ) + 𝑎 ( 𝑓 ( 𝑏 ) − 𝑓 ( 𝑎 ) ) 1 0 𝑏 𝑦 𝑑 𝑎 𝑦 = 𝑏 ( 𝑏 − 𝑎 ) 𝑓 ( 𝑎 ) + 𝑎 ( 𝑓 ( 𝑏 ) − 𝑓 ( 𝑎 ) ) 𝑎 − 𝑏 / 𝑎 − 1 l o g 𝑏 − l o g 𝑎 = 𝑏 𝑓 ( 𝑏 ) − 𝑎 𝑓 ( 𝑎 ) − ( 𝑓 ( 𝑏 ) − 𝑓 ( 𝑎 ) ) 𝐿 ( 𝑎 , 𝑏 ) = ( 𝑏 − 𝐿 ( 𝑎 , 𝑏 ) ) 𝑓 ( 𝑏 ) + ( 𝐿 ( 𝑎 , 𝑏 ) − 𝑎 ) 𝑓 ( 𝑎 ) . ( 3 . 7 )

Therefore, Theorem 1.1 follows from inequalities (3.6) and (3.7).

Proof of Theorem 1.2. From Lemmas 2.5 and 2.6 together with Theorem 1.1 we clearly see that 1 𝜓 ( 𝐼 ( 𝑎 , 𝑏 ) ) + ≥ 1 2 𝐼 ( 𝑎 , 𝑏 ) 𝑏 − 𝑎 𝑏 𝑎 1 𝜓 ( 𝑥 ) + 2 𝑥 𝑑 𝑥 = l o g Γ ( 𝑏 ) − l o g Γ ( 𝑎 ) + 1 𝑏 − 𝑎 , 1 2 𝐿 ( 𝑎 , 𝑏 ) ( 3 . 8 ) 𝜓 ( 𝐼 ( 𝑎 , 𝑏 ) ) + + 1 2 𝐼 ( 𝑎 , 𝑏 ) 1 2 𝐼 2 ≤ 1 ( 𝑎 , 𝑏 ) 𝑏 − 𝑎 𝑏 𝑎 1 𝜓 ( 𝑥 ) + + 1 2 𝑥 1 2 𝑥 2 = 𝑑 𝑥 l o g Γ ( 𝑏 ) − l o g Γ ( 𝑎 ) + 1 𝑏 − 𝑎 + 1 2 𝐿 ( 𝑎 , 𝑏 ) = 1 2 𝑎 𝑏 l o g Γ ( 𝑏 ) − l o g Γ ( 𝑎 ) + 1 𝑏 − 𝑎 + 1 2 𝐿 ( 𝑎 , 𝑏 ) 1 2 𝐺 2 . ( 𝑎 , 𝑏 ) ( 3 . 9 )

Therefore, Theorem 1.2 follows from (3.8) and (3.9).

Proof of Theorem 1.3. From Lemmas 2.5 and 2.6 together with Theorem 1.1 we get 1 𝑏 − 𝑎 𝑏 𝑎 1 𝜓 ( 𝑥 ) + 2 𝑥 𝑑 𝑥 ≥ 𝑏 − 𝐿 ( 𝑎 , 𝑏 ) 1 𝑏 − 𝑎 𝜓 ( 𝑏 ) + + 2 𝑏 𝐿 ( 𝑎 , 𝑏 ) − 𝑎 1 𝑏 − 𝑎 𝜓 ( 𝑎 ) + , 1 2 𝑎 ( 3 . 1 0 ) 𝑏 − 𝑎 𝑏 𝑎 1 𝜓 ( 𝑥 ) + + 1 2 𝑥 1 2 𝑥 2 𝑑 𝑥 ≤ 𝑏 − 𝐿 ( 𝑎 , 𝑏 ) 1 𝑏 − 𝑎 𝜓 ( 𝑏 ) + + 1 2 𝑏 1 2 𝑏 2 + 𝐿 ( 𝑎 , 𝑏 ) − 𝑎 1 𝑏 − 𝑎 𝜓 ( 𝑎 ) + + 1 2 𝑎 1 2 𝑎 2 . ( 3 . 1 1 )
Inequalities (3.10) and (3.11) lead to l o g Γ ( 𝑏 ) − l o g Γ ( 𝑎 ) ≥ 𝑏 − 𝑎 𝑏 − 𝐿 ( 𝑎 , 𝑏 ) 𝐿 𝑏 − 𝑎 𝜓 ( 𝑏 ) + ( 𝑎 , 𝑏 ) − 𝑎 𝐿 𝑏 − 𝑎 𝜓 ( 𝑎 ) + 2 ( 𝑎 , 𝑏 ) − 𝐺 2 ( 𝑎 , 𝑏 ) 2 𝐿 ( 𝑎 , 𝑏 ) 𝐺 2 ( 𝑎 , 𝑏 ) , ( 3 . 1 2 ) l o g Γ ( 𝑏 ) − l o g Γ ( 𝑎 ) ≤ 𝑏 − 𝑎 𝑏 − 𝐿 ( 𝑎 , 𝑏 ) 𝜓 𝐿 𝑏 − 𝑎 ( 𝑏 ) + ( 𝑎 , 𝑏 ) − 𝑎 𝜓 𝐿 𝑏 − 𝑎 ( 𝑎 ) + 2 ( 𝑎 , 𝑏 ) − 𝐺 2 ( 𝑎 , 𝑏 ) 2 𝐿 ( 𝑎 , 𝑏 ) 𝐺 2 + ( 𝑎 , 𝑏 ) 𝐿 ( 𝑎 , 𝑏 ) 𝐴 ( 𝑎 , 𝑏 ) − 𝐺 2 ( 𝑎 , 𝑏 ) 6 𝐺 4 . ( 𝑎 , 𝑏 ) ( 3 . 1 3 )

Therefore, Theorem 1.3 follows from (3.12) and (3.13).

Remark 3.1. Making use of a computer and the mathematica software we can show that the bounds in Theorems 1.2 and 1.3 are stronger than that in inequalities (1.9) and (1.10) for some 𝑎 and 𝑏 . In fact, if we let 𝑀 1 ( 𝑎 , 𝑏 ) = ( ( 𝑏 − 𝐿 ( 𝑎 , 𝑏 ) ) / ( 𝑏 − 𝑎 ) ) 𝜓 ( 𝑏 ) + ( ( 𝐿 ( 𝑎 , 𝑏 ) − 𝑎 ) / ( 𝑏 − 𝑎 ) ) 𝜓 ( 𝑎 ) , 𝑀 2 ( 𝑎 , 𝑏 ) = 𝜓 ( 𝐿 ( 𝑎 , 𝑏 ) ) , 𝑀 3 ( 𝑎 , 𝑏 ) = 𝜓 ( 𝐼 ( 𝑎 , 𝑏 ) ) − ( ( 𝐼 ( 𝑎 , 𝑏 ) − 𝐿 ( 𝑎 , 𝑏 ) ) / ( 2 𝐼 ( 𝑎 , 𝑏 ) 𝐿 ( 𝑎 , 𝑏 ) ) ) − ( ( 𝐼 2 ( 𝑎 , 𝑏 ) − 𝐺 2 ( 𝑎 , 𝑏 ) ) / ( 1 2 𝐼 2 ( 𝑎 , 𝑏 ) 𝐺 2 ( 𝑎 , 𝑏 ) ) ) , 𝑀 4 ( 𝑎 , 𝑏 ) = ( ( 𝑏 − 𝐿 ( 𝑎 , 𝑏 ) ) / ( 𝑏 − 𝑎 ) ) 𝜓 ( 𝑏 ) + ( ( 𝐿 ( 𝑎 , 𝑏 ) − 𝑎 ) / ( 𝑏 − 𝑎 ) ) 𝜓 ( 𝑎 ) + ( ( 𝐿 2 ( 𝑎 , 𝑏 ) − 𝐺 2 ( 𝑎 , 𝑏 ) ) / ( 2 𝐿 ( 𝑎 , 𝑏 ) 𝐺 2 ( 𝑎 , 𝑏 ) ) ) , 𝑁 1 ( 𝑎 , 𝑏 ) = 𝜓 ( 𝐿 ( 𝑎 , 𝑏 ) ) + l o g 𝐼 ( 𝑎 , 𝑏 ) / 𝐿 ( 𝑎 , 𝑏 ) , 𝑁 2 ( 𝑎 , 𝑏 ) = 𝜓 ( 𝐼 ( 𝑎 , 𝑏 ) ) − ( ( 𝐼 ( 𝑎 , 𝑏 ) − 𝐿 ( 𝑎 , 𝑏 ) ) / ( 2 𝐼 ( 𝑎 , 𝑏 ) 𝐿 ( 𝑎 , 𝑏 ) ) ) and 𝑁 3 ( 𝑎 , 𝑏 ) = ( ( 𝑏 − 𝐿 ( 𝑎 , 𝑏 ) ) / ( 𝑏 − 𝑎 ) ) 𝜓 ( 𝑏 ) + ( ( 𝐿 ( 𝑎 , 𝑏 ) − 𝑎 ) / ( 𝑏 − 𝑎 ) ) 𝜓 ( 𝑎 ) + ( 𝐿 2 ( 𝑎 , 𝑏 ) − 𝐺 2 ( 𝑎 , 𝑏 ) ) / ( 2 𝐿 ( 𝑎 , 𝑏 ) 𝐺 2 ( 𝑎 , 𝑏 ) ) + ( 𝐿 ( 𝑎 , 𝑏 ) 𝐴 ( 𝑎 , 𝑏 ) − 𝐺 2 ( 𝑎 , 𝑏 ) ) / ( 6 𝐺 4 ( 𝑎 , 𝑏 ) ) , then we have Tables 1 and 2 via elementary computation.

tab1
Table 1: Comparison of 𝑀 3 ( 𝑎 , 𝑏 ) and 𝑀 4 ( 𝑎 , 𝑏 ) with 𝑀 1 ( 𝑎 , 𝑏 ) and 𝑀 2 ( 𝑎 , 𝑏 ) for some 𝑎 and 𝑏 .
tab2
Table 2: Comparison of 𝑁 2 ( 𝑎 , 𝑏 ) and 𝑁 3 ( 𝑎 , 𝑏 ) with 𝑁 1 ( 𝑎 , 𝑏 ) for some 𝑎 and 𝑏 .

Remark 3.2. We clear see that the lower bound in Theorem 1.3 is stronger than that in inequality (1.9) for all 𝑎 , 𝑏 > 0 .

Acknowledgments

The authors wish to thank the anonymous referee for their very careful reading of the manuscript and fruitful comments and suggestions. This research is partly supported by N S Foundation of China under Grant 60850005 and N S Foundation of Zhejiang Province under Grants D7080080 and Y607128.

References

  1. J. Hadamard, “Étude sur les propriétés des fonctions entières et en particulier d'une fonction considérée par Riemann,” Journal de Mathématiques Pures et Appliquées, vol. 58, pp. 171–215, 1893.
  2. C. P. Niculescu, “The Hermite-Hadamard inequality for convex functions on global NPC space,” Journal of Mathematical Analysis and Applications, vol. 356, no. 1, pp. 295–301, 2009.
  3. S.-H. Wu, “On the weighted generalization of the Hermite-Hadamard inequality and its applications,” The Rocky Mountain Journal of Mathematics, vol. 39, no. 5, pp. 1741–1749, 2009.
  4. M. Alomari and M. Darus, “On the Hadamard's inequality for log-convex functions on the coordinates,” Journal of Inequalities and Applications, vol. 2009, Article ID 283147, 13 pages, 2009.
  5. C. Dinu, “Hermite-Hadamard inequality on time scales,” Journal of Inequalities and Applications, vol. 2008, Article ID 287947, 24 pages, 2008.
  6. M. Bessenyei, “The Hermite-Hadamard inequality on simplices,” American Mathematical Monthly, vol. 115, no. 4, pp. 339–345, 2008.
  7. M. Mihăilescu and C. P. Niculescu, “An extension of the Hermite-Hadamard inequality through subharmonic functions,” Glasgow Mathematical Journal, vol. 49, no. 3, pp. 509–514, 2007.
  8. M. Bessenyei and Z. Páles, “Characterization of convexity via Hadamard's inequality,” Mathematical Inequalities & Applications, vol. 9, no. 1, pp. 53–62, 2006.
  9. G.-S. Yang, D.-Y. Hwang, and K.-L. Tseng, “Some inequalities for differentiable convex and concave mappings,” Computers & Mathematics with Applications, vol. 47, no. 2-3, pp. 207–216, 2004.
  10. M. Sun and X. Yang, “Generalized Hadamard's inequality and r-convex functions in Carnot groups,” Journal of Mathematical Analysis and Applications, vol. 294, no. 2, pp. 387–398, 2004.
  11. L. Wang, “On extensions and refinements of Hermite-Hadamard inequalities for convex functions,” Mathematical Inequalities & Applications, vol. 6, no. 4, pp. 659–666, 2003.
  12. A. M. Mercer, “Hadamard's inequality for a triangle, a regular polygon and a circle,” Mathematical Inequalities & Applications, vol. 5, no. 2, pp. 219–223, 2002.
  13. S. S. Dragomir and C. E. M. Pearce, “Quasilinearity & Hadamard's inequality,” Mathematical Inequalities & Applications, vol. 5, no. 3, pp. 463–471, 2002.
  14. S. S. Dragomir, Y. J. Cho, and S. S. Kim, “Inequalities of Hadamard's type for Lipschitzian mappings and their applications,” Journal of Mathematical Analysis and Applications, vol. 245, no. 2, pp. 489–501, 2000.
  15. G.-S. Yang and K.-L. Tseng, “On certain integral inequalities related to Hermite-Hadamard inequalities,” Journal of Mathematical Analysis and Applications, vol. 239, no. 1, pp. 180–187, 1999.
  16. S. S. Dragomir and R. P. Agarwal, “Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula,” Applied Mathematics Letters, vol. 11, no. 5, pp. 91–95, 1998.
  17. S. S. Dragomir and R. P. Agarwal, “Two new mappings associated with Hadamard's inequalities for convex functions,” Applied Mathematics Letters, vol. 11, no. 3, pp. 33–38, 1998.
  18. C. E. M. Pearce, J. Pečarić, and V. Šimić, “Stolarsky means and Hadamard's inequality,” Journal of Mathematical Analysis and Applications, vol. 220, no. 1, pp. 99–109, 1998.
  19. P. M. Gill, C. E. M. Pearce, and J. Pečarić, “Hadamard's inequality for r-convex functions,” Journal of Mathematical Analysis and Applications, vol. 215, no. 2, pp. 461–470, 1997.
  20. J. E. Pečarić, F. Proschan, and Y. L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, vol. 187 of Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1992.
  21. G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, “Generalized convexity and inequalities,” Journal of Mathematical Analysis and Applications, vol. 335, no. 2, pp. 1294–1308, 2007.
  22. W. Gautschi, “Some elementary inequalitie