Local Extreme Points and a Young-Type Inequality

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1 Alied Mathematical Sciences Vol. 08 no HIKARI Ltd htts://doi.org/0.988/ams Local Extreme Points a Young-Te Inequalit Loredana Ciurdariu Deartment of Mathematics Politehnica Universit of Timisoara P-ta. Victoriei No Timisoara Romania Coright c 08 Loredana Ciurdariu. This article is distriuted under the Creative Commons Attriution License which ermits unrestricted use distriution reroduction in an medium rovided the original work is roerl cited. Astract In this aer is resented a Young-te inequalit then as an alication is given a corresonding Holder-te inequalit for isotonic linear functionals. Mathematics Suject Classiication 6D5 Kewords: Young-te inequalities arithmetic mean geometric mean isotonic linear functionals The classical inequalit of Young is. Introduction a ν ν < νa + ( ν where a are distinct ositive real numers 0 < ν < see [4]. In [] are given new results which extend man generalizations of Young s inequalit given efore. The following inequalit is a refinement of the lefth side of a refinement of the inequalit of Young roved in 00 0 Kittaneh Manasrah in [] [3]. Man generalizations refinements of Young s inequalit are resented also in [0] [8] [9] references therein. Theorem A([] Let λ ν τ e real numers with λ 0 < ν < τ <. Then ( ν ( λ A ν (a λ G ν (a λ λ ν < τ A τ (a λ G τ (a < λ τ

2 66 Loredana Ciurdariu for all ositive distinct real numers a. Moreover oth ounds are shar. The following imortant definition is given in [3] [5] we need to recall it here in order to hel us to give new Young-te inequalities for isotonic linear functionals in Section 3. Let E e a nonemt set L e a class of real-valued functions f : E R having the following roerties: (L If f g L a R then (af + g L. (L If f(t = for all t E then f L. An isotonic linear functional is a functional A : L R having the following roerties: (A If f g L a R then A(af + g = aa(f + A(g. (A If f L f(t 0 for all t E then A(f 0. The maing A is said to e normalised if (A3 A( =. New inequalities concerning isotonic linear functionals can e also found in [7] [3] [5] [6] referinces therein.. Local extreme oints a Young-te inequalit for three numers In this section is given a new Young-te inequaliti for three ositive numers which satisfies some conditions in Theorem using the Lemma where are stated several conditions for finding the local extreme oint for a secial function. Lemma. Let 3 3 e strictl ositive real numers which satisfies the conditions = + + = 3 ( (. (i If < ( [ ( + ( ] > (. then A( is a local minimum oint for the function f(x = x + + ( x x + + x 3 3 defined on the interval (0 (0.

3 Local extreme oints a Young-te inequalit 67 (ii If > ( [ ( + ( ] > (. then A( is a local maximum oint for the function f(x = x + + ( x x + + x 3 3 defined on the interval (0 (0. Proof. (i We consider the function f(x = x x ( x + + x 3 where the numers 3 3 satisfies the hothesis x are strictl ositive real numer with x > 0 > 0. First it is necessar to find the stationar oints of f on (0 (0 for that we comute its first derivative f f. We have x f x = x + x f = x + x then we otain the following sstem ( x = x ( x = ( x Using now the hothesis > 0 we get from the equation ( ( x = 0 that x = where satisf the hothesis eing aritrar numers. Last equation ecomes x = when x > 0.

4 68 Loredana Ciurdariu Therefore the last sstem will e Then we have ( = ( ( = ( = ( ( = when or the solution x = =. So we otain in the second case the stationar oint A(. First case when it is not interesting here ecause our hothesis are not satisfied i. e. from last sstem we have = ( (which is alread a restriction of in this wa the second equation of last sstem in checked ut this is not our hothesis. We stud now if A( is an extreme oint for the function f on the interval (0 (0. For that we comute the second derivative of the function then its hessian matrix in A(. We have f x = ( x + ( x f x ( = ( f = ( x + ( x f ( = ( + ( also f x = x + ( f x ( = x f x ( = (.

5 Local extreme oints a Young-te inequalit 69 Now we can write the hessian matrix in A( ( ( H( = ( ( + ( if = ( > 0 ( = [ ( + ( ] ( > 0 then A( is the local extreme oint for the function f defined efore. For (ii the roof is the same Examle. (i We take into account the articular case for the function f when = 5 = 6 3 = 30 9 = 4 = 5 3 = 0 see also in Figures. We can easil notice that the conditions from hothesis (i are fulfilled for the function f so that the oint A( is a local minimum oint for f. (ii Now if we relace 4 7 in revious articular case we can easil see that the conditions from hothesis (ii are satisfied for the function f so the oint A( is a local maximum oint for f. Theorem. Let M > 3 3 e ositive real numers which satisfies the conditions = = 3 3 > > ( > > 0. (i If x are two real numers with < x < M < < M then the following inequalit holds: x + + ( x > x + + x 3. 3 (ii Moreover if a c are three real numers a > 0 > 0 c > 0 so that c < a < Mc c < < Mc then the following inequalit takes lace: a + + ( c a c 3 > a + + c a 3 c 3. 3 Proof. Using Lemma we know that A( is a local minimum oint for the function f on the interval ( M ( M which it is the interior of the close interval [ M] [ M]. We stud how will e the function on the frontier of the aove interval. We see that the frontier of this interval from R is given the sets {x = [ M]} {x = M [ M]} {x [.M] = } {x [ M] = M}.

6 70 Loredana Ciurdariu Figure. The function f(x on [0 8] [0 8] when = 5 = 6 3 = 30 9 = 4 = 5 3 = 0. When x = [ M] then f( = ( + ( This function is increasing as a function of variale from hothesis of the aove theorem then f( < f( ecause <. Therefore we find that f( > f( = 0. Last function is increasing ecause its first derivative f ( = ( ( Now for = x [ M] we have > f(x = x + x ( (.. > 0.

7 Local extreme oints a Young-te inequalit 7 Figure. The function f(x on [ 8] [ 8] when = 5 = 6 3 = 30 9 = 4 = 5 3 = 0. This function is increasing ecause its first derivative f (x = (x x > 0 see hothesis of our revious theorem. Thus we also have f(x > f( = 0. If x [ M] = M then we otain f(x M = M ( + 3 ( x M this function is increasing in x when x [ M] ecause From here we get f (x M = (x M x M > 0. f(x M > f( M > 0 x M we otained this inequalit efore see the case when x = [ M].

8 7 Loredana Ciurdariu Last case when x = M [ M] we have the function f(m = ( + ( 3 + M M 3 3 which is increasing as a function of variale ecause its first derivative f (M = ( + M M = = [ ( ( ] M M > 0. We used here that > M From the second case we get > >. f(m = + M M > 0 then f(m > f(m > 0. Therefore the ointa( is the gloal minimum of the function f on the interval [ M] [ M]. Taking into account hothesis from Lemma (i denoting a c 3 3 we get c > a < <. Condition > 0 from the roof of Lemma ecomes ( [ ( ( + ] > or ( a [( + a( ] > ( calculus we have: ( a > a ( i.e. the condition ( > from our hothesis. (ii We relace x [ M] a [ M] ecause a [ M] c c c [ M] the inequalit from (i ecomes: c a c + c + ( a ( [ > a 3 c c c + c + ( a ( ] 3 c c multiling c > 0 we get the desired inequalit.

9 Local extreme oints a Young-te inequalit 73 Examle. The articular case from Examle (i satisfies the conditions of Theorem (i then the oint A( is the gloal minimum for the function f the inequalit from Theorem (i takes lace. 3. Holder-te inequalit for three functions The following result is otained as a consequence of Theorem (ii for isotonic linear functionals eing a Holder-te inequalit in the case of three functions. Theorem. Let M > 3 3 e ositive real numers which satisfies the conditions = = 3 3 > > ( > > 0 L satisfing conditions L L A satisfing A A on the set E. Considering the nonnegative functions f g h with 3 fgh f g h 3 f g h 3 L A(f > 0 A(g > 0 A((h 3 > 0 h if in addition 3 < f < M h 3 h 3 < g < M h 3 we A(h 3 A(f A(h 3 A(h 3 A(g A(h 3 will have A(f gh A (f A (g > A 3 (h 3 A A(f 3 g h 3 (f A (g A 3 (h 3 Proof. We use inequalit from Theorem (ii for a = f = g A(f A(g c = h 3 we have A(h 3 > f + g + A(f A(g 3 f g + + A(f A(g 3 h 3 A(h 3 h 3 A(h 3 Now using hothesis condition A we get > A(f + A(g + A(h 3 A(f A(g 3 A(h 3 A(f + A(g + A(h 3 A(f A(g 3 A(h 3 or calculus we otain the desired inequalit. fgh A (f A (g A 3 f g h 3 3 (h 3 > A (f A (g A 3 (h 3 A(f gh A (f A (g A A(f 3 g h 3 3 (h 3 > A (f A (g A 3 (h 3..

10 74 Loredana Ciurdariu As a articular case when instead of the isotonic linear functional A(f we consider as in [3] f(xdx Theorem ecomes: a Remark. Let M > 3 3 e ositive real numers which satisfies the conditions = = 3 3 > > ( > > 0 Considering the continuous functions f g h > 0 on the interval [a ] with M h 3 (x a h 3 (xdx h 3 (x < f (x a h 3 (xdx a f (xdx we will have < M h 3 (x a h 3 (x h 3 (xdx < g (x a h 3 (xdx a f(xg(xh(xdx ( f a (xdx ( a g (xdx ( > a h 3 (xdx 3 3 > a f (xg (xh 3 (xdx ( f a (xdx ( a g (xdx (. a h 3 (xdx 3 a g (xdx < References [] H. Alzer C. M. Fonseca A. Kovacec Young-te inequalities their matrix analogues Linear Multilinear Algera. 63 (05 no htts://doi.org/0.080/ [] D. Andrica C. Badea Gruss inequalit for ositive linear functionals Periodica Math. Hung. 9 ( htts://doi.org/0.007/f [3] M. Anwar R. Bii M. Bohner J. Pecaric Integral Inequalities on Time Scales via the Theor of Isotonic Linear Functionals Astract Alied Analsis 0 (0-6 Article ID htts://doi.org/0.55/0/ [4] M. Bohner A. Peterson Dnamic Equations on Time Scales: An Introduction with Alications Sringer Birkhauser Boston 00. htts://doi.org/0.007/ [5] S.S. Dragomir A surve of Jessen s Te Inequalities for Positive Functionals RGMIA Res. Re. Coll. (0 46. [6] S.S. Dragomir A Gruss Te Inequalit for Isotonic Linear Functionals Alications RGMIA Res. Re. Coll. (00 0. [7] S.S. Dragomir Some results for isotonic functionals via an inequalit due to Liao Wu Zhao RGMIA Res. Re. Coll. (05. [8] S.S. Dragomir New refinements reverses of Hermite-Hadamard inequalit alications to Young s oerator inequalit RGMIA Res. Re. Coll. 9 (06 Art. 4. [9] S.S. Dragomir Some asmmetric reverses of Young s scalar oerator inequalities with alications RGMIA Res. Re. Coll. 9 (06 Art 44. [0] S. Furuichi N. Minculete Alternative reverse inequalities for Young s inequalit Journal of Mathematical Inequalities 5 (0 no htts://doi.org/0.753/jmi-05-5 [] G.S. Guseinov Integration on time scales J. Math. Anal. Al 85 ( htts://doi.org/0.06/s00-47x(

11 Local extreme oints a Young-te inequalit 75 [] F. Kittaneh Y. Manasrah Reverse Young Heinz inequalities for matrices Linear Multilinear Algera 59 (0 no htts://doi.org/0.080/ [3] F. Kittaneh Y. Manasrah Imroved Young Heinz inequalities for matrices J. Math. Anal. Al. 36 ( htts://doi.org/0.06/j.jmaa [4] W.H. Young On classes of summale functions their Fourier series Proceedings of the Roal Societ A: Mathematical Phsical Engineering Sciences 87 ( htts://doi.org/0.098/rsa Received: Setemer 4 08; Pulished: Octoer 8 08