X h Inernaional Saisics Days Conference (ISDC 206), Giresun, Turkey New Osrowski Tye Ineualiies for Harmonically Quasi-Convex Funcions Tuncay Köroğlu,*, İmda İşcan 2, Mehme Kun 3,3 Karadeniz Technical Universiy, Faculy of Science, Dearmen of Mahemaics, 6080, Trzon, Turkey 2 Giresun Universiy, Faculy of Science and Ars, Dearmen of Mahemaics, 28200, Giresun, Turkey kor@ku.edu.r, 2 imdai@yahoo.com, 3 mkun@ku.edu.r Absrac In his aer, we have some new Osrowki ye ineualiies and some error esimaes ou he midoin formula for funcions whose derivaives in solue values a cerain owers are harmonically uasi-convex. Keywords: Osrowki ye ineualiies, midoin ye ineualiies, harmonically uasiconvex funcion.. Inroducion The following resul is well known in he lieraure as Osrowski s ineualiy 8; Theorem 2, 9. Le f: a, b R be a differenile maing on (a, b) wih he roery ha f () M for all (a, b). Then f(x) f()d (b a)m + (x 2 )2 4 (b a) (.) 2 for all x a, b. The consan is he bes ossible i means ha i canno be relaced by a smaller 4 consan. The ineualiy (.) can be exressed in he following form: f(x) b a b M f()d a +(b x) 2 b a (x a)2 2. (.2) For some resuls which generalize, imrove and exend he ineualiies (.) and (.2) we refer he reader o he recen aers (see, 2, 4, 5, 8, 9, 0). In 3, I scan gave definiion of harmonically convex funcions and Hermie-Hadamard ye ineualiy for harmonically convex funcions as follows: Definiion 3. Le I R\{0} be a real inerval. A funcion f: I R is said o be harmonically convex, if f ( xy x+( )y ) f(y) + ( )f(x) (.3) for all x, y I and 0,. If he ineualiy in (.3) is reversed, hen f is said o be harmonically concave. Theorem 2 3. Le f: I R\{0} R be a harmonically convex funcion and a, b I wih a < b. If f La, b hen he following ineualiies holds: f ( 2 ) f(x) x 2 f(a)+f(b) dx. 2 769
X h Inernaional Saisics Days Conference (ISDC 206), Giresun, Turkey In, Zhang e al. gave definiion of harmonically uasi-convex funcions as follows: Definiion 2 A funcion I (0, ) 0, ) is said o be harmonically uasi-convex, if f ( xy x+( )y for all x, y I and 0,. ) su{f(x), f(y)} (.4) In 6, Köroğlu e al. gave he following Lemma o rove some Osrowski ye ineualiies and some error esimaes of midoin ye ineualiies for harmonically convex funcions. Lemma. Le f: I R\{0} R be a differenile funcion on I such ha a, b I wih a < b. If f La, b hen f(x) b a b a du = (b a) 0, 0, () = {, (, for all x a, b. () (a+( )b) 2 f ( a+( )b ) d (.5) We consider he following secial funcions which are called as bea and hyergeomeric funcion resecively in he lieraure β(x, y) = Γ(x)Γ(y) = Γ(x+y) 0 x ( ) y d, x, y > 0, 2 F (a, b; c; z) = β(b,c b) 0 b ( ) c b ( z) a d, c > b > 0, z < (see7). We will use he Lemma o rove some Osrowski ye ineualiies and some error esimaes of midoin ye ineualiies for harmonically uasi-convex funcions. 2. New Osorowski Tye Ineualiies Theorem 3 Le f: I (0, ) R be a differenile funcion on I such ha a, b I wih a < b and f L. If f is harmonically uasi-convex on a, b, hen for all x a, b, we have (b a)su{ f (a), f (b) }T (a, b, x) + T 2 (a, b, x) (2.) T (a, b, x) = 2 ( (x a) )2 2 F (2,2; 3; ( a b )), ( (x a) 2b 2 )2 2 F (2,; 3; ( a )) b T 2 (a, b, x) = b ( (x a) ) 2F (2,; 2; ( a b )). + ( (x a) 2 )2 2 F (2,2; 3; ( a )) b Proof. By using Lemma and harmonically uasi-convexiy of f, we have b a b a 770
X h Inernaional Saisics Days Conference (ISDC 206), Giresun, Turkey (b a) 0 + 0 (a+( )b) 2 f ( (a+( )b) (a+( )b) 2 f ( a+( )b ) d a+( )b (b a)su{ f (a), f (b) } 2 d + ) d 2 (a+( )b) 2 d. (2.2) Calculaing aearing inegrals wih hyergeomeric funcions, we have 0 (x a) (a+( )b) 2 d = ( )2 0 (a+( )b) 2 d = 0 u ( ( a b ) u) 2 du = 2 ( (x a) )2 2 F (2,2; 3; ( a b )) = T (a, b, x), (2.3) + 0 d (a+( )b) 2 0 (a+( )b) 2 d (a+( )b) 2 d ( (x a) 2b 2 )2 2 F (2,; 3; ( a )) b = b ( (x a) ) 2F (2,; 2; ( a b )) + ( (x a) 2 )2 2 F (2,2; 3; ( a )) b = T 2 (a, b, x) (2.4) A combinaion of (2.2)-(2.4) we have (2.). This comlees he roof. Corollary In addiion o he condiions of he Theorem 3, if we choose:. f (x) M, for all x a, b, we have he following Osrowski s ye ineualiy f ( 2 ) (b a)m T (a, b, x), (2.5) +T 2 (a, b, x) 2. x = 2, we have he following midoin ye ineualiy for harmonically uasi-convex funcions (b a)su{ f (a), f (b) } T 2 (a, b, ) +T 2 (a, b, 2 (2.6) ). Theorem 4 Le f: I (0, ) R be a differenile funcion on I such ha a, b I wih a < b and f L. If f is harmonically uasi-convex on a, b for, hen for all x a, b, we have (b a)su{ f (a), f (b) } 77
X h Inernaional Saisics Days Conference (ISDC 206), Giresun, Turkey ( 2 ( )2 ) (T3 (a, b, x)) + ( 2 ( )2 + ) (T4 (a, b, x)) (2.7) 2 T 3 (a, b, x) = 2b2 ( )2 2 F (2, 2; 3; ( a )), b 2b 2 2F (2, ; 3; ( a b )) T 4 (a, b, x) = + 2b2 ( )2 2 F (2, 2; 3; ( a )) b, b 2 2F (2, ; 2; ( a b )) Proof. By using Lemma, ower mean ineualiy and harmonically uasi-convexiy of f, we have (b a) 0 + (a+( )b) 2 f ( ( 0 (a+( )b) 2 f ( a+( )b a+( )b ) d ) d (b a) (su{ f (a), f (b) } 0 (a+( )b) + ( ( ) (su{ f (a), f (b) } = (b a)su{ f (a), f (b) } ( 2 ( )2 ) ( 0 (a+( )b) + ( 2 ( )2 + ) 2 ( 2 ( ) (a+( )b) 2 ( ) (a+( )b) 2 2 (2.8) Calculaing aearing inegrals wih hyergeomeric funcions, we have 0 (a+( )b) 2 d = 2b2 ( )2 2 F (2, 2; 3; ( a )) b = T 3 (a, b, x), (2.9) = 0 ( ) (a+( )b) 2 d ( ) d (a+( )b) ( ) 2 0 (a+( )b) 2 d 772
X h Inernaional Saisics Days Conference (ISDC 206), Giresun, Turkey 2b 2 2F (2, ; 3; ( a b )) = + 2b2 ( )2 2 F (2, 2; 3; ( a )) b b 2 2F (2, ; 2; ( a b )) = T 4 (a, b, x). (2.0) A combinaion of (2.8)-(2.0) we have (2.7). This comlees he roof. Corollary 2 In addiion o he condiions of he Theorem 4, if we choose:. f (x) M, for all x a, b, we have he following Osrowski s ye ineualiy (b a)m ( 2 ( )2 ) (T3 (a, b, x)) + ( 2 ( )2, (2.) + ) (T4 (a, b, x)) 2 2. x = 2, we have he following midoin ye ineualiy for harmonically uasi-convex funcions f ( 2 ) (b a)su{ f (a), f (b) } ( 8 ) (T 3 (a, b, 2 )) )) + (T 4 (a, b, 2. (2.2) Theorem 5 Le f: I (0, ) R be a differenile funcion on I such ha a, b I wih a < b and f L. If f is harmonically uasi-convex on a, b for > and + =, hen for all x a, b, we have ( + ) (b a)su{ f (a), f (b) } ( )+ + ( (b x)a )+ T 5 (a, b, x) = b 2 2F (2, ; 2; ( a b )), T 6 (a, b, x) = b 2 2F (2, ; 2; ( a b )). 2F (2, ; 2; ( a )) b b 2 (T 5 (a, b, x)) (2.3) (T 6 (a, b, x)) Proof. By using Lemma, Hölder ineualiy and harmonically uasi-convexiy of f, we have 773
X h Inernaional Saisics Days Conference (ISDC 206), Giresun, Turkey (b a) 0 + (a+( )b) 2 f ( (a+( )b) 2 f ( a+( )b (b a)su{ f (a), f (b) } ( 0 + ( ( 0 ( ) ( (a+( )b) (b a)su{ f (a), f (b) } ( + ( )+ ) + ( + ((b x)a )+ ) ) d a+( )b 2 (a+( )b) ( 0 (a+( )b) ( (a+( )b) 2 2 ) d 2. (2.4) Calculaing aearing inegrals wih hyergeomeric funcions, we have 0 (a+( )b) 2 d = b 2 2F (2, ; 2; ( a )) b = T 5 (a, b, x), (2.5) = (a+( )b) 2 d = 0 b 2 b 2 2F (2, ; 2; ( a b )) d (a+( )b) 2 0 2F (2, ; 2; ( a )) b = T 6 (a, b, x), (2.6) (a+( )b) 2 d A combinaion of (2.4)-(2.6) we have (2.3). This comlees he roof. Corollary 3 In addiion o he condiions of he Theorem 5, if we choose:. f (x) M, for all x a, b, we have he following Osrowski s ye ineualiy ( + ) (b a)m ( )+ + ( (b x)a )+ (T 5 (a, b, x)), (2.7) (T 6 (a, b, x)) 2. x = 2, we have he following midoin ye ineualiy for harmonically uasi-convex funcions 774
X h Inernaional Saisics Days Conference (ISDC 206), Giresun, Turkey f ( 2 ) 2 ( 2(+) ) (b a)su{ f (a), f (b) } (T 5 (a, b, 2 )) + (T6 (a, b, 2 )). (2.8) Theorem 6 Le f: I (0, ) R be a differenile funcion on I such ha a, b I wih a < b and f L. If f is harmonically uasi-convex on a, b for > and + =, hen for all x a, b, we have (T 7 (a, b, x)) ( ) (b a)su{ f (a), f (b) } +(T 8 (a, b, x)) ( (b x)a (2.9) ) T 7 (a, b, x) = T 8 (a, b, x) = (+)b2 ( )+ 2 F (2, + ; + 2; ( a )), b ( ) (a+( )b) 2 d. Proof. By using Lemma, Hölder ineualiy and harmonically uasi-convexiy of f, we have (b a) 0 + (a+( )b) 2 f ( (a+( )b) 2 f ( (b a)su{ f (a), f (b) } ( 0 + ( (a+( )b) ( ) (a+( )b) 2 2 ( 0 ( a+( )b a+( )b ) d ) d Since he aearing inegrals are as he following, we have 0 (a+( )b) 2 d = (+)b2 (. (2.20) )+ 2 F (2, + ; + 2; ( a )) b 775
X h Inernaional Saisics Days Conference (ISDC 206), Giresun, Turkey = T 7 (a, b, x), (2.2) ( ) (a+( )b) 2 d = T 8(a, b, x). (2.22) A combinaion of (2.20)-(2.22) we have (2.9). This comlees he roof. Corollary 4 In addiion o he condiions of he Theorem 6, if we choose:. f (x) M, for all x a, b, we have he following Osrowski s ye ineualiy (b a)m (T 7(a, b, x)) ( ) + (T 8 (a, b, x)) ( (b x)a ), (2.23) 2. x = 2, we have he following midoin ye ineualiy for harmonically uasi-convex funcions f ( 2 ) ( 2 ) (b a)su{ f (a), f (b) } References (T 7 (a, b, 2 )) )) + (T 8 (a, b, 2. (2.24) M. Alomari, M. Darus, Some Osrowski's ye ineualiies for convex funcions wih alicaions, RGMIA Res.Re. Collec., 3() (200), Aricle 3, -4. 2 S. S. Dragomir, T. M. Rassias, Osrowski ye ineualiies and alicaions in numerical inegraion, Kluwer Academic Publishers, 2002. 3 İ. İşcan, Hermie-Hadamard ye ineualiies for harmonically convex funcions, Hace. J. Mah. Sa., 43 (6) (204), 935-942. 4 İ. İşcan, Osrowski ye ineualiies for harmonically s-convex funcions, Konural Jurnal of Mahemaics, Volume 3, No (205) 63-74. 5 İ. İşcan, S. Numan, Osrowski ye ineualiies for harmonically uasi-convex funcions, Elec. J. Mah. Anal. A., 2(2) (204) 89-98. 6 T. Köroğlu, İ. İşcan, M. Kun, New Osrowski ye ineualiies for harmonically convex funcions, doi:0.340/rg.2..4269.3364, Availle online a hs://www.researchgae.ne/ublicaion/306000839, (206). 7 A. A. Kilbas, H. M. Srivasava, J. J. Trujillo, Theory and Alicaions of Fracional Differenial Euaions, Elsevier, Amserdam, 2006. 8 A. Osrowski, Über die Absoluweichung einer differenienbaren Funkionen von ihren Inegralmielwer. Commen. Mah. Hel, 0 (938), 226--227. 9 M. E. Özdemir, Ç. Yıldız, New Osrowski Tye Ineualiies for Geomerically Convex Funcions, In. J. Modern Mah. Sci., 8() (203) 27-35. 0 E. Se, M. E. Özdemir, M. Z. Sarıkaya, New ineualiies of Osrowski's ye for s-convex funcions in he second sense wih alicaions, Faca Uni. Ser. Mah. Inform. 27() (202) 67-82. T.-Y. Zhang, A.-P. Ji, F. Qi, Inegral ineualiies of Hermie-Hadamard ye for harmonically uasiconvex funcions. Proc. Jangjeon Mah. Soc, 6(3) (203) 399-407. 776