On the Hermite-Hadamard Inequality and Other Integral Inequalities Involving Two Functions

  • Erhan Set1Email author,

    Affiliated with

    • MEmin Özdemir2 and

      Affiliated with

      • SeverS Dragomir3

        Affiliated with

        Journal of Inequalities and Applications20102010:148102

        DOI: 10.1155/2010/148102

        Received: 25 September 2009

        Accepted: 31 March 2010

        Published: 24 May 2010

        Abstract

        We establish some new Hermite-Hadamard-type inequalities involving product of two functions. Other integral inequalities for two functions are obtained as well. The analysis used in the proofs is fairly elementary and based on the use of the Minkowski, Hölder, and Young inequalities.

        1. Introduction

        Integral inequalities have played an important role in the development of all branches of Mathematics.

        In [1, 2], Pachpatte established some Hermite-Hadamard-type inequalities involving two convex and log-convex functions, respectively. In [3], Bakula et al. improved Hermite-Hadamard type inequalities for products of two http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq1_HTML.gif -convex and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq2_HTML.gif -convex functions. In [4], analogous results for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq3_HTML.gif -convex functions were proved by Kirmaci et al.. General companion inequalities related to Jensen's inequality for the classes of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq4_HTML.gif -convex and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq5_HTML.gif -convex functions were presented by Bakula et al. (see [5]).

        For several recent results concerning these types of inequalities, see [612] where further references are listed.

        The aim of this paper is to establish several new integral inequalities for nonnegative and integrable functions that are related to the Hermite-Hadamard result. Other integral inequalities for two functions are also established.

        In order to prove some inequalities related to the products of two functions we need the following inequalities. One of inequalities of this type is the following one.

        Barnes-Gudunova-Levin Inequality (see [1315] and references therein)

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq6_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq7_HTML.gif be nonnegative concave functions on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq8_HTML.gif . Then, for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq9_HTML.gif we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ1_HTML.gif
        (1.1)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ2_HTML.gif
        (1.2)
        In the special case http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq10_HTML.gif we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ3_HTML.gif
        (1.3)
        with
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ4_HTML.gif
        (1.4)

        To prove our main results we recall some concepts and definitions.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq11_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq12_HTML.gif be two positive http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq13_HTML.gif -tuples, and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq14_HTML.gif . Then, on putting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq15_HTML.gif the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq16_HTML.gif th power mean of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq17_HTML.gif with weights http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq18_HTML.gif is defined [16] by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ5_HTML.gif
        (1.5)
        Note that if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq19_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ6_HTML.gif
        (1.6)

        (see, e.g., [10, page 15]).

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq20_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq21_HTML.gif . The http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq22_HTML.gif -norm of the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq23_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq24_HTML.gif is defined by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ7_HTML.gif
        (1.7)

        and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq25_HTML.gif is the set of all functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq26_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq27_HTML.gif .

        One can rewrite the inequality (1.1) as follows:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ8_HTML.gif
        (1.8)

        For several recent results concerning http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq28_HTML.gif -norms we refer the interested reader to [17].

        Also, we need some important inequalities.

        Minkowski Integral Inequality (see page 1 in [18])

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq29_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq30_HTML.gif . Then
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ9_HTML.gif
        (1.9)

        Hermite-Hadamard's Inequality (see page 10 in [10])

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq31_HTML.gif be a convex function on interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq32_HTML.gif of real numbers and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq33_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq34_HTML.gif . Then the following Hermite-Hadamard inequality for convex functions holds:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ10_HTML.gif
        (1.10)
        If the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq35_HTML.gif is concave, the inequality (1.10) can be written as follows:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ11_HTML.gif
        (1.11)

        For recent results, refinements, counterparts, generalizations, and new Hermite-Hadamard-type inequalities, see [1921].

        A Reversed Minkowski Integral Inequality (see page 2 in [18])

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq36_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq37_HTML.gif be positive functions satisfying
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ12_HTML.gif
        (1.12)
        Then, putting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq38_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ13_HTML.gif
        (1.13)

        One of the most important inequalities of analysis is Hölder's integral inequality which is stated as follows (for its variant see [10, page 106]).

        Hölder Integral Inequality

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq39_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq40_HTML.gif If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq41_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq42_HTML.gif are real functions defined on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq43_HTML.gif and if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq44_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq45_HTML.gif are integrable functions on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq46_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ14_HTML.gif
        (1.14)

        with equality holding if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq47_HTML.gif almost everywhere, where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq48_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq49_HTML.gif are constants.

        Remark 1.1.

        Observe that whenever, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq50_HTML.gif is concave on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq51_HTML.gif the nonnegative function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq52_HTML.gif is also concave on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq53_HTML.gif . Namely,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ15_HTML.gif
        (1.15)
        that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ16_HTML.gif
        (1.16)
        and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq54_HTML.gif ; using the power-mean inequality (1.6), we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ17_HTML.gif
        (1.17)

        For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq55_HTML.gif , similarly if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq56_HTML.gif is concave on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq57_HTML.gif the nonnegative function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq58_HTML.gif is concave on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq59_HTML.gif .

        2. The Results

        Theorem 2.1.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq60_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq61_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq62_HTML.gif , be nonnegative functions such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq63_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq64_HTML.gif are concave on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq65_HTML.gif . Then
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ18_HTML.gif
        (2.1)
        and if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq66_HTML.gif , then one has
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ19_HTML.gif
        (2.2)

        Here http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq67_HTML.gif is the Barnes-Gudunova-Levin constant given by (1.1).

        Proof.

        Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq68_HTML.gif are concave functions on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq69_HTML.gif , then from (1.11) and Remark 1.1 we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ20_HTML.gif
        (2.3)
        By multiplying the above inequalities, we obtain (2.4) and (2.5)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ21_HTML.gif
        (2.4)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ22_HTML.gif
        (2.5)
        If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq70_HTML.gif , then it easy to show that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ23_HTML.gif
        (2.6)

        Thus, by applying Barnes-Gudunova-Levin inequality to the right-hand side of (2.4) with (2.6), we get (2.1).

        Applying the Hölder inequality to the left-hand side of (2.5) with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq71_HTML.gif , we get (2.2).

        Theorem 2.2.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq72_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq73_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq74_HTML.gif be positive functions with
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ24_HTML.gif
        (2.7)
        Then
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ25_HTML.gif
        (2.8)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq75_HTML.gif

        Proof.

        Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq76_HTML.gif are positive, as in the proof of the inequality (1.13) (see [18, page 2]), we have that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ26_HTML.gif
        (2.9)
        By multiplying the above inequalities, we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ27_HTML.gif
        (2.10)

        Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq77_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq78_HTML.gif by applying the Minkowski integral inequality to the right hand side of (2.10), we obtain inequality (2.8).

        Theorem 2.3.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq79_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq80_HTML.gif be as in Theorem 2.1. Then the following inequality holds:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ28_HTML.gif
        (2.11)

        Proof.

        If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq81_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq82_HTML.gif are concave on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq83_HTML.gif , then from (1.11) we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ29_HTML.gif
        (2.12)
        which imply that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ30_HTML.gif
        (2.13)
        On the other hand, if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq84_HTML.gif from (1.6) we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ31_HTML.gif
        (2.14)
        or
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ32_HTML.gif
        (2.15)
        which imply that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ33_HTML.gif
        (2.16)
        Combining (2.13) and (2.16), we obtain the desired inequality as
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ34_HTML.gif
        (2.17)
        that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ35_HTML.gif
        (2.18)
        To prove the following theorem we need the following Young-type inequality (see [7, page 117]):
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ36_HTML.gif
        (2.19)

        Theorem 2.4.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq85_HTML.gif be functions such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq86_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq87_HTML.gif are in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq88_HTML.gif , and
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ37_HTML.gif
        (2.20)
        Then
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ38_HTML.gif
        (2.21)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ39_HTML.gif
        (2.22)

        and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq89_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq90_HTML.gif

        Proof.

        From http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq91_HTML.gif we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ40_HTML.gif
        (2.23)
        From (2.19) with (2.23) we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ41_HTML.gif
        (2.24)
        Using the elementary inequality http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq92_HTML.gif , ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq93_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq94_HTML.gif ) in (2.24), we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ42_HTML.gif
        (2.25)

        This completes the proof of the inequality in (2.21).

        Declarations

        Acknowledgment

        The authors thank the careful referees for some good advices which have improved the final version of this paper.

        Authors’ Affiliations

        (1)
        Department of Mathematics, K. K. Education Faculty, Atatürk University
        (2)
        Graduate School of Natural and Applied Sciences, Ağrı İbrahim Çeçen University
        (3)
        Research Group in Mathematical Inequalities & Applications, School of Engineering & Science, Victoria University

        References

        1. Pachpatte BG: On some inequalities for convex functions. RGMIA Research Report Collection E 2003., 6:
        2. Pachpatte BG: A note on integral inequalities involving two log-convex functions. Mathematical Inequalities & Applications 2004, 7(4):511–515.MathSciNetView Article
        3. Bakula MK, Özdemir ME, Pečarić J: Hadamard type inequalities for -convex and -convex functions. Journal of Inequalities in Pure and Applied Mathematics 2008., 9(4, article 96):
        4. Kirmaci US, Bakula MK, Özdemir ME, Pečarić J: Hadamard-type inequalities for -convex functions. Applied Mathematics and Computation 2007, 193(1):26–35. 10.1016/j.amc.2007.03.030MathSciNetView Article
        5. Bakula MK, Pečarić J, Ribičić M: Companion inequalities to Jensen's inequality for -convex and -convex functions. Journal of Inequalities in Pure and Applied Mathematics 2006., 7(5, article 194):
        6. Bakula MK, Pečarić J: Note on some Hadamard-type inequalities. Journal of Inequalities in Pure and Applied Mathematics 2004., 5(3, article 74):
        7. Dragomir SS, Agarwal RP, Barnett NS: Inequalities for beta and gamma functions via some classical and new integral inequalities. Journal of Inequalities and Applications 2000, 5(2):103–165. 10.1155/S1025583400000084MathSciNet
        8. Hardy GH, Littlewood JE, Pólya G: Inequalities. Cambridge Mathematical Library, Cambridge , UK; 1998:xii+324.
        9. Kirmaci US, Özdemir ME: Some inequalities for mappings whose derivatives are bounded and applications to special means of real numbers. Applied Mathematics Letters 2004, 17(6):641–645. 10.1016/S0893-9659(04)90098-5MathSciNetView Article
        10. Mitrinović DS, Pečarić JE, Fink AM: Classical and New Inequalities in Analysis, Mathematics and Its Applications (East European Series). Volume 61. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1993:xviii+740.View Article
        11. Özdemir ME, Kırmacı US: Two new theorem on mappings uniformly continuous and convex with applications to quadrature rules and means. Applied Mathematics and Computation 2003, 143(2–3):269–274. 10.1016/S0096-3003(02)00359-4MathSciNetView Article
        12. Pachpatte BG: Inequalities for Differentiable and Integral Equations. Academic Press, Boston, Mass, USA; 1997.
        13. Pečarić J, Pejković T: On an integral inequality. Journal of Inequalities in Pure and Applied Mathematics 2004., 5(2, article 47):
        14. Pečarić JE, Proschan F, Tong YL: Convex Functions, Partial Orderings, and Statistical Applications, Mathematics in Science and Engineering. Volume 187. Academic Press, Boston, Mass, USA; 1992:xiv+467.
        15. Pogány TK: On an open problem of F. Qi. Journal of Inequalities in Pure and Applied Mathematics 2002., 3(4, article 54):
        16. Bullen PS, Mitrinović DS, Vasić PM: Means and Their Inequalities, Mathematics and Its Applications (East European Series). Volume 31. D. Reidel, Dordrecht, The Netherlands; 1988:xx+459.
        17. Kirmaci US, Klaričić M, Özdemir ME, Pečarić J: On some inequalities for -norms. Journal of Inequalities in Pure and Applied Mathematics 2008., 9(1, article 27):
        18. Bougoffa L: On Minkowski and Hardy integral inequalities. Journal of Inequalities in Pure and Applied Mathematics 2006., 7(2, article 60):
        19. Alomari M, Darus M: On the Hadamard's inequality for log-convex functions on the coordinates. Journal of Inequalities and Applications 2009, 2009:-13.
        20. Dinu C: Hermite-Hadamard inequality on time scales. Journal of Inequalities and Applications 2008, 2008:-24.
        21. Dragomir SS, Pearce CEM: Selected Topics on Hermite-Hadamard Inequalities and Applications. RGMIA Monographs, Victoria University, Melbourne, Australia; 2000.

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        © Erhan Set et al. 2010

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