INTEGRAL INEQUALITY REGARDING r-convex AND

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1 J Koren Mth Soc 47, No, pp DOI 434/JKMS47373 INTEGRAL INEQUALITY REGARDING r-convex AND r-concave FUNCTIONS WdAllh T Sulimn Astrct New integrl inequlities concerning r-conve nd r-concve functions re presented Introduction The following open question ws proposed in [5] Under wht conditions does the inequlity f + d f hold for nd? The uthors in [4], [], nd [] hve een delt grdully with this inequlity ssuming different conditions, ut I think the ide of [] is the est In [], the uthors gve n nswer y estlishing the following Theorem If the function f stisfies then 3 ftdt, [, ], f + d for every rel nd > f d Very recently, Liu, Cheng, nd Li estlished more generl cse y giving the following result Theorem Let f e continuous function on [, ] stisfying 4 f min, tdt t min, dt, [, ] Received June, 8; Revised Decemer 6, 8 Mthemtics Suject Clssifiction 6D5 Key words nd phrses integrl inequlity, conve nd concve functions 373 c The Koren Mthemticl Society

2 374 WAADALLAH T SULAIMAN Then the inequlity 5 f + d f d, holds for every positive rel numer >, > The gmm function is denoted y Γp nd is defined y Γp p e d, p > A function f : [, ] R is sid to e conve if 6 ft + ty tf + tfy,, y [, ], t [, ] If the inequlity is reversed, then f is sid to e concve A positive function f is log-conve on rel intervl [, ] if for ll, y [, ] nd t [, ], we hve 7 ft + ty f t + f t y A positive function f is r-conve on [, ] if for ll, y [, ] nd t [, ], { tf 8 ft + ty r + tf r y /r, r, f t f t y, r If the ove inequlity reverses, then f is r-concve Clerly, the -conve functions re simply the log-conve functions nd - conve functions re ordinry conve functions Hdmrd inequlity is s follows + 9 f fd f + f, provided f is conve Very recently, concerning inequlity 3, out more thn ppers hve een pulished Since Hdmrd inequlity deling with conve nd concve functions is importnt in nlysis, it is resonle to incorporte inequlities similr to 3 with Hdmrd inequlity to find new inequlities tht re proly importnt in nlysis nd pplictions A lemm nd Hdmrd inequlity Lemm Let, >, t [, ] Then t t + t t + Proof Set Then, we hve f + t t t t f t t t t t t, if,

3 INTEGRAL INEQUALITY REGARDING r-convex AND r-concave 375 nd [f ] [t t t t + t t t t ] t t This shows tht f ttins its minimum t which is zero Therefore f f Theorem Let f : [, ] R e positive r-conve function r Then f r stisfies the Hdmrd inequlity Tht is, + f r f r d f r + f r Furthermore, if f is -conve, then is stisfied, nd lso we hve 3 f r d f r f r ln f r ln f r If f is r-concve or -concve, then nd 3 re reversed Proof If f is r-conve, then f r is conve, nd therefore it stisfies the Hdmrd inequlity If f is -conve, then + f r + + f r r f / f / + d / f r d f r + d f r d f r t + tdt f r t + tdt + f tr f tr dt + f r + f r / f r t + tdt f tr f tr dt f tr f tr + f tr f tr dt f r + f r dt y Lemm

4 376 WAADALLAH T SULAIMAN Also, the -conveity of f implies f r d f r t + t dt f tr f tr dt rt f f r dt f f r f r ln f r ln f r We stte nd prove the following: 3 Min results Theorem 3 Let f, g e non-negtive continuous functions defined on [, ], g is -concve with g, [, ] nd let, > If 3 then 3 f tdt g tdt, [, ], f + d f ln d g + g + + g Γ + / If g is -concve with g, g/g is non-incresing nd 3 is stisfied, then 33 f + d g g f g g+ g + Proof In view of Theorem, we hve f t dtd f tdt g tdt +/ + d d

5 INTEGRAL INEQUALITY REGARDING r-convex AND r-concave 377 g + g d g g + g d g + g + + g Also, vi chnging the order of integrtion, we hve f t t dtd f tdt d ln f tdt t Collecting the ove results, we otin ln f tdt g + g + t Applying the AG inequlity, + f + + On putting h ln / or + f + + f + f ln Integrting the ove gives 34 Now, on putting ln + g + h+ f h, h >, we otin + ln+/ f ln, f ln ln +/ f + d f ln u, we hve ln +/ d f ln d d ln +/ u +/ e u du Γ + /, d

6 378 WAADALLAH T SULAIMAN nd hence f + f ln d g + g + + g Γ + / If g is -concve nd g, then y Theorem, f t ln g ln g dtd f ln g ln g tdt d g tdt g g d g g g d ln g ln g d g g+ g + By chnging the order of integrtion, we hve f t ln g ln g dtd f tdt t ln g ln g d t g c f tdt d gc for some c, < c <, y the men vlue theorem g c gc Collecting the ove, we otin t f tdt t f tdt gc g c g g g g+ g + g g+ g + Similrly, s we did in the previous result, we hve f + f f +/

7 INTEGRAL INEQUALITY REGARDING r-convex AND r-concave 379 Integrting the ove to otin f + g g f g g+ g + +/ + Theorem 3 Let f, g e non-negtive continuous functions defined on [, ], g is -conve or -conve with g, [, ], nd let, > If 3 is stisfied, then 35 f + d f ln d f d Γ + / + Proof By virtue of Theorem, we hve f t dtd f tdt f + d Also, y chnging the order of integrtion, we hve, s efore Therefore, we hve 36 f t dtd ln f tdt t ln d f tdt t t + f dt If we proceeding ectly s we did in the proof of Theorem 3, using the ove estimtion, we hve f + d f + + f ln d d Γ + / f d Γ + /

8 38 WAADALLAH T SULAIMAN If g is -conve with g, [, ], then y mking use of 34 nd 36, we otin f + d f ln d + f d Γ + / + f d Γ + / Theorem 33 Let f, g e non-negtive continuous functions defined on [, ], g is -conve with g, nd let < < < If 37 then 38 f d / f tdt g tdt g + g + [, ], + g + Γ / Also, if g is -conve with g nd g/g is non-incresing, we hve f g g + g + d 39 g g + / Proof By Theorem, f t dtd f tdt d g + g d Also, y chnging the order of integrtion, we hve f t dtd g + g g d g + g+ g + f tdt f t ln t d dt t

9 INTEGRAL INEQUALITY REGARDING r-convex AND r-concave 38 The ove result implies f t ln dt g + g + + g t Mking use of the AG inequlity, we hve f k f k, k > On putting k ln /, we otin f ln / f ln, or f /f ln + ln / Integrting the ove inequlity gives f d / f ln / g + g + d + Concerning inequlity 39, if we re ssuming I ln / d + g + Γ / f t ln g ln g dtd, then s we delt efore, it is not difficult to show tht nd I g+ g + g, I g d tf tdt, gd for some d, < d <, y the men vlue theorem, which together implies tf tdt gd g + g + g d g + g + g g By using the AG inequlity s efore, we get g g f /f + /

10 38 WAADALLAH T SULAIMAN Integrting the ove gives f d gc g c g + g + + / g Theorem 34 Let f, g e non-negtive continuous functions defined on [, ], g is -concve with g, nd let < < < If 3 is stisfied, then + 3 f d / f d + Γ / Proof By Theorem we hve As efore f t dtd f t dtd Therefore, we get f t ln dt t f + f t ln Mking use of the AG inequlity, we hve s efore f d / / f ln f + d + f tdt d d dt t t + f dt d + Γ / ln / d Acknowledgement The uthor is so indeted to the referee who red through the pper very well nd mentioned mny scientificlly nd grmmticlly mistkes References [] K Boukerriou nd A Guezne Lkoud, On n open question regrding n integrl inequlity, J Inequl Pure Appl Mth 8 7, no 3, Article 77, 3 pp [] L Bougoff, Note on n open prolem, J Inequl Pure Appl Mth 8 7, no, Article 58, 4 pp [3] W-J Liu, G-S Cheng, nd C-C Li, Further development of n open prolem concerning n integrl inequlity, J Inequl Pure Appl Mth 9 8, no, Article 4, 5 pp

11 INTEGRAL INEQUALITY REGARDING r-convex AND r-concave 383 [4] W-J Liu, C-C Li, nd J W Dong, On n open prolem concerning n integrl inequlity, J Inequl Pure Appl Mth 8 7, no 3, Article 74, 5 pp [5] Q A Ngo, D D Thng, T T Dt, nd D A Thn, Notes on n integrl inequlity, J Inequl Pure Appl Mth 7 6, no 4, Article, 5 pp Deprtment of Computer Engineering College of Engineering University of Mosul Mosul, Irq E-mil ddress: wdsulimn@hotmilcom