In this paper, some inequalities connected with the left and right hand side of Hermite-

Hadamard type inequalities are established for functions whose the first and second

derivatives absolute values are quasi-convex

Alawiah Ibrahim,

UKM Journal Article Repository

Journal of Quality Measurement and Analysis              JQMA 14(2) 2018, 91-98 Jurnal Pengukuran Kualiti dan Analisis                          QUASI-CONVEXITY AND HERMITE-HADAMARD TYPE INEQUALITIES  (Kekuasicembungan dan Ketaksamaan Jenis Hermite-Hadamard)  ALAWIAH IBRAHIM  ABSTRACT In this paper, some inequalities connected with the left and right hand side of Hermite-Hadamard type inequalities are established for functions whose the first and second derivatives absolute values are quasi-convex. Keywords: Hermite-Hadamard inequality; convex function; quasi-convex function   ABSTRAK Dalam makalah ini, beberapa ketaksamaan bersabit dengan  ketaksamaam jenis Hermite-Hadamard sebelah kiri dan kanan diperoleh untuk fungsi yang nilai mutlak terbitan peringkat pertama dan keduanya adalah fungsi kuasi-cembung.  Kata kunci: ketaksamaan Hermite-Hadamard; fungsi-cembung; fungsi kuasi-cembung              1. Introduction Let I be an interval of real numbers. A function  :f I    is said to be convex  on  I  if the inequality   (1 ) ( ) (1 t) ( )f tx t y tf x f y                                 (1)  holds for  all ,x y I  and [0,1]t . If the inequality  (1) is reversed,  then f  is said to be concave.  The following double inequality,  1 ( ) ( )( )2 2baa b f a f af f x dxb a                                   (2)  is well known in the literature as the Hermite-Hadamard integral inequality (see Dragomir & Agarwal 1998), which provides the estimates of the mean value of a continuous convex function defined on an real interval [ , ]a b  with a b . Inequality (2) has attracted many researchers over the years to demostrate new proofs, refinements, extensions and  generalization, see for instance Niculescu and Persson (2003), Ciurdariu (2012),  Iscan (2013), González et al. (2015), Khan et al. (2018)  and the references cited therein.  Definition 1.1.  A function :f I    is said to be quasi-convex  on I  if     (1 ) max ( ), ( )f tx t y f x f y                                         (3)  Alawiah Ibrahim  92  holds for all ,x y I and [0,1]t . If  f  is quasi-concave, then inequality (3) is reversed.  Dragomir and Argawal (1998), Pearce and Pěcarić (2000), Sarikaya and Aktan (2011) provided the inequalities connected with the right part of (3) for functions whose first and second derivatives are convex.  Similar results also have been establised by Ion (2007) and Alomari et al. (2010) for the class of functions whose derivatives absolute values are quasi-convex, where some of the results are pointed out as follows.  Theorem 1.1.  (Ion 2007) Assume ,a b  with a b   and : [ , ]f a b   is a differentiable function on  ( , ),a b  then we have the inequalities         11 111( )sup ( ) , ( ) (4)4[ , ],( ) ( ) 1( )( )2 sup ( ) , ( ) , (5)2( 1) -  1- [ , ].ppppp pp ppbafor f is quasi convex onpf is quasi convexob af a f ba bf a f bf x dxb ab a f a f bpfor n a b               Theorem 1.2.  (Alomari et al. 2010)  Let :f I    be a twice differentiable function on the interior points of  I,  ,a b I  with a b  and  f   be integrable on [ , ]a b . Then we have    11 111 12232( )s  -  1- [ ,up ( ) , ( ) (6)12[ , ],( ) ( ) 1 ( ) (1 )( )2 8 2 ( )sup ( ) ,]( ).( ) , 7p pppp pppppbafor f is quasi convex onpf is quasi convexon a bb af a f ba bf a f b b a pf x dxb a pf a f bfor                        Theorem 1.3.  (Alomari et al. 2010) Let  [ , ]L a b  be class of continuous functions defined on  [ , ]a b . Let : [0, )f I     be a differentiable function on I such that [ , ]f L a b , where ,a b I  with a b . If  1 , 1ppf p   is quasi-convex on [ , ]a b . Then we have  Quasi-convexity and Hermite-Hadamard type inequalities     93            1111111( ) ( ) 1 ( )( ) max ,2 24( 1)(8)max , .2pppp ppppppp ppbaf a f b b a a bf x dx f f bb a pa bf f a                                        The main purpose of this study  is to provide some more inequalities related with  the Hermite-Hadamard type inequality for twice differentiable functions whose the absolute values of the first and second derivatives are quasi-convex. In particular, we generalise the result of Ion (2007). 2. Some  Inequalities of  Hermite-Hadamard Type for Quasi-Convex Functions In order to prove our main results, we need the following integral equalities.   Lemma 2.1.  (Iqbal et al. 2012) Let :f I    be differentiable funtcion on the interior points of  I, where ,a b I with a b .  If [ , ]f L a b  and , [0, )    with  0   , then  ( ) ( ) 1( )baf a f bf x dxb a                    20 0( ) t( )b a t t t tt f a b dt f a b dt                                            (9)  Lemma 2.2.  (Xi & Qi 2013) Let :f I    be a twice differentiable function on the interior points of  I and R .  If [ , ]f L a b  where ,a b I with a b , then  ( ) ( ) 1(1 ) ( )2 2baf a f b a bf f x dxb a                     120( )(2 ) (1 ) (1 )16 2 2b a a b a bt t f t a t f t t b dt                      .         (10)  Lemma 2.3.  (Xi & Qi 2013) Let 0 1  , [0,1]t  and 1r   . Then,  2 112 10(2 ) (2 1)(1 2 ) 1, 0 ,( 1)( 2) 22(2 ) (2 1)(2 1) 1, 1.( 1)( 2) 2r rrr rrr rt t dtrr r                                                   (11)  Alawiah Ibrahim  94  We are now in a position to establish some new integral inequalities of Hermite-Hadamard type for  twice differentiable  and quasi-convex functions.  Theorem 2.1.  Assume ,a b  with a b and : [ , ]f a b  is a differentiable function on ( , )a b .  Then for , [0, )    with  0   ,  the following inequalities hold true  ( ) ( ) 1( )baf a f bf x dxb a               11 1112 222 22max ( ) , ( ) (12)2- [ , ],( )( )max ( ) , ( ) , 1 (13)( ) ( 1)- [ , ].ppp pp ppppb af a f bfor f is quasi convexon a bb af a f b ppfor f is quasi convexon a b                                      Proof.  By Lemma 2.1 with  the fact that f   is a quasi-convex function on [ , ]a b  and  some elementary integration culculus we have  ( ) ( ) 1( )baf a f bf x dxb a                  20 0( )b a t t t tt f a dt t f a b dt                                                                                                              20 0max ( ) , ( )( )b af a f b t dt t dt                      2 22max ( ) , ( )( ) 2b af a f b         .  Also, from Lemma 2.1 and applying the weighted  Hölder integral inequality, one obtains  ( ) ( ) 1( )baf a f bf x dxb a         111120 00 0( ),qpqpqutp tqtp tb at dt f a dtt dt f a b dt                                      Quasi-convexity and Hermite-Hadamard type inequalities     95   where  1p   with  1 1 1p q  .  Using the fact that qf   is quasi convex function on [ , ]a b ,  then we have  ( ) ( ) 1( )baf a f bf x dxb a              11 1 1120 0 0 0max ( ) , ( )( )qp p qquq q p pb a f a f b t dt dt t dt dt                                    1 111 11 12max ( ) , ( )( ) 1 1p pqq qp pq qb a uf a f b up p                      112 22( )( )max ( ) , ( )( ) ( 1)qpq qb af a f bp     . The proof of  Theorem 2.1 is completed.   Remark 2.1.   If  in Theorem 2.1 we choose 1   , then we recapture the Ion’s results (4) and (5).  Theorem 2.2.  Assume ,a b  with a b and : [ , ]f a b   is a twice differentiable function on [ , ]a b  such that ( )f x  is integrable.  If  f   and  1 ( 1)ppf p   are quasi-convex functions on [ , ]a b , then for 0 1  ,  the following inequalities hold true:  ( ) ( ) 1(1 ) ( )2 2baf a f b a bf f x dxb a               11111212( )( ) max ( ) , max , (14)16 2 2[ , ],( ) 1( ) max ( ) ,16 2 2mapppppp pp ppb a a b a bH f a f f f bfor f is quasi convex on a bb a a bH f a f                                                     1111x , , 1 (15)2, [ , ],pppp ppppa bf f b pfor f is quasi convex on a b                         where Alawiah Ibrahim  96     318 3 1 1, 0 ,3 2( )3 1 1, 1,3 2H                         (16)  and  2 12 1(2 ) (2 1)(1 2 ) 1, 0 ,( 1)( 2) 2( )(2 ) (2 1)(2 1) 1, 1.( 1)( 2) 2p pp p ppp pHpp p                                               (17)  Proof:  From  Lemma 2.2  we have  ( ) ( ) 1(1 ) ( )2 2baf a f b a bf f x dxb a                   20( )(2 ) (1 ) (1 )16 2 2tb a a b a bt t f t a t f t t b dt                             (18)    Since f   is a quasi convex function on [ , ]a b , thus from (18) we get  ( ) ( ) 1(1 ) ( )2 2baf a f b a bf f x dxb a              20( )max ( ) , max , (2 )16 2 2tb a a b a bf a f f f b t t dt                                            21 2 2( )( ) max ( ) , max , ,16a b a bb aH f a f f f b         where from Lemma 2.3 we have that    318 3 1 1, 0 ,3 2( )3 1 1, 1.3 2H         Hence, we obtain  the desired result (14). Agains, from  Lemma 2.2  and  applying the  weighted  Hölder’s  inequality for integrals, we have  Quasi-convexity and Hermite-Hadamard type inequalities     97  ( ) ( ) 1(1 ) ( )2 2baf a f b a bf f x dxb a                     1 11 11 1220 01 120( )2 (1 )162 (1 )p qp qqp a bqp a bob at t dt t f t a t dtt t dt t f t t b dt                                           111220( )( ) (1 )16qpqa bpb aH t f t a t dt                       112(1 )qqa bot f t t b dt         for , 1p q    with  1 11p q    and  10( ) 2ppH t t dt   .  Therefore,  by the quasi-convexity of qf  , we obtain  ( ) ( ) 1(1 ) ( )2 2baf a f b a bf f x dxb a                  1111222( ) 1( ) max ( ) ,16 2max ( ) ,q qpqqq a bpqqa bb aH f a ff f b                    1 1111111222( ) 1( ) max ( ) ,16 2max ( ) , ,p ppp ppp pppp ppppa bpa bb aH f a ff f b                   where ( )pH   as given  in (17). Thus, we complete the proof of Theorem 2.2.  References Alomari M., Darus M. &  Dragomir S. S. 2010. New  Inequalities of  Hermite-Hadamard type for  functions whose second derivatives absolute values are quasi-convex. Tamkang Journal of Mathematics  41(4): 357-359. Ciurdariu  L.  2012. A  note  concerning  several   Hermite-Hadamard   type   inequalities  for   different  types  of convex functions. International Journal of Mathematical Analysis 6(33): 1623-1637. Dragomir  S. S & Argawal  R. P. 1998. 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Centre for Modelling and Data Science Faculty of Science and Technology Universiti Kebangsaan Malaysia 43600 UKM Bangi Selangor DE, MALAYSIA E-mail:  alaibra@ukm.edu.my*                                   ___________________ *Corresponding author 

Quasi-convexity and Hermite-Hadamard type inequalities

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Quasi-convexity and Hermite-Hadamard type inequalities

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