Some integral inequalities of the Hermite Hadamard type for log-convex functions on co-ordinates

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1 Avilble online t J. Nonliner Si. Appl. 9 06), Reserh Artile Some integrl inequlities o the Hermite Hdmrd type or log-onvex untions on o-ordintes Yu-Mei Bi, Feng Qi b,, College o Mthemtis, Inner Mongoli University or Ntionlities, Tonglio City, Inner Mongoli Autonomous Region, Chin. b Deprtment o Mthemtis, College o Siene, Tinjin Polytehni University, Tinjin City, 30060, Chin. Institute o Mthemtis, Henn Polytehni University, Jiozuo City, Henn Provine, 500, Chin. Communited by Sh. Wu Abstrt In the pper, the uthors estblish some new integrl inequlities or log-onvex untions on o-ordintes. These newly-estblished inequlities re onneted with integrl inequlities o the Hermite Hdmrd type or log-onvex untions on o-ordintes. 06 All rights reserved. Keywords: Log-onvex untions, o-ordintes, integrl inequlity, Hermite Hdmrd type. 00 MSC: 6A5, 6D5, 6D0, 6E60, A55.. Introdution Let : I R R be onvex untion on n intervl I nd let, b I suh tht < b. Then the double inequlity ) + b ) + b) x) b holds. This double inequlity is known in the literture s the Hermite Hdmrd integrl inequlity. Deinition.. I positive untion : I R R + = 0, ) stisies λx + λ)y) x) λ y) λ or ll x, y I nd λ 0,, then we sy tht is logrithmilly onvex or simply, log-onvex) untion on I. I the bove inequlity is reversed, then we sy tht is log-onve untion. Corresponding uthor Emil ddresses: biym008@sohu.om Yu-Mei Bi), qieng68@gmil.om, qieng68@hotmil.om Feng Qi) Reeived

2 Y.-M. Bi, F. Qi, J. Nonliner Si. Appl. 9 06), Equivlently, untion is log-onvex on I i nd only i is positive nd its logrithm ln is onvex on I. Moreover, i the seond derivtive exists on I, then is log-onvex i nd only i ) 0. A orresponding version o the Hermite Hdmrd integrl inequlity or log-onvex untions ws given in 5 s ollows. Theorem. 5). Suppose tht :, b R R + is log-onvex untion on, b. Then ) + b b x) L), b)), where Lx, y) is the logrithmi men y x Lx, y) = ln y ln x, x y, x, x = y. In 3,, the so-lled onvex untions on o-ordintes were introdued s ollows. Deinition.3 3, ). A untion : =, b, d R R is sid to be onvex on o-ordintes on with < b nd < d i the prtil mppings y :, b R, y u) = y u, y) nd x :, d R, x v) = x x, v) re onvex or ll x, b) nd y, d). Deinition. 3, ). A untion : =, b, d R R is sid to be onvex on o-ordintes on with < b nd < d i the inequlity tx + t)z, λy + λ)w) tλx, y) + t λ)x, w) + t)λz, y) + t) λ)z, w) holds or ll t, λ 0, n, y), z, w). An inequlity o the Hermite Hdmrd type or onvex untion on o-ordintes on retngle rom the plne R ws estblished in 3, s ollows. Theorem.5 3,, Theorem.). Let : =, b, d R R be onvex on o-ordintes on with < b nd < d. Then one hs + b, + d ) b x, + d ) + ) + b b d, y d y b )d ) b b x, y) d y x, ) + x, d) + d, ) + b, ) +, d) + b, d)., y) + b, y) d y In, Alomri nd Drus introdued lss o log-onvex untions on o-ordintes s ollows. Deinition.6 ). A untion : =, b, d R R + is lled log-onvex on o-ordintes on with < b nd < d i tx + t)z, λy + λ)w) x, y) tλ x, w) t λ) z, y) t)λ z, w) t) λ) holds or ll t, λ 0, n, y), z, w).

3 Y.-M. Bi, F. Qi, J. Nonliner Si. Appl. 9 06), Remrk.7. I nd g re both log-onvex on o-ordintes on, then their omposite g is lso log-onvex on o-ordintes on. An inequlity o the Hermite Hdmrd type or log-onvex untions on o-ordintes on retngle rom the plne R ws estblished by Alomri nd Drus in s ollows. Theorem.8, Theorem 3.3). Suppose tht : =, b, d R R + is log-onvex on oordintes on or < b nd < d. Let A =, )b, d), d) b, ), B =, nd C = b, ), d) b, d) b, d). Then the inequlity, A = B = C =, B C ln B ln C, A =, HC), B =, HB), C =, C ln C, A = B =, I = b )d ) x, y) d y b, d) B ln B, A = C =, γ + ln ln A) + Ei, ln A), B = C =, ln A B ln B + AB, A, B, C > 0, lnab) 0 C β AB lnab) d β, otherwise holds, where γ is the Euler onstnt, Hx) = nd Ei, ln A) + ln ln x Ei, lnax)) ln lnax) ln A is the exponentil integrl untion. e t Eix) = V.P. d t x t ln ln A ln ln A), < ln x + ln A ln A < 0, 0, otherwise; For more nd detiled inormtion on this topi, plese reer to the newly published ppers, 6 5 nd plenty o reerenes therein.. Some new integrl inequlities o the Hermite Hdmrd type In this setion, we prove some new inequlities o the Hermite Hdmrd type or log-onvex untions on o-ordintes.

4 Y.-M. Bi, F. Qi, J. Nonliner Si. Appl. 9 06), Theorem.. Let : =, b, d R R + or < b nd < d be log-onvex on o-ordintes on. Then one hs b )d ) x, y) d y where Lu, v) is the logrithmi men. b L x, ), x, d) ) + L, y), b, y)) d y b d b d x, ) + x, d) +, y) + b, y) d y b d ) ) L, ), b, )) + L, d), b, d)) + L, ),, d) + L b, ), b, d), ) + b, ) +, d) + b, d), Proo. For ll x, y > 0, it is known tht Lx, y) x+y. Setting y = λ + λ)d or ll 0 λ, using the log-onvexity o, nd by the rithmeti-geometri men inequlity, we obtin b )d ) x, y) d y = b b = b 0 b ) 0 x, λ + λ)d) d λ x, ) λ x, d) λ d λ Lx, ), x, d)) x, ) + x, d). Sine x, ), ) t b, ) t n, d), d) t b, d) t or eh t 0,, we hve b ) By similr rgument, we n obtin b )d ) x, + x, d) {, ) t b, ) t +, d) t b, d) t} 0 = L, ), b, )) + L, d), b, d)) The proo o Theorem. is thus omplete., ) + b, ) +, d) + b, d). x, y) d y d L, y), b, y)) d y, y) + b, y) d y d ) ) ) L, ),, d) + L b, ), b, d), ) + b, ) +, d) + b, d). Exmple.. The untion x, y) = x y + is log-onvex on o-ordintes on =,. In Theorem.8, sine A = B = C =, we hve b )d ) x, y) d y = 0 9 < = b, d).

5 Y.-M. Bi, F. Qi, J. Nonliner Si. Appl. 9 06), By Theorem., we obtin b )d ) = b x, y) d y = 0 9 < 3 L x, ), x, d) ) + d L, y), b, y)) d y <. Theorem.3. Let : =, b, d R R + with < b nd < d be log-onvex on o-ordintes on. Then + b, + d ) { b x, + d ) + b x, + d ) / b + ) ) + b + b / d, y, + d y d y} b x, + d ) + ) + b b d, y d y.) d { x, y)x, + d y) / b )d ) + x, y) + b x, y) /} d y b )d ) x, y) d y. Proo. Utilizing the log-onvexity o leds to + b, + d ) = t + t)b + t) + tb, + d + + d )) t + t)b, + d ) t) + tb, + d ) /.) or ll 0 t. On using the hnge o the vrible x = t + t)b or 0 t, integrting the inequlity.) over t on 0,, nd by the rithmeti-geometri men inequlity, we proure + b, + d ) t + t)b, + d ) t) + tb, + d ) / d t 0 = x, + d ) + b x, + d ) /.3) b x, + d ). b Using the log-onvexity o, we ind x, + d ) x, λ + λ)d)x, λ) + λd) / or ll 0 λ n, b. Integrting the inequlity.) with respet to x, λ) on, b 0, nd using the inequlity.3) give b x, + d ) / x, λ + λ)d)x, λ) + λd) d λ b b = 0 b )d ) b )d ) x, y) x, + d y ) / d y x, y) d y..).5)

6 Y.-M. Bi, F. Qi, J. Nonliner Si. Appl. 9 06), Similrly, we obtin + b, + d ) d d b )d ) b )d ) ) + b, y ) + b, y d y ) + b /, + d y d y x, y) + b x, y ) / d y x, y) d y. A ombintion o.3),.5), nd the lst inequlity gives the desired inequlity.). Theorem.3 is thus proved. Mking use o Theorem.3, we derive the ollowing orollry. Corollry.. Let, g : =, b, d R R + with < b nd < d be log-onvex on o-ordintes on. Then + b, + d ) + b g, + d ) { b x, + d ) g x, + d ) + b x, + d ) g + b x, + d ) / b + ) ) ) ) + b + b + b + b / d, y g, y, + d y g, + d y d y} b x, + d ) g x, + d ) + ) ) + b + b b d, y g, y d y d { x, y)gx, y)x, + d y)gx, + d y) / b )d ) + x, y)gx, y) + b x, y)g + b x, y) /} d y b )d ) x, y) d y. Theorem.5. Let : =, b, d R R + with < b nd < d be log-onvex on o-ordintes on. Then + b, + d ) b x, + d ) + b x, + d ) / b + ) ) + b + b / d, y, + d y d y d / x, y)x, + d y) + b x, y) + b x, + d y) d y b )d ) d { x, y)x, + d y) / + x, y) + b x, y) /} d y b )d ) b )d ) x, y) d y.

7 Y.-M. Bi, F. Qi, J. Nonliner Si. Appl. 9 06), Proo. Sine is log-onvex on o-ordintes on, using the inequlities.3),.5), nd the rithmetigeometri inequlity igures out + b, + d ) x, + d ) + b x, + d ) / b x, λ + λ)d)x, λ) + λd) b = 0 + b x, λ + λ)d) + b x, λ) + λd) / d λ b )d ) + b x, + d y) / d y b )d ) Similrly, we obtin + b, + d ) d b )d ) b )d ) x, y)x, + d y) + b x, y) { x, y)x, + d y) / + x, y) + b x, y) /} d y x, y) d y. ) ) + b + b /, y, + d y d y + b x, + d y) / d y b )d ) b )d ) Hene, the proo o Theorem.5 is omplete. x, y)x, + d y) + b x, y) { x, y)x, + d y) / + x, y) + b x, y) /} d y x, y) d y. Corollry.6. Let, g : =, b, d R R + with < b nd < d be log-onvex on o-ordintes on. Then + b, + d ) + b g, + d ) b x, + d ) g x, + d ) + b x, + d ) g + b x, + d ) / b + ) ) ) ) + b + b + b + b / d, y g, y, + d y g, + d y d y d x, y)gx, y)x, + d y)gx, + d y) b )d ) + b x, y)g + b x, y) + b x, + d y)g + b x, + d y) / d y b )d ) { x, y)gx, y)x, + d y)gx, + d y) / + x, y)gx, y) + b x, y)g + b x, y) /} d y b )d ) x, y) d y.

8 Y.-M. Bi, F. Qi, J. Nonliner Si. Appl. 9 06), Theorems. nd.3 n be improved s ollows. Corollry.7. Under the onditions o Theorems. nd.3, i x, y) = x)g y) or x, y), then ) ) + b + d g b + d b )d ) x) + b x) ) / + d )g g x)g + d y) ) / + d d y ) x)g y) + b x)g + d y) / d y x)g y) d y b )d ) b L x)g ), x)g d) ) b L )g y), b)g y)) d y + d g ) + g d) b x) + ) + b) d g y) d y b d L ), b))g ) + g d) + ) + b)lg ), g d)) ) + b)g ) + g d). 3. Conlusions By the rithmeti-geometri inequlity nd other tehniques, we estblish some new integrl inequlities or log-onvex untions on o-ordintes. These newly-estblished inequlities re onneted with integrl inequlities o the Hermite Hdmrd type or log-onvex untions on o-ordintes. Aknowledgment This work ws prtilly supported by the Ntionl Nturl Siene Foundtion o Chin under Grnt No nd by the Foundtion o the Reserh Progrm o Siene nd Tehnology t Universities o Inner Mongoli Autonomous Region under Grnt No. NJZY680, Chin. The uthors ppreite the ommuniting editor nd nonymous reerees or their kind help, reul orretions, nd vluble omments on the originl version o this pper. Reerenes M. Alomri, M. Drus, On the Hdmrd s inequlity or log-onvex untions on the oordintes, J. Inequl. Appl., ), 3 pges.,.6,,.8 S.-P. Bi, F. Qi, S.-H. Wng, Some new integrl inequlities o Hermite-Hdmrd type or α, m; P )-onvex untions on o-ordintes, J. Appl. Anl. Comput., 6 06), S. S. Drgomir, On the Hdmrds inequllity or onvex untions on the o-ordintes in retngle rom the plne, Tiwnese J. Mth., 5 00), ,.3,.,,.5 S. S. Drgomir, C. E. M. Pere, Seleted topis on Hermite Hdmrd inequlities nd pplitions, RGMIA Monogrphs, Vitori University, 00).,.3,.,,.5 5 P. M. Gill, C. E. M. Pere, J. Pečrić, Hdmrd s inequlity or r-onvex untions, J. Mth. Anl. Appl., 5 997), 6 70.,. 6 X.-Y. Guo, F. Qi, B.-Y. Xi, Some new inequlities o Hermite-Hdmrd type or geometrilly men onvex untions on the o-ordintes, J. Comput. Anl. Appl., 06), 55.

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