New Ostrowski Type Inequalities for Harmonically Quasi-Convex Functions

Transcription

1 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 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

3 X h Inernaional Saisics Days Conference (ISDC 206), Giresun, Turkey (b a) (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

4 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

5 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

6 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

7 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

8 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, S. S. Dragomir, T. M. Rassias, Osrowski ye ineualiies and alicaions in numerical inegraion, Kluwer Academic Publishers, İ. İşcan, Hermie-Hadamard ye ineualiies for harmonically convex funcions, Hace. J. Mah. Sa., 43 (6) (204), İ. İşcan, Osrowski ye ineualiies for harmonically s-convex funcions, Konural Jurnal of Mahemaics, Volume 3, No (205) İ. İşcan, S. Numan, Osrowski ye ineualiies for harmonically uasi-convex funcions, Elec. J. Mah. Anal. A., 2(2) (204) T. Köroğlu, İ. İşcan, M. Kun, New Osrowski ye ineualiies for harmonically convex funcions, doi:0.340/rg , Availle online a hs:// (206). 7 A. A. Kilbas, H. M. Srivasava, J. J. Trujillo, Theory and Alicaions of Fracional Differenial Euaions, Elsevier, Amserdam, A. Osrowski, Über die Absoluweichung einer differenienbaren Funkionen von ihren Inegralmielwer. Commen. Mah. Hel, 0 (938), M. E. Özdemir, Ç. Yıldız, New Osrowski Tye Ineualiies for Geomerically Convex Funcions, In. J. Modern Mah. Sci., 8() (203) 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) 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)