On Hadamard-Type Inequalities Involving Several Kinds of Convexity

  • Erhan Set1Email author,

    Affiliated with

    • MEmin Özdemir1 and

      Affiliated with

      • SeverS Dragomir2, 3

        Affiliated with

        Journal of Inequalities and Applications20102010:286845

        DOI: 10.1155/2010/286845

        Received: 14 May 2010

        Accepted: 23 August 2010

        Published: 26 August 2010

        Abstract

        We do not only give the extensions of the results given by Gill et al. (1997) for log-convex functions but also obtain some new Hadamard-type inequalities for log-convex http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq1_HTML.gif -convex, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq2_HTML.gif -convex functions.

        1. Introduction

        The following inequality is well known in the literature as Hadamard's inequality:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ1_HTML.gif
        (1.1)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq3_HTML.gif is a convex function on the interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq4_HTML.gif of real numbers and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq5_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq6_HTML.gif This inequality is one of the most useful inequalities in mathematical analysis. For new proofs, note worthy extension, generalizations, and numerous applications on this inequality; see ([16]) where further references are given.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq7_HTML.gif be on interval in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq8_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq9_HTML.gif is said to be convex if, for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq10_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq11_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ2_HTML.gif
        (1.2)

        (see [5], Page 1). Geometrically, this means that if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq12_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq13_HTML.gif are three distinct points on the graph of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq14_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq15_HTML.gif between http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq16_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq17_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq18_HTML.gif is on or below chord http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq19_HTML.gif

        Recall that a function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq20_HTML.gif is said to be log-convex function if, for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq21_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq22_HTML.gif , one has the inequality (see [5], Page 3)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ3_HTML.gif
        (1.3)

        It is said to be log-concave if the inequality in (1.3) is reversed.

        In [7], Toader defined http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq23_HTML.gif -convexity as follows.

        Definition 1.1.

        The function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq24_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq25_HTML.gif is said to be http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq26_HTML.gif -convex, where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq27_HTML.gif , if one has
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ4_HTML.gif
        (1.4)

        for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq28_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq29_HTML.gif We say that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq30_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq31_HTML.gif -concave if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq32_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq33_HTML.gif -convex.

        Denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq34_HTML.gif the class of all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq35_HTML.gif -convex functions on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq36_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq37_HTML.gif (if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq38_HTML.gif ). Obviously, if we choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq39_HTML.gif Definition 1.1 recaptures the concept of standard convex functions on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq40_HTML.gif

        In [8], Miheşan defined http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq41_HTML.gif -convexity as in the following:

        Definition 1.2.

        The function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq42_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq43_HTML.gif , is said to be http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq44_HTML.gif -convex, where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq45_HTML.gif , if one has
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ5_HTML.gif
        (1.5)

        for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq46_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq47_HTML.gif .

        Denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq48_HTML.gif the class of all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq49_HTML.gif -convex functions on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq50_HTML.gif for which http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq51_HTML.gif . It can be easily seen that for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq52_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq53_HTML.gif -convexity reduces to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq54_HTML.gif -convexity and for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq55_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq56_HTML.gif -convexity reduces to the concept of usual convexity defined on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq57_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq58_HTML.gif .

        For recent results and generalizations concerning http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq59_HTML.gif -convex and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq60_HTML.gif -convex functions, see ([912]).

        In the literature, the logarithmic mean of the positive real numbers http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq61_HTML.gif is defined as the following:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ6_HTML.gif
        (1.6)

        (for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq62_HTML.gif , we put http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq63_HTML.gif ).

        In [13], Gill et al. established the following results.

        Theorem 1.3.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq64_HTML.gif be a positive, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq65_HTML.gif -convex function on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq66_HTML.gif . Then
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ7_HTML.gif
        (1.7)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq67_HTML.gif is a logarithmic mean of the positive real numbers as in (1.6).

        For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq68_HTML.gif a positive http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq69_HTML.gif -concave function, the inequality is reversed.

        Corollary 1.4.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq70_HTML.gif be positive http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq71_HTML.gif -convex functions on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq72_HTML.gif . Then
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ8_HTML.gif
        (1.8)
        If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq73_HTML.gif is a positive http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq74_HTML.gif -concave function, then
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ9_HTML.gif
        (1.9)

        For some recent results related to the Hadamard's inequalities involving two http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq75_HTML.gif -convex functions, see [14] and the references cited therein. The main purpose of this paper is to establish the general version of inequalities (1.7) and new Hadamard-type inequalities involving two http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq76_HTML.gif -convex functions, two http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq77_HTML.gif -convex functions, or two http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq78_HTML.gif -convex functions using elementary analysis.

        2. Main Results

        We start with the following theorem.

        Theorem 2.1.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq79_HTML.gif    http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq80_HTML.gif be http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq81_HTML.gif -convex functions on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq82_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq83_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq84_HTML.gif . Then the following inequality holds:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ10_HTML.gif
        (2.1)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq85_HTML.gif is a logarithmic mean of positive real numbers.

        For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq86_HTML.gif a positive http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq87_HTML.gif -concave function, the inequality is reversed.

        Proof.

        Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq88_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq89_HTML.gif are http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq90_HTML.gif -convex functions on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq91_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ11_HTML.gif
        (2.2)
        for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq92_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq93_HTML.gif Writing (2.2) for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq94_HTML.gif and multiplying the resulting inequalities, it is easy to observe that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ12_HTML.gif
        (2.3)

        for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq95_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq96_HTML.gif

        Integrating inequality (2.3) on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq97_HTML.gif over http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq98_HTML.gif , we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ13_HTML.gif
        (2.4)
        As
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ14_HTML.gif
        (2.5)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ15_HTML.gif
        (2.6)

        the theorem is proved.

        Remark 2.2.

        By taking http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq99_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq100_HTML.gif in Theorem 2.1 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq101_HTML.gif we obtain (1.7).

        Corollary 2.3.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq102_HTML.gif be http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq103_HTML.gif -convex functions on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq104_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq105_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq106_HTML.gif . Then
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ16_HTML.gif
        (2.7)
        If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq107_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq108_HTML.gif are positive http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq109_HTML.gif -concave functions, then
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ17_HTML.gif
        (2.8)

        Proof.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq110_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq111_HTML.gif be positive http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq112_HTML.gif -convex functions. Then by Theorem 2.1 we have that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ18_HTML.gif
        (2.9)

        for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq113_HTML.gif , whence (2.7). Similarly we can prove (2.8).

        Remark 2.4.

        By taking http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq114_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq115_HTML.gif in (2.7) and (2.8) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq116_HTML.gif we obtain the inequalities of Corollary 1.4.

        We will now point out some new results of the Hadamard type for log-convex, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq117_HTML.gif -convex, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq118_HTML.gif -convex functions, respectively.

        Theorem 2.5.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq119_HTML.gif be http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq120_HTML.gif -convex functions on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq121_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq122_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq123_HTML.gif Then the following inequalities hold:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ19_HTML.gif
        (2.10)

        Proof.

        We can write
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ20_HTML.gif
        (2.11)
        Using the elementary inequality http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq124_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq125_HTML.gif reals) and equality (2.11), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ21_HTML.gif
        (2.12)
        Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq126_HTML.gif are http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq127_HTML.gif -convex functions, we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ22_HTML.gif
        (2.13)

        for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq128_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq129_HTML.gif .

        Rewriting (2.12) and (2.13), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ23_HTML.gif
        (2.14)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ24_HTML.gif
        (2.15)
        Integrating both sides of (2.14) and (2.15) on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq130_HTML.gif over http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq131_HTML.gif , respectively, we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ25_HTML.gif
        (2.16)

        Combining (2.16), we get the desired inequalities (2.10). The proof is complete.

        Theorem 2.6.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq132_HTML.gif be http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq133_HTML.gif -convex functions on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq134_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq135_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq136_HTML.gif Then the following inequalities hold:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ26_HTML.gif
        (2.17)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq137_HTML.gif is a logarithmic mean of positive real numbers.

        Proof.

        From inequality (2.14), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ27_HTML.gif
        (2.18)

        for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq138_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq139_HTML.gif

        Using the elementary inequality http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq140_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq141_HTML.gif reals) on the right side of the above inequality, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ28_HTML.gif
        (2.19)
        Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq142_HTML.gif are http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq143_HTML.gif -convex functions, then we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ29_HTML.gif
        (2.20)
        Integrating both sides of (2.19) and (2.20) on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq144_HTML.gif over http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq145_HTML.gif , respectively, we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ30_HTML.gif
        (2.21)

        Combining (2.21), we get the required inequalities (2.17). The proof is complete.

        Theorem 2.7.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq146_HTML.gif be such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq147_HTML.gif is in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq148_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq149_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq150_HTML.gif is nonincreasing http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq151_HTML.gif -convex function and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq152_HTML.gif is nonincreasing http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq153_HTML.gif -convex function on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq154_HTML.gif for some fixed http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq155_HTML.gif then the following inequality holds:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ31_HTML.gif
        (2.22)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ32_HTML.gif
        (2.23)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ33_HTML.gif
        (2.24)

        Proof.

        Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq156_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq157_HTML.gif -convex function and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq158_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq159_HTML.gif -convex function, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ34_HTML.gif
        (2.25)
        for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq160_HTML.gif . It is easy to observe that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ35_HTML.gif
        (2.26)
        Using the elementary inequality http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq161_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq162_HTML.gif reals), (2.25) on the right side of (2.26) and making the charge of variable and since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq163_HTML.gif is nonincreasing, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ36_HTML.gif
        (2.27)
        Analogously we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ37_HTML.gif
        (2.28)

        Rewriting (2.27) and (2.28), we get the required inequality in (2.22). The proof is complete.

        Theorem 2.8.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq164_HTML.gif be such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq165_HTML.gif is in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq166_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq167_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq168_HTML.gif is nonincreasing http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq169_HTML.gif -convex function and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq170_HTML.gif is nonincreasing http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq171_HTML.gif -convex function on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq172_HTML.gif for some fixed http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq173_HTML.gif Then the following inequality holds:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ38_HTML.gif
        (2.29)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ39_HTML.gif
        (2.30)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ40_HTML.gif
        (2.31)

        Proof.

        Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq174_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq175_HTML.gif -convex function and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq176_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq177_HTML.gif -convex function, then we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ41_HTML.gif
        (2.32)
        for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq178_HTML.gif . It is easy to observe that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ42_HTML.gif
        (2.33)
        Using the elementary inequality http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq179_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq180_HTML.gif reals), (2.32) on the right side of (2.33) and making the charge of variable and since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq181_HTML.gif is nonincreasing, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ43_HTML.gif
        (2.34)
        Analogously we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ44_HTML.gif
        (2.35)

        Rewriting (2.34) and (2.35), we get the required inequality in (2.29). The proof is complete.

        Remark 2.9.

        In Theorem 2.8, if we choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq182_HTML.gif , we obtain the inequality of Theorem 2.7.

        Authors’ Affiliations

        (1)
        Department of Mathematics, K.K. Education Faculty, Atatürk University
        (2)
        Research Group in Mathematical Inequalities & Applications, School of Engineering & Science, Victoria University
        (3)
        School of Computational and Applied Mathematics, University of the Witwatersrand

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        © Erhan Set et al. 2010

        This article is published under license to BioMed Central Ltd. 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.