Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 32 (2016), ISSN

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1 Aca Mahemaica Academiae Paedagogicae Nyíregyháziensis 3 6, ISSN 76-9 INTEGRAL INEQUALITIES OF HERMITE HADAMARD TYPE FOR FUNCTIONS WHOSE DERIVATIVES ARE STRONGLY α-preinvex YAN WANG, BO-YAN XI, AND FENG QI Asrac. In he paper, he auhors inroduce a new noion srongly α- preinvex funcion, esalish an inegral ideniy for he newly inroduced funcion, and find some Hermie Hadamard ype inegral inequaliies for a funcion ha he power of he asolue value of is firs derivaive is srongly α-preinvex.. Inroducion Le us recall some definiions of various convex funcions. Definiion.. A funcion f : I R =, R is said o e convex if. fλx λy λfx λfy holds for all x, y I and λ, ]. Definiion.,, ]. A se S R n is said o e invex wih respec o he map η : S S R n if for every x, y S and, ]. y ηx, y S. Definiion.3 ]. Le S R n e an invex se wih respec o η : S S R n. For every x, y S, he η-pah P xv joining he poins x and v = x ηy, x is defined y.3 P xv = {z z = x ηy, x,, ]}. Mahemaics Sujec Classificaion. 6D5; Secondary 6A5, 6B, A55. Key words and phrases. Hermie Hadamard ype inegral inequaliies; invex se; srongly α-preinvex funcion. This work was parially suppored y he Naional Naural Science Foundaion of China under Gran No. 363 and y he Inner Mongolia Auonomous Region Naural Science Foundaion Projec under Gran No. 5MS3, China. 79

2 Y. WANG, B.-Y. XI, AND F. QI Definiion. ]. Le S R n e an invex se wih respec o η : S S R n. A funcion f : S R is said o e preinvex wih respec o η, if for every x, y S and, ],. fy ηx, y fx fy. Definiion.5 ]. For f : a, ] R, if.5 fλx λy λfx λfy cλ λx y is valid for all x, y a, ] and λ, ],c >, hen we say ha fx is a srongly convex funcion on a, ]. Le us reformulae some inequaliies of Hermie Hadamard ype for he aove menioned convex funcions. Theorem. 5, Theorem.]. Le f : I R R e a differeniale mapping and a, I wih a <. If f x is convex on a, ], hen.6 fa f fx d x a a f a f. a Theorem., Theorem.]. Le A R e an open invex suse wih respec o θ : A A R and le f : A R e a differeniale funcion. If f is preinvex on A, hen for every a, A wih θa, we have.7 f f θa, θa, θa, fx d x θa, f a f ]. For more informaion on Hermie Hadamard ype inequaliies for various convex funcions, please refer o recenly pulished aricles 3,, 6, 7,,, 3] and closely relaed references herein. In his aricle, we will inroduce a new noion α-preinvex funcion, esalish an inegral ideniy for such a kind of funcions, and find some Hermie Hadamard ype inegral inequaliies for a funcion ha he power of he asolue value of is firs derivaive is α-preinvex.. A new definiion and wo lemmas The so-called srongly α-preinvex funcion may e inroduced as follows. Definiion.. Le S R n e an invex se wih respec o η : S S R n. A funcion f : S R is said o e srongly α-preinvex wih respec o η for α, ] and c >,if for every x, y S and, ],. fy ηx, y α fx α fy c x y. For esalishing our new inegral inequaliies of Hermie Hadamard ype for srongly α-preinvex funcions, we need he following inegral ideniy.

3 INTEGRAL INEQUALITIES OF HERMITE HADAMARD TYPE Lemma.. Le A R e an open invex suse wih respec o θ : A A R and le a, A wih θa,. If f : A R is a differeniale funcion and f is inegrale on he θ-pah P c, c = θa,, hen θa, θa, = θa, fx d x f θa, f θa, f ] θa, d. Proof. Since a, A and A is an invex se wih respec o θ, for every, ], we have θa, A. Inegraing y par gives f θa, f ] θa, d = f θa, f θa, d θa, f θa, f ] θa, d = θa, f θa, f θa, θa, = θa, θa, θa, θa, θa, fx d x f θa, ] fx d x f θa,. θa, The proof of Lemma. is compleed. By he same way, we oain anoher lemma. θa,]/ ] fx d x Lemma.. Le A R e an open invex suse wih respec o θ : A A R and le a, A wih θa,. If f : A R is a differeniale funcion and f is inegrale on he θ-pah P c, c = θa,, hen f θa, = θa, θa, fx d x f θa, f ] θa, d. 3. Some new inegral inequaliies of Hermie Hadamard ype We are now in a posiion o esalish some Hermie Hadamard ype inegral inequaliies for a funcion ha he power of he asolue value of is firs derivaive is srongly α-preinvex.

4 Y. WANG, B.-Y. XI, AND F. QI Theorem 3.. Le A R e an open invex suse wih respec o θ : A A R, a, A wih θa,. Suppose f : A R e a differeniale funcion and f is inegrale on he θ-pah P c, c = θa,, some α, ]. If f q is srongly α-preinvex on A for q, hen θa, 3. fx d x f θa, θa, θa, α α /q{ 3 3 α f a q 9 α α 3 α f q α α ca ] /q 3 f a q 3 α α f q 5 α α ca ] /q }. Proof. Since θa, A for every, ], y Lemma. and Hölder s inequaliy, we have θa, fx d x f θa, θa, θa, f θa, f ] θa, d { θa, /q d f q ] /q θa, d /q d f q ] /q } θa, d { /q θa, 3 α f a q α f q ] /q ca d /q α f a q α ] /q } f q ca d θa, /q{ = 3 α α q α f a q 3 α α α f q 3 α α ca ] /q f a q α α f q 5 3 α α ca ] /q }. The proof of Theorem 3. is compleed.

5 INTEGRAL INEQUALITIES OF HERMITE HADAMARD TYPE 3 Corollary. Under he condiions of Theorem 3., if α = q =, we have θa, θa, fx d x f θa, θa, f a q f q ca ]. Theorem 3.. Le A R e an open invex suse wih respec o θ : A A R, a, A wih θa,. Suppose f : A R e a differeniale funcion and f is inegrale on he θ-pah P c, c = θa,, some α, ]. If f q is srongly α-preinvex on A for q >, hen θa, 3. fx d x f θa, θa, θa, q /q q /q q α α q ] /q { α f a q α α f q 3 α α ca ] /q } f a q α α f q 3 α α ca. Proof. Since θa, A for every, ], y Lemma., Hölder s inequaliy, we have θa, 3.3 fx d x f θa, θa, θa, f θa, f ] θa, d { θa, /q q/q d f q ] /q θa, d /q q/q d f q ] /q } θa, d /q { θa, q q /q α q f a q q α f q ] /q ca d α α ] /q } f a q f q ca d /q θa, q /q{ = q /q q α α q

6 Y. WANG, B.-Y. XI, AND F. QI ] /q α f a q α α f q 3 α α ca ] /q } f a q α α f q 3 α α ca. The proof of Theorem 3. is complee. Corollary. Under he condiions of Theorem 3., if α =, we have θa, fx d x f θa, θa, /q θa, q /q{ q /q q q f a q 3 f q 3 ca ] /q f a q 3 f q 3 ca ] /q }. Theorem 3.3. Le A R e an open invex suse wih respec o θ : A A R, a, A wih θa,. Suppose f : A R e a differeniale funcion and f is inegrale on he θ-pah P c, c = θa,, some α, ]. If f q is srongly α-preinvex on A for q, hen f θa, 3. fx d x θa, θa, α α α /q{ α3 α α 3 f a q α α α α3 α α 3] f q 5 α 3 α α ca ] /q 3 /q α 3 f a q 3 α α α α 3 f q 3 α α α ca ] /q }. Proof. Since θa, A for every, ], y Lemma., Hölder s inequaliy, we have f θa, fx d x θa, θa, f θa, f ] θa, d { θa, /q d f q ] /q θa, d /q d f q ] /q } θa, d

7 INTEGRAL INEQUALITIES OF HERMITE HADAMARD TYPE 5 { /q θa, α α f a q f q /q /q 3 α ca d ] f a q α ] /q } f q ca d θa, /q{ = α3 α α 3 f a q α α α α α α α3 α α 3] f q 5 α 3 α α ca ] /q 3 /q α 3 f a q 3 α α α α 3 f q 3 α α α ca ] /q }. The proof of Theorem 3.3 is complee. Corollary 3. Under he condiions of Theorem 3.3, if α = q =, we have f θa, fx d x θa, θa, f a q 7 f q ca ]. Theorem 3.. Le A R e an open invex suse wih respec o θ : A A R, a, A wih θa,. Suppose f : A R e a differeniale funcion and f is inegrale on he θ-pah P c, c = θa,,some α, ]. If f q is srongly α-preinvex on A for q >, hen f θa, 3.5 fx d x θa, θa, α α /q{ α f a q α α 3 /q f a q α f q 3 α α ca ] /q α α f q 3 α α ca ] /q }. Proof. Since θa, A for every, ], y Lemma., Hölder s inequaliy, we have f θa, fx d x θa, θa, f θa, f ] θa, d { θa, /q q/q d f q ] /q θa, d

8 6 Y. WANG, B.-Y. XI, AND F. QI /q q/q d f q ] /q } θa, d { /q θa, α α f a q f q /q /q 3 α ca d ] f a q α ] /q } f q ca d θa, /q{ = α f a q α α α α α f q 3 α α ca ] /q 3 /q f a q α α f q 3 α α ca ] /q }. The proof of Theorem 3. is complee. Corollary. Under he condiions of Theorem 3., if α =, we have f θa, fx d x θa, θa, /q{ 3 f a q 3 f q 3 ca ] /q 3 /q f a q 3 f q 3 ca ] /q }. Remark. On April, Professor S. S. Dragomir, Ausralia poined ou ha here are errors appeared in he paper 9] as follows. Definiion. from 9] has no meaning if i sands for he imaginary uni. So, no inequaliies like in he hypohesis of Theorem 3. or eq. 3. and ohers can e saed for general ϕ. References ] T. Anczak. Mean value in invexiy analysis. Nonlinear Anal., 6:73, 5. ] A. Barani, A. G. Ghazanfari, and S. S. Dragomir. Hermie-Hadamard inequaliy for funcions whose derivaives asolue values are preinvex. J. Inequal. Appl., pages :7, 9,. 3] S. S. Dragomir. On Hadamard s inequaliies for convex funcions. Mah. Balkanica N.S., 63:5, 99. ] S. S. Dragomir. Two mappings in connecion o Hadamard s inequaliies. J. Mah. Anal. Appl., 67:9 56, 99. 5] S. S. Dragomir and R. P. Agarwal. Two inequaliies for differeniale mappings and applicaions o special means of real numers and o rapezoidal formula. Appl. Mah. Le., 5:9 95, 99.

9 INTEGRAL INEQUALITIES OF HERMITE HADAMARD TYPE 7 6] S. S. Dragomir, J. Pečarić, and L. E. Persson. Some inequaliies of Hadamard ype. Soochow J. Mah., 3:335 3, ] S. S. Dragomir, J. E. Pečarić, and J. Sándor. A noe on he Jensen-Hadamard inequaliy. Anal. Numér. Théor. Approx., 9:9 3, 99. ] D. A. Ion. Some esimaes on he Hermie-Hadamard inequaliy hrough quasi-convex funcions. An. Univ. Craiova Ser. Ma. Inform., 3:3, 7. 9] W.-D. Jiang, D.-W. Niu, and F. Qi. Some inequaliies of Hermie Hadamard ype for r-ϕ-preinvex funcions. Tamkang J. Mah., 5:3 3,. ] B. T. Polyak. Exisence heorems and convergence of minimizing sequences in exremum prolems wih resicions. Sovie Mah. Dokl., 357:7 75, 966. ] F. Qi, Z.-L. Wei, and Q. Yang. Generalizaions and refinemens of Hermie-Hadamard s inequaliy. Rocky Mounain J. Mah., 35:35 5, 5. ] B.-Y. Xi, R.-F. Bai, and F. Qi. Hermie-Hadamard ype inequaliies for he m- and α, m-geomerically convex funcions. Aequaiones Mah., 3:6 69,. 3] B.-Y. Xi and F. Qi. Some inegral inequaliies of Hermie-Hadamard ype for convex funcions wih applicaions o means. J. Func. Spaces Appl., pages Ar. ID 93,,. ] X. M. Yang and D. Li. On properies of preinvex funcions. J. Mah. Anal. Appl., 56:9,. Received January 7, 5. Yan Wang, College of Mahemaics, Inner Mongolia Universiy for Naionaliies, Tongliao Ciy, Inner Mongolia Auonomous Region, 3, China address: sella@vip.qq.com Bo-Yan Xi, College of Mahemaics, Inner Mongolia Universiy for Naionaliies, Tongliao Ciy, Inner Mongolia Auonomous Region, 3, China address: aoyinu7@imun.edu.cn Feng qi corresponding auhor, Deparmen of Mahemaics, College of Science, Tianjin Polyechnic Universiy, Jiaozuo Ciy, Henan Province, 5, China address: qifeng6@homail.com