Journal of Mathematical Analysis and Applications 250, doi: jmaa , available online at http:

Transcription

1 Joural of Mathematical Aalysis ad Applicatios 5, 886 doi:6jmaa766, available olie at o Fuctioal Equalities ad Some Mea Values Shoshaa Abramovich Departmet of Mathematics, Uiersity of Haifa, Haifa 395, Israel abramos@mathhaifaacil ad Josip Pecaric Faculty of Textile Techology, Uiersity of Zagreb, Peirottijea 6, Zagreb, Croatia pecaric@hazuhr Submitted by Themistocles M Rassias Received November 4, 999 INTRODUCTION Meas as a geeralizatio of arithmetic, geometric, ad harmoic meas are dealt with extesively i mathematical literature; see 8 to quote a few I this paper we defie a mea value of a, ad b, as a fuctio which satisfies the followig: A M: R R R, B Ma, b Mb, a, C Ma, a a, D a Ma, b b for ba Some iterestig mea values were cosidered recetly by Haruki ad Rassias 4 I 5 the followig classes of meas were cosidered: ab M a, b; h h H h s, H a, b H a, b a b 8-47X $35 Copyright by Academic Press All rights of reproductio i ay form reserved

2 8 ABRAMOVICH AND PECARIC h x x,,, ad h x log x, i Theorem 3 i the case that s si ta cos tb ad i Theorem 3 i the case that s a si t b cos t Note that his meas for h x i Theorem 3 ad for h x x, x, log x i Theorem 3 satisfy coditios A, B, ad C oly I this paper we preset geeralizatios of his results We itroduce a fuctio Ma, b, hs h H hs where M a, b s sa, b, t ad Ma, b is a give mea value that is, a mea value satisfyig A to D We deal with a fuctioal equatio which leads to a characterizatio of our meas We also show that this mea ad Kim s meas are ot meas i classical sese sice they do ot satisfy the very importat coditio D I fact, we show that istead of coditio D, such meas satisfy D a Ma, b, hs b for ba T, so we shall say that these meas are o a limited iterval THE RESULTS THEOREM Let g: R R, h: R R be strictly mootoic fuctios with cotiuous secod deriaties Let g ad h be the ierse fuctios of g ad h, respectiely Let sa, b, t be a positive cotiuous fuctio i a, b, ad t, with cotiuous first ad secod derivatives with respect to a, where sc, c, t s c, c, t Dc is idepedet of t, ad s c, c, t a is ot idepedet of t a The, ž H / ž H / h h s a, b, t g g s a, b, t, for all real a, b iff g x chx c, c Proof We will show that the techique used i 4, 5 for specific meas works i the much more geeral case stated here Istead of provig we will prove that H We will deote ž / h s a, b, t h g g s a, b, t H f a, b h s a, b, t ž ž H // k a, b h g g s a, b, t

3 FUNCTIONAL EQUALITIES AND MEANS 83 It is easy to see that f c, c k c, c h D c 3 Differetiatig f a, b ad ka, b with respect to a ad settig a c, b c, we get f k c, c h D H s a c, c, t c, c 4 a a Computig f a a, b ad k a a, b we get that f c, c h D H sa c, c, t h D H saa c, c, a k c, c a ½ g D H a ž H a / 5 g D h D H sa c, c, t h D H saa c, c, t Therefore, i order for to be satisfied, we get that s c, c, t s c, c, t k f c, c, t c, c, t 5 a a a H a h D s c, c, t s c, c, t D a a g h D H s c, c, t H s c, c, t 6 g D Accordig to the Cauchy Schwarz iequality, a H a s c, c, t s c, c, t as log as s c, c, t is ot idepedet of t Therefore we get from 6 a h D g D h D g D

4 84 ABRAMOVICH AND PECARIC ad hece g D ch D c 7 Thus we proved that if f a, b ka, b, 7 holds To prove that if 7 holds the holds is trivial THEOREM Let be a positie iteger, the for eery a, b I H a si t b cos t k! k! k k Ý a b 8! k k! k! k Proof It is easy to verify that I H 9 a si t b cos t ' ab ad that I I I ž / a b The proof is by iductio usig 9 ad We will omit it Result Replacig a with a, b with b i 8, we get that J H si t cos t ž a b / k! k! k k Ý a b,! ž k / k! k! k where is a positive iteger THEOREM 3 Let g: R R, h: R R be strictly mootoic fuctios i R with cotiuous secod deriaties Let g ad h be the iverse fuctios of g ad h, respectively The a b M a, b, g s g ž H g s ab / a b h H h s ab M a, b; h s,

5 FUNCTIONAL EQUALITIES AND MEANS 85 where si k t cos t ž a b / s holds for all ozero a ad b iff g x chx c, where c ad c are arbitrary umbers Proof This theorem follows from Theorem by replacig sa, b, t by THEOREM 4 Let be a iteger, ad let h ad s be as i Theorem 3 The M a, b; h s a b h H h s ab ž Ý / k a b ab! k k! k! a b k!k! k k iff h x c x c 3 Proof The theorem follows easily from Theorem ad the result of Theorem THEOREM 5 Let h: R R be a strictly mootoic fuctio i R with a cotiuous deriatie M Let Ma, b be a mea value such that c, c q, q a The there exists a umber T such that a h h s b M a, b for ba T, s si ta cos tb, ad i this iterval M a, b; h h h s 4 M a, b is a mea value o a limited iterval M M x x Proof Computig a, x; h we get that a, a; h q Hece Ma, x; h is icreasig i x i the eighborhood of x a Therefore a Ma, a; h Ma, b; h i some iterval ba R

6 86 ABRAMOVICH AND PECARIC M y Because Ma, b Mb, a, we get from y, b; h also that M b, b; h q Therefore Ma, b; h Mb, b; h b i y some iterval ba Q Hece there is a iterval ba T such that a M a, b; h b, 5 ad coditio D i the defiitio of mea value is satisfied i the iterval ba T a It is obvious that Ma, b; h also satisfies coditios A, B, ad C there Therefore Ma, b; h is a mea value i the iterval ba T Now we show that iequality 5 for h x x,, 3, holds oly for ba T, where T is a fiite real umber THEOREM 6 For ba, where ba is large eough M a, b; h b, 6 where h x x,, 3,, Ma, b, ad s si ta cos tb ab a b Proof We proved i Theorem 4 that Ma, b, x satisfies equality 3 It is easy to see that whe ba, the Ma, b, x b too Therefore 6 holds for ba large eough Hece i these cases Ma, b; h is a mea value oly o a limited iterval REFERENCES P S Bulle, D S Mitriovic, ad P S Vasic, Meas ad Their Iequalities, Reidel, DordrechtBostoLacasterTokyo, 988 H Haruki, New characterizatio of arithmeticgeometric mea of Gauss ad other well kow mea values, Publ Math Debrece 38 99, H Haruki ad T M Rassias, A ew aalogue of Gauss fuctioal equatio, Iterat J Math Sci 8 995, H Haruki ad T M Rassias, New characterizatio of some mea values, J Math Aal Appl 996, Y-H Kim, O some further extesios of the characterizatios of mea values by H Haruki ad Th Rassias, J Math Aal Appl , D S Mitriovic, J E Pecaric, ad A M Fik, Classical ad New Iequalities i Aalysis, Kluwer Academic, DordrechtBostoLodo, Gh Toader, Some mea values related to the arithmeticgeometric mea, J Math Aal Appl 8 998, Gh Toader ad Th Rassias, New properties of some mea values, J Math Aal Appl 3 999,