Some Hermite-Hadamard type inequalities for functions whose exponentials are convex
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1 Stud. Univ. Beş-Bolyi Mth. 6005, No. 4, Some Hermite-Hdmrd type inequlities for functions whose exponentils re convex Silvestru Sever Drgomir nd In Gomm Astrct. Some inequlities of Hermite-Hdmrd type for functions whose exponentils re convex re otined. Mthemtics Suject Clssifiction 00: 6D5, 5D0. Keywords: Convex functions, Hermite-Hdmrd inequlity, specil mens.. Introduction The following integrl inequlity + f f t dt f + f,. which holds for ny convex function f :, ] R, is well known in the literture s the Hermite-Hdmrd inequlity. There is n extensive mount of literture devoted to this simple nd nice result which hs mny pplictions in the Theory of Specil Mens nd in Informtion Theory for divergence mesures, from which we would like to refer the reder to the monogrph ] nd the references therein. We denote y Expconv I the clss of ll functions defined on the intervl I of rel numers such tht exp f is convex on I. If Conv I is the clss of convex functions defined on I then we hve the following fct: Proposition.. We hve the strict inclusion Conv I Expconv I. Proof. If f is convex, then expf is log-convex nd therefore convex on I nd the inclusion is proved. For r the function f r x = r x, x > 0 is concve on 0,. We hve exp f r x = x r is convex function, therefore f r Expconv I \ Conv I.
2 58 Silvestru Sever Drgomir nd In Gomm We oserve tht for twice differentile functions g on I, the interior of I we hve tht exp g x = g x] + g x exp g x, x I, therefore g Expconv I if nd only if g x] + g x 0 for ny x I.. Some Hermite-Hdmrd type inequlities Now, if g Expconv I, then y the Hermite-Hdmrd inequlity for exp g we hve for, I with < tht + exp g exp g t dt exp g + exp g ].. By Jensen s integrl inequlity for the exp function we lso hve for ny integrle function h :, ] R tht exp h t dt exp h t dt.. We define the logrithmic men s if =, L = L, := if,, > 0. We cn improve the inequlity. for convex functions s follows: Theorem.. Let f : I R e convex function on I nd, I with <. Then we hve for f f the inequlities + exp f exp f t dt exp f t dt.3 exp f exp f f f exp f + exp f ]. Proof. The first inequlity follows y Hermite-Hdmrd inequlity for the convex function f. The second inequlity follows y.. It is know tht if g is log convex, then y ] g t dt L g, g..4 Since f is convex, then g = exp f is log-convex nd y.4 we get the third inequlity in.3.
3 Some Hermite-Hdmrd type inequlities 59 A recent pper connected with such results is 4]. Consider the identric men of two positive numers if =, I = I, := e if, We oserve tht I, = for, > 0,. The following result holds: udu, > 0. Theorem.. Assume tht f Expconv I nd, I with <. Then we hve exp f t dt I exp f, exp f.5 nd + exp f exp ] exp f x + exp f + x dx exp f x dx. Proof. Since f Expconv I, then exp f λ + λ λ exp f + λ exp f for ny λ 0, ], which is equivlent to.6 f λ + λ λ exp f + λ exp f ].7 for ny λ 0, ]. Integrting.7 on 0, ] we get f t dt = 0 nd the inequlity in.5 is proved. = 0 f λ + λ dλ.8 λ exp f + λ exp f ] dλ exp f exp f = I exp f, exp f exp f exp f udu
4 530 Silvestru Sever Drgomir nd In Gomm From.7 we hve ] x + y exp f x + exp f y f for ny x, y I. From.9 we hve ] + exp f x + exp f + x f.9.0 for ny x, ]. Integrting the inequlity.0 over x on, ] we get the first inequlity in.6. By the Jensen s inequlity for the concve function we hve ] exp f x + exp f + x dx. = = exp f x + exp f + x ] dx exp f x + exp f + x] dx exp f x dx nd the second inequlity in.6 is proved. If we consider Toder s men defined s see for instnce 5] nd 7] for mny reltions of this men with other mens if =, E = E, :=, R. log I exp, exp if, we cn write.5 in n equivlent form s f t dt E exp f, exp f.. Remrk.3. If the function g : I 0, is convex on I, then f = g Expconv I nd for, I with < we hve, y.5 nd.6, the following inequlities exp g t dt I g, g.3
5 Some Hermite-Hdmrd type inequlities 53 nd + g exp g x dx. ] g x + g + x dx.4 3. Relted results The following relted result lso holds: Theorem 3.. Assume tht f Expconv I nd, I with <. Then we hve for ny x, ]. In prticulr, we hve f x + f x exp f x exp f x f y dy 3. ] exp f y] dy f + f f y dy exp f exp f ] exp f y] dy. Proof. Since the function exp f is convex, it hs lterl derivtives in ech point of, nd f = exp f does the sme. Then for ny x, y, we hve nd dividing y exp f y > 0 we get exp f x exp f y f y x y exp f y exp f x exp f y] f y x y 3.3 for ny x, y,. Integrting 3.3 over y on, ] nd dividing y we get exp f x = = exp f y] dy 3.4 f y x y dy ] f y x y + f y dy ] f y dy f x f x for ny x, ], which is equivlent to the desired inequlity 3..
6 53 Silvestru Sever Drgomir nd In Gomm Corollry 3.. With the ssumptions of Theorem 3. we hve f + f exp f + exp f Proof. If we tke x = nd x = in 3.4 we get f y dy 3.5 ] exp f y] dy. nd exp f exp f exp f y] dy exp f y] dy f y dy f f y dy f. Adding these inequlities nd dividing y two we get ] exp f + exp f exp f y] dy f y dy f + f, which is equivlent to the desired inequlity 3.5. Corollry 3.3. With the ssumptions of Theorem 3. nd if x 0 := f f f y dy, ], 3.6 f f where f f, then we hve f f f y dy exp f y] dy exp f. 3.7 f f Proof. Follows y 3. y tking x = x 0 defined in 3.6. The inequlity 3.7 cn e found in Sándor s pper 3] where x 0 considered in 3.6 is in fct men clled y him s generted y derivtives of functions. This men is extended in 9] see lso 6], nd generlized mny results relted to integrl inequlities. See lso 8] for more results. Remrk 3.4. Since x 0 = f y ydy f y dy, then sufficient condition for 3.6 to hold is tht f is monotonic nondecresing or nonincresing on the whole intervl, ].
7 Some Hermite-Hdmrd type inequlities 533 Remrk 3.5. If the function g : I 0, is convex on I, then f = g Expconv I nd for, I with < we hve, y 3., 3. nd 3.5, the following inequlities nd If g ] x g x g ] x g x g g + g g + g g x 0 := g + g g] where g g, then we hve g] g y dy g y dy 3.8 ], g y dy 3.9 ] g y dy, g y dy 3.0 ] g y dy. g y dy, ], 3. g g g y dy g] g] g gydy g g. 3. Acknowledgement. The uthors would like to wrmly thnk the nonymous referee for pointing out some essentil references nd mking vlule comments tht hve een implemented in the finl version of the pper. References ] Drgomir, S.S., Perce, C.E.M., Selected Topics on Hermite-Hdmrd Inequlities nd Applictions, RGMIA Monogrphs, 000. ] Gill, P.M., Perce, C.E.M., Pečrić, J., Hdmrd s inequlity for r convex functions, J. of Mth. Anl. nd Appl., 5997, ] Sándor, J., On mens generted y derivtives of functions, Inter. J. Mth. Educ. Sci. Technol., 8997, ] Sándor, J., On upper Hermite-Hdmrd inequlities for geometric-convex nd logconvex functions, Notes Numer Th. Discr. Mth., 004, no. 5, 5-30.
8 534 Silvestru Sever Drgomir nd In Gomm 5] Sándor, J., Toder, Gh., On some exponentil mens, Preprint, Beş-Bolyi Univ., Cluj, 990, ] Sándor, J., Toder, Gh., Some generl mens, Czechoslovk Mth. J., , ] Sándor, J., Toder, Gh., On some exponentil mens. Prt II, Intern. J. Mth. Mth. Sci., 006, ID ] Song, Y., Long, B., Chu, Y., On Toder-Sndor men, Intern. Mth. Forum, 803, no., ] Toder, Gh., Sándor, J. Inequlities for generl integrl mens, J. Inequl. Pure & Appl. Mth., 7006, no., Art. 3. Silvestru Sever Drgomir Mthemtics, College of Engineering & Science Victori University, PO Box 448 Melourne City, MC 800, Austrli School of Computtionl & Applied Mthemtics University of the Witwtersrnd, Privte Bg 3 Johnnesurg 050, South Afric e-mil: sever.drgomir@vu.edu.u In Gomm Mthemtics, College of Engineering & Science Victori University, PO Box 448 Melourne City, MC 800, Austrli e-mil: in.gomm@vu.edu.u
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