Fejér-Type Inequalities (I)

  • Kuei-Lin Tseng1,

    Affiliated with

    • Shiow-Ru Hwang2 and

      Affiliated with

      • SS Dragomir3, 4Email author

        Affiliated with

        Journal of Inequalities and Applications20102010:531976

        DOI: 10.1155/2010/531976

        Received: 3 May 2010

        Accepted: 3 December 2010

        Published: 15 December 2010

        Abstract

        We establish some new Fejér-type inequalities for convex functions.

        1. Introduction

        Throughout this paper, let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq1_HTML.gif be convex, and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq2_HTML.gif be integrable and symmetric to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq3_HTML.gif . We define the following functions on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq4_HTML.gif that are associated with the well-known Hermite-Hadamard inequality [1]
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ1_HTML.gif
        (1.1)
        namely
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ2_HTML.gif
        (1.2)

        For some results which generalize, improve, and extend the famous integral inequality (1.1), see [26].

        In [2], Dragomir established the following theorem which is a refinement of the first inequality of (1.1).

        Theorem A.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq5_HTML.gif be defined as above, and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq6_HTML.gif be defined on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq7_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ3_HTML.gif
        (1.3)
        Then, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq8_HTML.gif is convex, increasing on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq9_HTML.gif , and for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq10_HTML.gif , one has
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ4_HTML.gif
        (1.4)

        In [6], Yang and Hong established the following theorem which is a refinement of the second inequality in (1.1).

        Theorem B.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq11_HTML.gif be defined as above, and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq12_HTML.gif be defined on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq13_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ5_HTML.gif
        (1.5)
        Then, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq14_HTML.gif is convex, increasing on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq15_HTML.gif , and for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq16_HTML.gif , one has
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ6_HTML.gif
        (1.6)

        In [3], Fejér established the following weighted generalization of the Hermite-Hadamard inequality (1.1).

        Theorem C.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq17_HTML.gif be defined as above. Then,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ7_HTML.gif
        (1.7)

        is known as Fejér inequality.

        In this paper, we establish some Fejér-type inequalities related to the functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq18_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq19_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq20_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq21_HTML.gif introduced above.

        2. Main Results

        In order to prove our main results, we need the following lemma.

        Lemma 2.1 (see [4]).

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq22_HTML.gif be defined as above, and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq23_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq24_HTML.gif . Then,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ8_HTML.gif
        (2.1)

        Now, we are ready to state and prove our results.

        Theorem 2.2.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq25_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq26_HTML.gif be defined as above. Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq27_HTML.gif is convex, increasing on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq28_HTML.gif , and for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq29_HTML.gif , one has the following Fejér-type inequality:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ9_HTML.gif
        (2.2)

        Proof.

        It is easily observed from the convexity of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq30_HTML.gif that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq31_HTML.gif is convex on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq32_HTML.gif . Using simple integration techniques and under the hypothesis of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq33_HTML.gif , the following identity holds on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq34_HTML.gif :
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ10_HTML.gif
        (2.3)
        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq35_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq36_HTML.gif . By Lemma 2.1, the following inequality holds for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq37_HTML.gif :
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ11_HTML.gif
        (2.4)
        Indeed, it holds when we make the choice
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ12_HTML.gif
        (2.5)

        in Lemma 2.1.

        Multipling the inequality (2.4) by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq38_HTML.gif , integrating both sides over http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq39_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq40_HTML.gif and using identity (2.3), we derive http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq41_HTML.gif . Thus http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq42_HTML.gif is increasing on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq43_HTML.gif and then the inequality (2.2) holds. This completes the proof.

        Remark 2.3.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq44_HTML.gif in Theorem 2.2. Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq45_HTML.gif and the inequality (2.2) reduces to the inequality (1.4), where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq46_HTML.gif is defined as in Theorem A.

        Theorem 2.4.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq47_HTML.gif be defined as above. Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq48_HTML.gif is convex, increasing on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq49_HTML.gif , and for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq50_HTML.gif , one has the following Fejér-type inequality:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ13_HTML.gif
        (2.6)

        Proof.

        By using a similar method to that from Theorem 2.2, we can show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq51_HTML.gif is convex on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq52_HTML.gif , the identity
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ14_HTML.gif
        (2.7)
        holds on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq53_HTML.gif , and the inequalities
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ15_HTML.gif
        (2.8)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ16_HTML.gif
        (2.9)

        hold for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq54_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq55_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq56_HTML.gif .

        By (2.7)–(2.9) and using a similar method to that from Theorem 2.2, we can show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq57_HTML.gif is increasing on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq58_HTML.gif and (2.6) holds. This completes the proof.

        The following result provides a comparison between the functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq59_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq60_HTML.gif .

        Theorem 2.5.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq61_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq62_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq63_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq64_HTML.gif be defined as above. Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq65_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq66_HTML.gif .

        Proof.

        By the identity
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ17_HTML.gif
        (2.10)

        on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq67_HTML.gif , (2.3) and using a similar method to that from Theorem 2.2, we can show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq68_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq69_HTML.gif . The details are omited.

        Further, the following result incorporates the properties of the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq70_HTML.gif .

        Theorem 2.6.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq71_HTML.gif be defined as above. Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq72_HTML.gif is convex, increasing on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq73_HTML.gif , and for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq74_HTML.gif , one has the following Fejér-type inequality:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ18_HTML.gif
        (2.11)

        Proof.

        Follows by the identity
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ19_HTML.gif
        (2.12)

        on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq75_HTML.gif . The details are left to the interested reader.

        We now present a result concerning the properties of the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq76_HTML.gif .

        Theorem 2.7.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq77_HTML.gif be defined as above. Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq78_HTML.gif is convex, increasing on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq79_HTML.gif , and for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq80_HTML.gif , one has the following Fejér-type inequality:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ20_HTML.gif
        (2.13)

        Proof.

        By the identity
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ21_HTML.gif
        (2.14)

        on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq81_HTML.gif and using a similar method to that for Theorem 2.2, we can show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq82_HTML.gif is convex, increasing on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq83_HTML.gif and (2.13) holds.

        Remark 2.8.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq84_HTML.gif in Theorem 2.7. Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq85_HTML.gif and the inequality (2.13) reduces to (1.6), where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq86_HTML.gif is defined as in Theorem B.

        Theorem 2.9.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq87_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq88_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq89_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq90_HTML.gif be defined as above. Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq91_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq92_HTML.gif .

        Proof.

        By the identity
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ22_HTML.gif
        (2.15)

        on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq93_HTML.gif , (2.12) and using a similar method to that for Theorem 2.2, we can show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq94_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq95_HTML.gif . This completes the proof.

        The following Fejér-type inequality is a natural consequence of Theorems 2.2–2.9.

        Corollary 2.10.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq96_HTML.gif be defined as above. Then one has
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ23_HTML.gif
        (2.16)

        Remark 2.11.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_IEq97_HTML.gif in Corollary 2.10. Then the inequality (2.16) reduces to
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ24_HTML.gif
        (2.17)

        which is a refinement of (1.1).

        Remark 2.12.

        In Corollary 2.10, the third inequality in (2.16) is the weighted generalization of Bullen's inequality [5]
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F531976/MediaObjects/13660_2010_Article_2182_Equ25_HTML.gif
        (2.18)

        Declarations

        Acknowledgment

        This research was partially supported by Grant NSC 97-2115-M-156-002.

        Authors’ Affiliations

        (1)
        Department of Applied Mathematics, Aletheia University
        (2)
        China University of Science and Technology
        (3)
        School of Engineering Science, VIC University
        (4)
        School of Computational and Applied Mathematics, University of the Witwatersrand

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