Several Applications of Young-Type and Holder s Inequalities

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1 Applied Mathematical Sciences, Vol. 0, 06, no. 36, HIKARI Ltd, Several Applications of Young-Type and Holder s Ineualities Loredana Ciurdariu Department of Mathematics Politehnica University of Timisoara P-ta. Victoriei, No., Timisoara, Romania Copyright c 06 Loredana Ciurdariu. This article is distriuted under the Creative Commons Attriution License, which permits unrestricted use, distriution, and reproduction in any medium, provided the original work is properly cited. Astract The aim of this paper is to present several applications of several new Young-type and Holder s ineualities given y Alzer, H., Fonseca, C. M. and Kovacec, A. to isotonic linear functional, power series and inner product. Mathematics Suject Classification: 6D0 Keywords: Young ineuality, selfadjoint operators, inner product. Introduction The famous Young s ineuality, as a classical result, state that: a ν ν νa + ν, when a and are positive numers, a and ν 0,. In these years, there are many interesting generalizations of this well-known ineuality and its reverse, see for example [,, 9, 8, ] many others and references therein. As in [], we consider A ν a, = νa + ν, and G ν a, = a ν ν. The following result, given in [8] is a refinement of the left-hand side of a refinement of the ineuality of Young proved in 00 and 0 y Kittaneh and Manasrah, [], [].

2 764 Loredana Ciurdariu Proposition. For all a, > 0 we have 3ν A 3 if 0 ν, and 3 3ν ν if 3 ν. a, G 3 a, A ν a, G ν a, A 3 a, G 3 a, A ν a, G ν a, More recently, in [] are given new results which extend many generalizations of Young s ineuality given efore. We recall these results elow in order to use them in the next sections. Theorem. Let λ, ν and τ e real numers with λ and 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 for all positive and distinct real numers a and. Moreover, oth ounds are sharp. Theorem. Let ν 0,. For all real numers a, with 0 a we have ν ν a log ν ν A ν a, G ν a, log a a and ν ν exp a A νa, G ν a, exp ν ν. a In each ineuality, the factor ν ν is the est possile.. The Young-type and Holder s ineualities for isotonic linear functional The following important definition is given in [], [6], [3] and it is necessary to recall it here.

3 Several applications of Young-type and Holder s ineualities 765 Definition. Let E e a nonempty set and L e a class of real-valued functions f : E R having the following properties: L If f, g L and a, R, then af + g L. L If ft = for all t E, then f L. An isotonic linear functional is a functional A : L R having the following properties: A If f, g L and a, R, then Aaf + g = aaf + Ag. A If f L and ft 0 for all t E, then Af 0. The mapping A is said to e normalised if A3 A =. Lemma. [0], Corollary 3.3 If f is - integrale on [a, then for an aritrary positive numer α the function f α is -integrale on [a,. Lemma. [0], Theorem 3.6 Let f and g e -integrale functions on [a,. then their product fg is -integrale on [a,. The following results are applications of recent Young-type ineualities given in [] in the case if isotonic linear functionals and also Holder-type ineualities in the same case using methods presented in [7],[6]. Also, are presented as particular cases these ineualities for the cases of the time scale Cauchy delta, Cauchy navla, α- diamond, multiple Riemann, and multiple Leesue integrals. Theorem 3. L satisfy conditions L, L and A satisfy A, A on the set E. If f p, g, fg, f p log g, g log g L, Af p > 0, Ag > 0, Af p f p Ag f and g are positive functions and 0 and Af p f p Ag f p Af p g [f p Af p A p log Af p f p Ag g then: Ag ] Afg A p f p A g ] g [g p Ag A log Af p f p Ag A fg exp p f p Ag g Af p A p f p A g

4 766 Loredana Ciurdariu A fg exp g Af p, p f p Ag where p > and + =, if in this second case, instead of f p log g Af p, p f p Ag g log g Af p L we have fg exp f p Ag, f p Ag p g Af p fg exp g Af p p f p Ag L. Proof. We take into account first ineuality from Theorem, ν ν a log ν ν A ν a, G ν a, log a a where we replace a y f p x and y g x and ν =. Af p Ag p Then we otain: p f p x Af p log g x f p x Af p Ag f p x p Af p + g x f p x g x Ag fgx A p f p A g g x Af p p Ag log Ag for any x E. Now we take the functional A in previous ineuality and we get the desired ineuality. For the second ineuality the same method will e used. Theorem 4. If L satisfy conditions L, L and A satisfy A, A on the set E. If f p, g, fg, f pτ, g τ L, Af p > 0, Ag > 0, pτ >, τ, + = and f and g are positive functions then: p [ Af ] pτ Ag τ Afg pτ A τ f p A τ g A p f p A g [ Af ] pτ Ag τ. τ A τ f p A τ g Moreover, if A, B : L R are two normalised isotonic functionals such that the conditions f pτ g τ, f p τ g τ, f p g p L appear instead of f pτ, g τ L then we have the following ineuality: pτ [τaf p Bg + τag Bf p Af pτ g τ Bf p τ g τ ] p Af p Bg + Ag Bf p AfgBf p g p pτ [τaf p Bg + τag Bf p Af pτ g τ Bf p τ g τ ].

5 Several applications of Young-type and Holder s ineualities 767 A particular case will e otained when, A = B, pτ [Af p Ag Af pτ g τ Af p τ g τ ] Af p Ag AfgAf p g p pτ [Af p Ag Af pτ g τ Af p τ g τ ]. Proof. If we use in ineuality from Theorem, a = f p x and = f p y then we g x g y get [ ν τ f p x τ g x + y τfp g y f ] pτ x f p τ y g τ x g τ y f p ν y g ν y ν f p x g x + y νfp g y f pν x g ν x ν [ τ f p x τ g x + y τfp g y f pτ x g τ x for any x, y E. We multiply last ineuality y g xg y > 0 and we get [ ν τ τf p xg y + τf p yg x f pτ x g τ x f p τ y g τ y ], f p τ y g τ y g xg y ] νf p xg y + νf p yg x f pν x f p ν y g ν x g ν y g xg y ν [ τf p xg y + τf p yg x f ] pτ x f p τ y τ g τ x g τ y g xg y, for any x, y E. Fix y E as in [7] and we have in the order of L that ν [ τf p g y + τf p yg f pτ f p τ yg τ g τ y ] τ νf p g y + νf p yg f pν f p ν yg ν g ν y ν [ τf p g y + τf p yg f pτ f p τ yg τ g τ y ], τ If we take now the functional A in previous ineuality, we have, [ τaf p g y + τf p yag Af pτ g τ f p τ yg τ y ] pτ p Af p g y + f p yag Afgf p yg p y [ τaf p g y + τf p yag Af pτ g τ f p τ yg τ y ], τ for any y E. Then we write this ineuality in the order of L and then we take the functional B in this ineuality otaining the desired result.

6 768 Loredana Ciurdariu Theorem 5. Let pτ >, τ with p > 0, τ > 0 and + =. If L satisfy p conditions L, L and A satisfy A, A on the set E. If f p, g, fg, f pτ g τ, f pk n g k n L, k = 0, n, Af p > 0, Ag > 0 and f and g are positive functions then: or [ f p n A τ pτ n A n f p [ f p n A [ A τ n pτ n τ n k=0 n g + τ A n g n ] g n p A n f p + A n g f p n n g τ + τ A n f p A n g ] Af pτ g τ A τ f p A τ g Afg A p f p A g n n Af pτ g τ A τ f p A τ g [ n ] n τ k τ n k k k=0 A k n f p A k n g A f pk n g k n Af pτ g τ A τ f p A τ g n pk n Af n g k n k p k n k k=0 A k n f p A k n g Afg A p f p A g [ n n τ k τ n k k A k n f p A k n g A f pk n g k n Af pτ g τ A τ f p A τ g Proof. The proof will e as in Theorem 3 and Theorem 4. Theorem 6. Let pτ >, τ with p > 0, τ > 0 and + =. If L satisfy p conditions L, L and A satisfy A, A on the set E. If f p, g, fg, f pτ g τ, f p k n g k n, f p τ g τ, f pk n g k n L, k = 0, n, Af p > 0, Ag > 0 and f and g are positive functions then we have: ν n n n [ τ τ k n k τ k Af p k n g k n Bf p k n g k n Af pτ g τ Bf p τ g τ ] k=0 n n k p n k g k=0 n ν n n [ τ k k=0 k p Af n g k n k Bf p k n g k n AfgBf p g p τ n k τ k Af p k n g k n Bf p k n g k n Af pτ g τ Bf p τ g τ ] ] ]

7 Several applications of Young-type and Holder s ineualities 769 In [8] Feng improved the left-hand side of the Kittaneh-Manasrah ineuality [, ]. Proposition. If L satisfy conditions L, L and A satisfy A, A on the set E and if f p, g, fg L, Af p > 0, Ag > 0, f and g are positive functions and p + = then: i If p 3 and f p 3 g 3 L 3 Af p 3 g 3 Afg p A 3 f p A 3 g A p f p A g. ii If p 3 and f p 3 g 3 L 3 Af p 3 g 3 Afg A 3 f p A 3 g A p f p A g. iii If p > 3 and f p 3 g 3, f p 3 g 3, f p g p p L then we have: Ag Bf p 3 p Af p p p 3 g 3 Bf 3 g 3 AfgBf g p. Moreover, if A = B then previous ineuality ecomes: p Ag Af p 3 p Af p p p 3 g 3 Af 3 g 3 AfgAf g p iv If p 3 and g 3 f p 3, g 3 f p 3, f p g p p L then we have: Bg Af p 3 Ag p p p 3 f 3 Bg 3 f 3 AfgBf g p. Moreover, if A = B then previous ineuality ecomes: Ag Af p 3 p Ag p p p 3 f 3 Ag 3 f 3 AfgAf g p. Conseuence. Theorem 4, 5 and Proposition can e rewritten for the time scale Cauchy delta, Cauchy nala, α-diamond, multiple Riemann, and multiple Leesue integrals. Remark. For example, first ineuality of Theorem 4 can e rewritten like elow: i Let f, g C rd [a, ], R e positive functions, pτ f pτ x x a τ a g x x a f p x x a g τ x x τ a f p x x fxgx x a p a g x x

8 770 Loredana Ciurdariu f pτ x x a a g τ x x τ τ τ, f a p x x a g x x where p, τ and as in Theorem 4. ii Let f, g C ld [a, ], R e positive functions, f pτ x x a a g τ x x a pτ τ τ f a p fxgx x x x a g x x f a p p x x a g x x f pτ x x a a g τ x x τ τ τ, f a p x x a g x x where p, τ and as in Theorem 4. iii Let f, g : [a, ] R e α -integrale and positive functions f pτ x a α x a g τ x α x pτ τ τ f a p x α x a g x α x a fxgx α x f a p p x α x a g x α x f pτ x a α x a g τ x α x τ τ f a p x α x a g x α x where p, τ and as in Theorem 4. τ, 3. The Young-type ineualities for inner product First, as in [6], it is necessary to recall that for selfadjoint operators A, B BH we write A B or B A if Ax, x > Bx, x > for every vector x H. We will consider for eginning A as eing a selfadjoint linear operator on a complex Hilert space H;.,. >. The Gelfand map estalishes a - isometrically isomorphism Φ etween the set CSpA of all continuous functions defined on the spectrum of A, denoted SpA, and the C - algera C A generated y A and the identity operator H on H as follows: For any f, f CSpA and for any α, β C we have i Φαf + βg = αφf + βφg; ii Φfg = ΦfΦg and Φf = Φf ; iii Φf = f := sup t SpA ft ;

9 Several applications of Young-type and Holder s ineualities 77 iv Φf 0 = H and Φf = A, where f 0 t = and f t = t for t SpA. Using this notation, as in [6] for example, we define fa := Φf for all f CSpA and we call it the continuous functional calculus for a selfadjoint operator A. It is known that if A is a selfadjoint operator and f is a real valued continuous function on SpA, then ft 0 for any t SpA implies that fa 0, i.e. fa is a positive operator on H. In addition, if and f and g are real valued functions on SpA then the following property holds: ft gt for any t SpA implies that fa ga in the operator order of BH. Using the definition from [6], we say that the functions f, g : [a, ] R are syncronous asyncronous on the interval [a, ] if they satisfy the following condition: ft fsgt gs 0 for each t, s [a, ]. Conseuence. Let A BH and B BH e two selfadjoint operators on H and ν 0,. i If 0 ν then we have 3 3ν [ 3 Ax, x > y, y > + 3 x, x > By, y > A 3 x, x > B 3 y, y > ] ν Ax, x > y, y > + ν x, x > By, y > A ν x, x > B ν y, y > ii If 3 ν then we otain [ ] 3 ν 3 Ax, x > y, y > + 3 x, x > By, y > A 3 x, x > B 3 y, y > ν Ax, x > y, y > + ν x, x > By, y > A ν x, x > B ν y, y > iii If λ = and 0 ν τ then we have ν τ [τ Ax, x > y, y > + τ x, x > By, y > Aτ x, x > B τ y, y >] ν Ax, x > y, y > + ν x, x > By, y > A ν x, x > B ν y, y > ν τ [τ Ax, x > y, y > + τ x, x > By, y > Aτ x, x > B τ y, y >] for each x, y H.

10 77 Loredana Ciurdariu iv Moreover, when B B K is an selfadjoint operator on pseudo-hilert space K, see [4] and [5], instead of a selfadjoint operator on H and λ = and 0 ν τ then we have ν τ [τ Ax, x > [y, y] + τ x, x > [By, y] Aτ x, x > [B τ y, y]] ν Ax, x > [y, y] + ν x, x > [By, y] A ν x, x > [B ν y, y] ν τ [τ Ax, x > [y, y] + τ x, x > [By, y] Aτ x, x[ B τ y, y]] for each x H and y K. Proof. The demonstration will e as in [6] and we prove only i. We fix a and apply the property for the first ineuality from Proposition, otaining: 3ν[ 3 A + 3 I A 3 3 ] νa + νi A ν ν for each R. Then we otain for each x H the following ineuality: 3ν[ 3 Ax, x > + 3 x, x > A 3 x, x > 3 ] ν Ax, x > + ν x, x > A ν x, x > ν and now we will apply again the property, this time for and we get 3ν[ 3 Ax, x > I + 3 B x, x > A 3 x, x > B 3 ] ν Ax, x > I + νb x, x > A ν x, x > B ν which is euivalent to: 3ν[ 3 Ax, x > y, y > + 3 By, y > x, x > A 3 x, x > B 3 y, y >] ν Ax, x > y, y > + ν By, y > x, x > A ν x, x > B ν y, y >. Remark. Let A BH and B BH e two selfadjoint operators on H and ν 0,. i If we consider in Proposition the elements x, y H with x = and y = then we otain the following ineualities: ν ν [ Ax, x > log By, y > + A log Ax, x > A log Ax, x > log By, y >] ν Ax, x > + ν By, y > A ν x, x > B ν y, y > ν ν [ B log By, y > + log Ax, x > By, y > log Ax, x > B log By, y >],

11 Several applications of Young-type and Holder s ineualities 773 ii If we consider in Conseuence the elements x, y H with x = and y = then we otain the following ineualities: a If p 3 then [ ] 3ν 3 Ax, x > + 3 By, y > A 3 x, x > B 3 y, y > ν Ax, x > + ν By, y > A ν x, x > B ν y, y >. If p 3 then [ ] 3 ν 3 Ax, x > + 3 By, y > A 3 x, x > B 3 y, y > ν Ax, x > + ν By, y > A ν x, x > B ν y, y >. c If λ = and 0 ν τ then ν τ [τ Ax, x > + τ By, y > Aτ x, x > B τ y, y >] ν Ax, x > + ν By, y > A ν x, x > B ν y, y > ν τ [τ Ax, x > + τ By, y > Aτ x, x > B τ y, y >]. References [] H. Alzer, C. M. Fonseca, A. Kovacec, Young-type ineualities and their matrix analogues, Linear and Multilinear Algera, 63 05, no. 3, [] D. Andrica and C. Badea, Gruss ineuality for positive linear functionals, Periodica Math. Hung., 9 988, [3] M. Anwar, R. Bii, M. Bohner and J. Pecaric, Integral Ineualities on Time Scales via the Theory of Isotonic Linear Functionals, Astract and Applied Analysis, 0 0, Article ID , [4] L. Ciurdariu, A. Craciunescu, On spectral representation on gramian normal operators on pseudo-hilert spaces, Analele Universitatii de Vest din Timisoara, Seria Matematica Informatica, XLV 007, no., [5] L. Ciurdariu, Ineualities for selfadjoint operators on Hilert spaces and pseudo-hilert spaces, RGMIA, Res. Rep. Coll., 8 05, paper no. 60. [6] S. S. Dragomir, Ceysev s type ineualities for functions of selfadjoint operators in Hilert spaces, Linear and Multilinear Algera, 58 00, [7] S. S. Dragomir, Several results for isotonic functionals via two new reverses of Young s ineuality, RGMIA, Res. Rep. Coll., to appear. [8] Y. Feng, Refinements of Young s ineualities for matrices, Far East J. Math. Sci., 63 0, [9] S. Furuichi, N. Minculete, Alternative reverse ineualities for Young s ineuality, Journal of Mathematical Ineualities, 5 0, no. 4,

12 774 Loredana Ciurdariu [0] G. S. Guseinov, Integration on time scales, J. Math. Anal. Appl, , [] F. Kittaneh, Y. Manasrah, Reverse Young and Heinz ineualities for matrices, Linear and Multilinear Algera, 59 0, no. 9, [] F. Kittaneh, Y. Manasrah, Improved Young and Heinz ineualities for matrices, J. Math. Anal. Appl., 36 00, Received: March 4, 06; Pulished: May 7, 06